From <@UBVM.CC.BUFFALO.EDU:owner-LISTSERV@UBVM.CC.BUFFALO.EDU> Mon Feb 6 16:04:58 1995 Received: from netaxs.com (root@netaxs.com [198.69.186.1]) by access.netaxs.com (8.6.9/8.6.9) with ESMTP id QAA27271 for ; Mon, 6 Feb 1995 16:04:58 -0500 Received: from UBVM.cc.buffalo.edu (ubvm.cc.buffalo.edu [128.205.2.1]) by netaxs.com (8.6.9/8.6.9) with SMTP id QAA01829 for ; Mon, 6 Feb 1995 16:04:44 -0500 Message-Id: <199502062104.QAA01829@netaxs.com> Received: from UBVM.CC.BUFFALO.EDU by UBVM.cc.buffalo.edu (IBM VM SMTP V2R2) with BSMTP id 4118; Mon, 06 Feb 95 16:03:42 EST Received: from UBVM.CC.BUFFALO.EDU (NJE origin LISTSERV@UBVM) by UBVM.CC.BUFFALO.EDU (LMail V1.2a/1.8a) with BSMTP id 9077; Mon, 6 Feb 1995 12:40:15 -0500 Date: Mon, 6 Feb 1995 12:39:52 -0500 From: "L-Soft list server at UBVM (1.8a)" Subject: File: "GEODESIC LOG9408" To: "Christopher J. Fearnley" Status: RO ========================================================================= Date: Mon, 1 Aug 1994 08:34:27 +0100 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Andy Wardley Subject: Re: LOOKING FOR SYNERGETICS II In-Reply-To: <9407311749.AA01178@Q.icl.co.uk> from "The Butterfly" at Jul 31, 94 03:45:14 pm The Butterfly writes: > Oh, and I saw that GEODOME.JPG file. VERY impressive! Nice Ray-trace, >Andy! (Although it looks like the dome may have been mirrored in the middle, >which throws off some of the geodesic patterning, but at 167 hours >render-time, I don't really blame you. ;^) ) Glad you like it! It wasn't actually mirrored but the skewing was down to a cheap bit of coding I did. The utility I wrote to create the geodome data set started off with an octahedron and then recursively sub-divided each facet into smaller triangles. The problem with this is that the sides don't divide equally and you end up with this effect. What I *should* have done, and indeed now have, is to start with a dodecahedron which tessellates evenly. The new utility creates a whole load of other shapes too. When I get round to finishing it, I'll upload it somewhere and let you know where so the ray tracing fans amongst us can play with it. Cheers Andy This Spot Is Allowed whatever K says. Here, have some Andy Wardley chocolate, it's Terry's. No smug bait for Derek. M#0 abw@oasis.icl.co.uk Badgers are your friends. OK, so they don't frink but have you seen them forage? DAMN! I've run out of sp ========================================================================= Date: Mon, 1 Aug 1994 10:05:51 +0200 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: News-Reader XIAOMING UNSUBSCRIBE xiaoming xu ========================================================================= Date: Mon, 1 Aug 1994 08:29:52 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Ross Keatinge Organization: Public Access Internet, Auckland New Zealand Subject: Trimtab Sorry if this appears twice. More newsreader said 'posting failed'. I would appreciate it if someone could tell me if there has been a new issue of "Trimtab" since the Spring 1993 issue. It seems an awful long time since I got this one from the BFI and I wonder if I should chase them up about it. Regards Ross Keatinge icosa@iconz.co.nz Auckland New Zealand ========================================================================= Date: Mon, 1 Aug 1994 06:03:19 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: jpegs Sorry that I have to do this, but I lost my access to the Internet for a few weeks and would like to know where I can download the tensegrity jpeg from and the geodesic sphere. Thanks. Steve Mather ========================================================================= Date: Mon, 1 Aug 1994 06:45:56 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: IVM Someone mentioned a while back (when I couldn't post) that movement, according to theoretical physics, is in increments (not yet determined.) I had wondered if there was such a theory because energy/matter is alleged to move in very specific amounts as well. My problem with this, though in many ways it makes sense, is that then matter/energy is making tiny jumps through space/time. Am I wrong? Do I not understand it right? Steve Mather ========================================================================= Date: Mon, 1 Aug 1994 15:06:40 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: jpegs First of all (though I be late) I appreciated all of the jpegs. Just a question. What was the curve? Anything in particular? (no flame intended, I just want to know if it has any practical application.) Steve Mather ========================================================================= Date: Mon, 1 Aug 1994 21:59:36 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: jpegs In-Reply-To: Message of Mon, 1 Aug 1994 15:06:40 GMT from On Mon, 1 Aug 1994 15:06:40 GMT said: >First of all (though I be late) I appreciated >all of the jpegs. >Just a question. What was the curve? Anything >in particular? (no flame intended, I just >want to know if it has any practical application.) The curve is not mine. I think it's Richard Hawkins'. It does demonstrate some of the relationships in the cube that are not necessarily obvious such as when you look at it topologically where each edge is a circle. I'm not sure of it's significance, but Bucky never doubted the significance of a pattern, so I won't :) I wish the image were larger and had different perspective, but the author has trouble with uuencoding (I'm working with him on it). > Steve Mather Christopher J. Fearnley cfearnl@pacs.pha.pa.us fearnlcj@duvm.ocs.drexel.edu fearnlcj@duvm.bitnet ========================================================================= Date: Tue, 2 Aug 1994 01:20:03 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Hawku Organization: America Online, Inc. (1-800-827-6364) Subject: Re: JPEGs Regarding the curVE file, the cube with circular rather than square faces is similar to the cuboctahedron (Vector Equilibrium) in that the vertices (or in this case tangents) are identical. The cuboctahedron (or VE without the radials) can be constructed from 4 hexagonal planes. Likewise the curVE model is constructed from 4 six-quadrant segments (blue,red,yellow,&green in the JPEG). The six quadrants each at 90 degrees to it's adjacents form a closed loop. Four of these loops make up the circumferentials of the model. The armatures (red and yellow in the JPEG) have their pivot point at the center of the system and are fixed to each other(grouped). The red armature revolves around each (six-quadrant) loop, switching where the loops intersect, through all 4 loops returning to the starting point ; while the yellow armature tracks the spherical triangles at the opposing ends of each of the 4 axes of rotation. Fuller's concept of 4D modeling alludes to the possibilities of 4 (or more) unique perpendiculars interacting in a system. The tetrahedron can have 4 unique perpendiculars (to it's faces), while the cube has only 3. This model is an attempt to demonstrate this concept. The yellow armature is in a precession relationship with the red. Its' movement is conical while the red revolves through 4 axes of rotation. I hope this helps to clarify. If not, perhaps I can upload an MPEG file somewhere (56 frames of animation). Richard Hawkins ========================================================================= Date: Tue, 2 Aug 1994 09:19:38 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: In-line cable clamps Mitch clarifies his situation. > The poster suggested to look up references to 'Wire Rope' suppliers and > manufacturers in the Yellow Pages; this will obviously vary from area to > area - the Greater Rochester Yellow Pages has one entry under Wire Rope. > (Unfortunately, this local supplier has nothing in the scale of what I'm > interested in.) > I haven't had time to do any foot(phone)work lately, but the last round of > calls I made to a variety of possible suppliers turned up nothing. > -------------------------------------------------------------------------- > Mitch C. Amiano > amiano@delphi.com > I'll keep looking then. ========================================================================= Date: Tue, 2 Aug 1994 09:26:23 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: LOOKING FOR SYNERGETICS II Pat writes: > > Well, here's a blatant plug, and a potential reference, since Chris F. > was asking about this via private email: My Synergy Ball is being produced by > Design Science Toys, of Tivoli, NY. They're the people who make the > Tensegritoys, Octabug, Hoberman Sphere, and a slew of other geodesic and > geometry based toys. They've got a nifty catalog which you can get by phoning > 1-800-227-2316. > > Just called and got my catalog sent to me. Thanks Pat. ========================================================================= Date: Tue, 2 Aug 1994 09:30:05 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: LOOKING FOR SYNERGETICS II Pat > > > > Nice to be back. Thanks for keeping this going, people. It's great to > >see the interest out there! > Chris > We've come along way baby! I remember when the list was little more than you > (with growing amounts of help) getting everyone in the dorms at SUNY-Buffalo > interested in building a floating city :) Now, we have an ftp site and WWW > stuff out there! [PS: what is it with us SUNY students abandoning the state > that gave us so much?] I don't think that I ever mentioned that I attended SUNY at Buffalo for a year myself. ========================================================================= Date: Tue, 2 Aug 1994 09:35:06 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: JPEGs [Clarification deleted] I hope this helps to clarify. If > not, perhaps I can upload an MPEG file somewhere (56 frames of animation). > Richard Hawkins > Clear concept or not I'd like to see the MPEG anyway. ========================================================================= Date: Thu, 4 Aug 1994 13:37:46 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: BFI@AOL.COM Subject: Please send information on joining Please send information on joining Thank you Tony DeVarco Executive Director Buckminster Fuller Institute ========================================================================= Date: Fri, 5 Aug 1994 00:14:37 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: Please send information on joining In-Reply-To: Message of Thu, 4 Aug 1994 13:37:46 EDT from On Thu, 4 Aug 1994 13:37:46 EDT said: >Please send information on joining > >Thank you > >Tony DeVarco >Executive Director >Buckminster Fuller Institute Did the move to Santa Barbara go well? I haven't received a Trimtab in ages - is one being planned soon? (Maybe my membership slipped - I guess I should check my checkbook :) It's good to have the institute involved - last month was one of the most exciting and info-filled months on the list ever (though we've been fairly active since last Sept/Oct and periodically active since 1989). I imagine this month will be equally active. BTW, I've done some work at getting the FAQ up-to-date through July '94 - but I need a few more weeks to tidy things up and fix a few bugs - for now everyone can get the FAQ from the list archives or anonymous ftp from switchboard.ftp.com in the /bucky directory. Do Enjoy! Christopher J. Fearnley UNIX SIG Leader at PACS cfearnl@pacs.pha.pa.us (Philadelphia Area Computer Society) fearnlcj@duvm.bitnet Design Science Revolutionary fearnlcj@duvm.ocs.drexel.edu Explorer in Universe 503 S 44th ST (215)349-9681 Philadelphia PA 1914-3907 ========================================================================= Date: Fri, 5 Aug 1994 19:13:34 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "" Subject: Re: IVM >Someone mentioned a while back (when I couldn't >post) that movement, according to theoretical >physics, is in increments (not yet determined.) An interesting analog of entropy... I conjectured that, given that all energy appears to be quantized, and the energy of an object is a function of its mass and velocity, velocity should be quantized. Right ?-) > >I had wondered if there was such a theory because >energy/matter is alleged to move in very specific >amounts as well. > >My problem with this, though in many ways it makes >sense, is that then matter/energy is making tiny >jumps through space/time. I would also conjecture that for objects of matter, the net effects of a large number of discrete 'tiny jumps' would appear to be continuity of movement. For photons, questions of distance seem to make less sense. > >Am I wrong? Do I not understand it right? > Steve Mather > ------------------------------------------------------------------------ Mitch C. Amiano amiano@delphi.com ========================================================================= Date: Sat, 6 Aug 1994 10:56:20 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: The Butterfly Organization: Evolutionary Acceleration, Inc. Subject: Re: Please send information on joining BFI@AOL.COM writes: -Please send information on joining -Thank you -Tony DeVarco -Executive Director -Buckminster Fuller Institute Here's an old file I just dug up from my archives. I'm surprised it's still around, actually... (Gotta clean those old directories...) ------------------------------------------------------------------------------- To subscribe, send mail to LISTSERV@UBVM.BITNET and in the body of your letter put the line: SUB GEODESIC When you want to post, send mail to GEODESIC@UBVM.BITNET ******NOT***** to LISTSERV@UBVM.BITNET! LISTSERV@UBVM.BITNET is for subscriptions, administrivia, archive requests, etc. GEODESIC@UBVM.BITNET is the actual discussion group. Anything sent to GEODESIC will go to all members. (And you don't want to look like a jerk having everyone see your "SUB GEODESIC John Q. Public" command! ;^) ) This list is also linked to USENET in the group bit.listserv.geodesic You can subscribe to the list, [thereby helping us to have an accurate idea how many people are reading the group] and still read the messages through news, if you like. (That's what I do. I prefer not getting every post in my e-mail, but I want people to be able to find my address, if necessary.) If you'd like to be on the membership list, but not get mail, then after you subscribe, send mail to LISTSERV@UBVM.BITNET with the message SET GEODESIC NOMAIL This will remain in effect until you use SET GEODESIC MAIL to change it again. If you want to receive copies of everything you send to the list, use the command SET GEODESIC REPRO. If you DON'T want copies, use SET GEODESIC NOREPRO. (You may want to save this file to forward on to people who are interested, as it tells what the list is about, and how to subscribe.) Pat ______________________________Think For Yourself_______________________________ Patrick G. Salsbury 1800 Market Street #23, San Francisco, CA 94102 Voicemail: 415/703-7177 ------------------------------------------------------------------------------- I've seen the wiring under the board. :) -- Pat ______________________________Think For Yourself_______________________________ Patrick G. Salsbury 1800 Market Street #23, San Francisco, CA 94102 Voicemail: 415/703-7177 ------------------------------------------------------------------------------- I've seen the wiring under the board. :) ========================================================================= Date: Sat, 6 Aug 1994 19:38:44 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Nyrath Organization: TimeHeart Subject: Dymaxion car in "Addam's Family" Pardon me, but was that a Fuller style Dymaxion car that cousin It was driving in the movie "The Addam's Family"? Long torpedo shape, three wheels, one in the back, two in the front. Sure looked like one to me... ========================================================================= Date: Sat, 6 Aug 1994 14:33:18 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: PI to 20 or 30 sig-figs ?? RE: Kirby Urner's July 8 posting: >I've come up with an algorithm for deriving pi that uses >no trig, just pythagoras. Using his algorithm, and after 33 iterations, I arrived at: 3.141592653589793239 [My program uses "long double" Think C variables on a Mac with no floating point chip.] I would like to check the validity of my program, but I don't have a definition of pi to that many places. So it would be helpful if someone could confirm this number for me. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Sat, 6 Aug 1994 14:38:00 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Kirby Hey, whatever happened to Kirby? ========================================================================= Date: Sun, 7 Aug 1994 00:05:30 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: Kirby In-Reply-To: Message of Sat, 6 Aug 1994 14:38:00 GMT from On Sat, 6 Aug 1994 14:38:00 GMT said: >Hey, whatever happened to Kirby? He posted that he was going to Africa with his family for a few weeks(?) He'll be back soon - judging from history he'll reply to all of the messages he has "missed." Chris Fearnley ========================================================================= Date: Sun, 7 Aug 1994 00:34:41 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: PI to 20 or 30 sig-figs ?? In-Reply-To: Message of Sat, 6 Aug 1994 14:33:18 -0700 from On Sat, 6 Aug 1994 14:33:18 -0700 Lee Wood said: >RE: Kirby Urner's July 8 posting: >>I've come up with an algorithm for deriving pi that uses >>no trig, just pythagoras. > >Using his algorithm, and after 33 iterations, I arrived at: > > 3.141592653589793239 Very good! Here is my computer output to 28 decimal places: pi = 3.141592653589793238462643383. If you need more accuracy, I can increase the precision. > > >[My program uses "long double" Think C variables on a Mac >with no floating point chip.] > >I would like to check the validity of my program, but I don't >have a definition of pi to that many places. So it would be >helpful if someone could confirm this number for me. > >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | >Lee_Wood@sfu.ca | INTJ spoken here. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= -- Christopher J. Fearnley UNIX SIG Leader at PACS cfearnl@pacs.pha.pa.us (Philadelphia Area Computer Society) fearnlcj@duvm.bitnet Design Science Revolutionary fearnlcj@duvm.ocs.drexel.edu Explorer in Universe 503 S 44th ST Linux Advocate Philadelphia PA 1914-3907 (215)349-9681 ========================================================================= Date: Sat, 6 Aug 1994 23:30:15 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: PI to 20 or 30 sig-figs ?? RE: Chris Fearnley's post: > Very good! Here is my computer output to 28 decimal places: > pi = 3.141592653589793238462643383. If you need more accuracy, I can > increase the precision. I noticed that, while trying to "simplify" the equation, the program became unstable - resulting in SERIOUS roundoff errors So, considering how temperamental floating point numbers are, it's quite possible that my PI = 3.141592653589793239 and your pi = 3.141592653589793238462643383 are both way off the mark. I was actually hoping that someone had some absolutely authoritative source which *defines* PI to 100 sig-figs (or more!). =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Sun, 7 Aug 1994 05:32:08 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: PI to 20 or 30 sig-figs ?? In-Reply-To: Message of Sat, 6 Aug 1994 23:30:15 -0700 from On Sat, 6 Aug 1994 23:30:15 -0700 Lee Wood said: >RE: Chris Fearnley's post: > >> Very good! Here is my computer output to 28 decimal places: >> pi = 3.141592653589793238462643383. If you need more accuracy, I can >> increase the precision. > >I noticed that, while trying to "simplify" the equation, >the program became unstable - resulting in SERIOUS roundoff errors > >So, considering how temperamental floating point numbers are, >it's quite possible that my > >PI = 3.141592653589793239 > >and your > >pi = 3.141592653589793238462643383 > >are both way off the mark. > Very unlikely - I used the PARI/gp unlimited precision calculator. I think it keeps 100 sig-figs (it spits it out awfully fast). I think it's a pretty authoritative result. So you did something right! I was not using an algorithm, but a mathematical package. > >I was actually hoping that someone had some absolutely >authoritative source which *defines* PI to 100 sig-figs >(or more!). >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | >Lee_Wood@sfu.ca | INTJ spoken here. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Christopher J. Fearnley UNIX SIG Leader at PACS cfearnl@pacs.pha.pa.us (Philadelphia Area Computer Society) fearnlcj@duvm.bitnet Design Science Revolutionary fearnlcj@duvm.ocs.drexel.edu Explorer in Universe 503 S 44th ST Linux Advocate Philadelphia PA 1914-3907 (215)349-9681 ========================================================================= Date: Sun, 7 Aug 1994 16:39:58 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: PI to 20 or 30 sig-figs ?? In Article <199408062133.OAA10992@whistler.sfu.ca> Lee Wood writes: >RE: Kirby Urner's July 8 posting: >>I've come up with an algorithm for deriving pi that uses >>no trig, just pythagoras. > >Using his algorithm, and after 33 iterations, I arrived at: > 3.141592653589793239 here's what I found, below your own: 3.14159265358979323846264338327950... >[My program uses "long double" Think C variables on a Mac >with no floating point chip.] > >I would like to check the validity of my program, but I don't >have a definition of pi to that many places. So it would be >helpful if someone could confirm this number for me. I don't know how many places that is. COUNT THEM YOURSELF! =) My eyes hurt now. Steve Mather ========================================================================= Date: Mon, 8 Aug 1994 08:52:47 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: PI to 20 or 30 sig-figs ?? The reason for my apparent obsession with the number of significant digits in my results is as following: In the first iteration, my hypotenuse is simply the square root of 2. And it is used to derive the new hypotenuse. newH = sqrt( pow(h/2,2) + pow(1-sqrt(1-pow(h/2,2)),2) ); i.e. h= 1.4142135623730950490 newH= 0.7653668647301795434 Notice that these results contain about 20 sig-figs. However, by the time I've done 32 iterations, my new hypotenuse has shrunk to only 10 sig-figs. newH=0.0000000007314590396 But my value for PI, based on that hypotenuse, shows 19 sig-figs. PI= 3.141592653589793239 It's been about 20 years since I took Physics-101, but I'm fairly certain that one of the points they tried to drive home was that one cannot generate sig-figs out of thin air. i.e. If one of the numbers in my calculation has only 10 sig-figs, then the end product of the calculation is only significant to 10 digits. And I should simply discard any digits beyond that. So, I should only be permitted to say: PI= 3.141592653 The fact that there is such a close correspondence between my longer value and the PI generated by Chris Fearnley's canned math routine, frankly, has me puzzled. Theoretically, I should not be that close to the correct value. canned pi= 3.141592653589793238462643383 My PI= 3.141592653589793239 Chris, I would be interested to see if any of those low-end digits change as you double or triple your precision. Steve Mather reports an astounding: 3.14159265358979323846264338327950... i.e. Steve's Pi= 3.14159265358979323846264338327950... Chris' pi= 3.141592653589793238462643383 My PI= 3.141592653589793239 Steve, did you use a canned routine or write a program? If the latter, I'd be *very* interested in seeing your source code. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Mon, 8 Aug 1994 13:26:28 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: PI to 20 or 30 sig-figs ?? In-Reply-To: Message of Mon, 8 Aug 1994 08:52:47 -0700 from On Mon, 8 Aug 1994 08:52:47 -0700 Lee Wood said: [Significant digits stuff deleted] > >The fact that there is such a close correspondence between my >longer value and the PI generated by Chris Fearnley's canned >math routine, frankly, has me puzzled. Theoretically, I should >not be that close to the correct value. I'm not following all this, but maybe your program is better than you thought? > >canned pi= 3.141592653589793238462643383 >My PI= 3.141592653589793239 > > >Chris, I would be interested to see if any of those low-end >digits change as you double or triple your precision. Here are 100 significant digits: pi = 3.1415926535897932384626433832795028841971693993 75105820974944592307816406286208998628034825342117068 > >i.e. >Steve's Pi= 3.14159265358979323846264338327950... >Chris' pi= 3.141592653589793238462643383 >My PI= 3.141592653589793239 > >Steve, did you use a canned routine or write a program? >If the latter, I'd be *very* interested in seeing your source code. Apperently the pari calulator has some algorithm for calculating higher sig digits here is the initial definition (from the source): #ifndef PI const double PI = 3.141592653589; /* pi */ #endif This function is probably where it happens: void constpi(long prec) I think I got PARI/gp (for DOS, Unix and I think Amiga and Mac too, but I'm not sure) from megrez.ceremab.u-bordeaux.fr (147.210.16.17) by ftp. If you like I could probably find the actual function (I bet they use Taylor polynomials) that calculates PI in pari/gp. But the program was written by number theorists for speed and accuracy not synergetics :) >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | >Lee_Wood@sfu.ca | INTJ spoken here. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Mon, 8 Aug 1994 13:35:45 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: Dymaxion car in "Addam's Family" > Subject: Dymaxion car in "Addam's Family" > To: Multiple recipients of list GEODESIC > > > Pardon me, but was that a Fuller style Dymaxion car that > cousin It was driving in the movie "The Addam's Family"? > Long torpedo shape, three wheels, one in the back, two in the > front. > Sure looked like one to me... > Which one, Addams 2?"Just call me Trimtab" RBF _______ / \ BDAMICO@GWUVM.GWU.EDU ___________/__________\____________ \ Trimtab: A tiny gear / |\ which moves the rudder / |__\ that turns great ships / --------------------- ========================================================================= Date: Mon, 8 Aug 1994 14:26:35 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: PI to 20 or 30 sig-figs ?? In Article <199408081552.IAA00398@whistler.sfu.ca> Lee Wood writes: >The reason for my apparent obsession with the number of >significant digits in my results is as following: > >In the first iteration, my hypotenuse is simply the square >root of 2. And it is used to derive the new hypotenuse. > >newH = sqrt( pow(h/2,2) + pow(1-sqrt(1-pow(h/2,2)),2) ); I don't think I understand how one is used to determine the next. Could you explain? (I hate ASCII characters =) >Steve Mather reports an astounding: >3.14159265358979323846264338327950... The University of Toledo's Gopher server has it to 1 million signifigant figures. Check w/them. See my other post (note-- not U of Mich.) Steve Mather ========================================================================= Date: Mon, 8 Aug 1994 13:55:46 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: PI to 20 or 30 sig-figs ?? In Article <199408081552.IAA00398@whistler.sfu.ca> Lee Wood writes: >The reason for my apparent obsession with the number of >significant digits in my results is as following: > >In the first iteration, my hypotenuse is simply the square >root of 2. And it is used to derive the new hypotenuse. > >newH = sqrt( pow(h/2,2) + pow(1-sqrt(1-pow(h/2,2)),2) ); > >i.e. >h= 1.4142135623730950490 >newH= 0.7653668647301795434 > >Notice that these results contain about 20 sig-figs. > > > >However, by the time I've done 32 iterations, my new >hypotenuse has shrunk to only 10 sig-figs. > >newH=0.0000000007314590396 > >But my value for PI, based on that hypotenuse, shows >19 sig-figs. > >PI= 3.141592653589793239 > > >It's been about 20 years since I took Physics-101, but I'm >fairly certain that one of the points they tried to drive >home was that one cannot generate sig-figs out of thin air. > >i.e. If one of the numbers in my calculation has only >10 sig-figs, then the end product of the calculation is >only significant to 10 digits. And I should simply discard >any digits beyond that. > >So, I should only be permitted to say: > >PI= 3.141592653 > > > >The fact that there is such a close correspondence between my >longer value and the PI generated by Chris Fearnley's canned >math routine, frankly, has me puzzled. Theoretically, I should >not be that close to the correct value. > >canned pi= 3.141592653589793238462643383 >My PI= 3.141592653589793239 > > >Chris, I would be interested to see if any of those low-end >digits change as you double or triple your precision. > > > >Steve Mather reports an astounding: >3.14159265358979323846264338327950... > >i.e. >Steve's Pi= 3.14159265358979323846264338327950... >Chris' pi= 3.141592653589793238462643383 >My PI= 3.141592653589793239 > >Steve, did you use a canned routine or write a program? >If the latter, I'd be *very* interested in seeing your source code. I found that number in gopher (sorry, didn't mean to imply I programmed it, I can barely program =) I think it was at the U of Michigan in Ann Arbor. I'll have to check it out. They've got it to a million places, and that's what I wrote down from it a long time ago. Gimme a couple days and I'll track the source down for you.... Steve Mather ========================================================================= Date: Mon, 8 Aug 1994 22:23:04 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: SYoke Organization: America Online, Inc. (1-800-827-6364) Subject: Re: Dymaxion car in "Addam's Family" In article <320os4$p51@clarknet.clark.net>, nyrath@clark.net (Nyrath) writes: Pardon me, but was that a Fuller style Dymaxion car that cousin It was driving in the movie "The Addam's Family"? Long torpedo shape, three wheels, one in the back, two in the front. Sure looked like one to me... No. That car was a Messerschmidt Tiger, a bubble car built by the same co. that built Nazi planes in WWII. The Germans built cars with three wheels because they went in a lower automobile tax bracket after WWII, and not many Germans after WWII could even afford these small cars. ========================================================================= Date: Mon, 8 Aug 1994 22:29:33 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Bob Cook Subject: Help design the Tucson Solar Village Tucson Solar Village GEODESIC Delphi Questionaire For all of you in GEODESIC interested in the challenges of developing an actual "eco-village", here is your once in a decade opportunity to help us think through the final pre-construction phase of the Tucson Solar Village Program. The Tucson Solar Village is an ongoing program sponsored by the Tucson- Pima County Metropolitan Energy Commission and the City of Tucson to demonstrate that communities can be designed, built, marketed and lived in, in a more sustainable manner by: 1- Greatly increasing the efficiency of how energy, water and other resources are used; 2- Significantly reducing our impact on the natural environment; 3- Providing a lifestyle rich in variety, opportunities, amenities and sense of community; 4- Reducing dependence upon public services for transportation, security and other infrastructure. This starts a 10-week intensive planning effort with the objective of developing a Comprehensive Implementation Strategy for approval by City and State governments with assurance of cooperation by all parties involved. This process has four parts: 1- A Delphi process with 3 successive questionnaires to identify critical implementation issues and strategies; 2- Two workshops to help analyze the results of the Delphi process and integrate them into a comprehensive plan; 3- Critical review and comment which will include local, state and national experts; and 4- Public Forum for final review and information-sharing. The Metropolitan Energy Commission is seeking your thoughtful assistance to help achieve its dual goals of successful development of Civano and transfer of those concepts to the larger community. Please take a few moments, focus on the need to bring the vision of the Solar Village into reality from a development perspective, and answer the attached questionnaire. This first questionnaire is intended to be very general and somewhat open-ended. The succeeding two questionnaires will incorporate responses from the previous ones and will become more detailed and specific. Please E-mail or fax your responses in time to be received by Tuesday, August 16, 1994 to: Robert Cook, Acting Chair, Solar Village Steering Committee, Tucson-Pima County Metropolitan Energy Commission E-mail: BCook@Pimacc.Pima.Edu or FAX (602) 748-4754 For any other information, I can be reached by voice mail at: (602) 748- 4743 or s-mail: P.O. Box 41144 Tucson AZ 85717 Your responses, together with those of other participants, will help form the basis of a second questionnaire which will be sent to you about August 20, 1994. Thank you very much for your participation. PROJECT STATUS >From its inception, The Solar Village has been a cooperative effort between the public and private sectors on many levels. There are several phases to the project. PHASE 1 ended in the Spring of 1992 with the preparation of a Master Development Plan for Civano, an 820 acre mixed use community for 5600 people within the City of Tucson. This development plan, together with appropriate zoning, was approved unanimously by the Mayor and Council and subsequently formally adopted by the Arizona State Land Department, which owns the property as part of the State Urban Lands Trust. The adopted documents include Covenants, Conditions and Restrictions (CC&Rs) which guide the future design and decision-making regarding physical development and future governance of the land. General performance targets for resource efficiency and amenities include areas such as energy demand, water consumption, solid waste flows, affordability, landscape/land use, job-creation/employment oppportunities, and sustainable transportation in the village. This plan and public involvement process was awarded the "Best Project of 1992" by the Arizona Planning Association. PHASE 2 is soon ending with the preparation of a series of performance standards which will guide the design and development of all buildings within Civano. PHASE 3 is beginning now with detailed implementation planning to assure Civano's successful development and ways to expand its concepts and vision to the broader community. Because the Solar Village concept departs from conventional approaches to development, there are many issues which still need to be resolved in order to achieve its potential. Currently, the Arizona State Land Department is preparing a final appraisal for Civano based upon the land value, the adopted Master Development Plan, the current zoning, and the completed Performance Standards. This appraisal is currently scheduled to be finalized by December, 1994. Within six months after this date, a public auction can be held if the State Land Department chooses. The Metropolitan Energy Commission is particularly concerned with fostering -Continued building of consensus and cooperation between the public and private sectors at all levels, -Active participation by forward-thinking, energetic, enthusiastic and pro-active community leaders, and -Recruitment and selection of developers and builders who understand and are committed to the same vision. THE TUCSON SOLAR VILLAGE PROGRAM COMPREHENSIVE IMPLEMENTATION STRATEGY DELPHI QUESTIONNAIRE: CYCLE 1, GENERAL QUESTIONS This is the first cycle of a 3-cycle Delphi process to identify key issues and strategies preparatory to planning Phase 3 of the Solar Village Program, leading to construction. We are interested in your creativity and judgement and are looking for good ideas. All responses will be confidential and none will be associated with a name or organization. Please respond to be received by Tuesday, August 16. Thank you very much. 1. The Civano project of the Tucson Solar Village proposes to bring together the best and most recent developments in many fields. What distinquishing features or innovations to make this special and attractive would you most like to see in such a development? 2. The challenge of implementing this innovative land development project requires a well thought-out approach. What set of activities are most needed to assure Civano's success and accomplish such goals as you identified in question 1? (To answer this you might consider, for example, technology, marketing, financing, and other feasibility issues) 3. The Solar Village Program also seeks to bring the benefits of Civano to all of Tucson. What are the most important aspects of Civano that you think could and should be implemented in the community at large? 4. Considering your responses to both questions 2 and 3, how can we bring together the best people with the right resources to get the job done? 5. What questions should be included in the next two cycles of the questionnaire? ========================================================================= Date: Tue, 9 Aug 1994 09:51:23 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "James E. Hoburg" Subject: Re: PI to 20 or 30 sig-figs ?? In-Reply-To: <199408090325.XAA11517@dialup.oar.net> Steve Mather writes: > I found that number in gopher (sorry, didn't mean to > imply I programmed it, I can barely program =) > I think it was at the U of Michigan in Ann Arbor. > I'll have to check it out. They've got it to > a million places, and that's what I wrote down from it > a long time ago. > > Gimme a couple days and I'll track the source down for > you.... Excerpted from the newsgroup sci.math FAQ => [...] 4Q: Where I can get pi up to a few hundred thousand digits of pi? Does anyone have an algorithm to compute pi to those zillion decimal places? A: MAPLE or MATHEMATICA can give you 10,000 digits of Pi in a blink, and they can compute another 20,000-1,000,000 overnight (range depends on hardware platform). It is possible to retrieve 1.25+ million digits of pi via anonymous ftp from the site wuarchive.wustl.edu, in the files pi.doc.Z and pi.dat.Z which reside in subdirectory doc/misc/pi. New York's Chudnovsky brothers have computed 2 billion digits of pi on a homebrew computer. How is pi calculated to many decimals ? There are essentially 3 different methods. 1) One of the oldest is to use the power series expansion of atan(x) atan(x)=x-x^3/3+x^5/5-... together with formulas like pi=16*atan(1/5)-4*atan(1/239). This gives about 1.4 decimals per term. 2) A second is to use formulas coming from Arithmetic-Geometric mean computations. A beautiful compendium of such formulas is given in the book of Borwein and Borwein: Pi and the AGM, Canadian Math. Soc. Series, John Wiley and Sons, New York, 1987. They have the advantage of converging quadratically, i.e. you double the number of decimals per iteration. For instance, to obtain 1 000 000 decimals, around 20 iterations are sufficient. The disadvantage is that you need FFT type multiplication to get a reasonable speed, and this is not so easy to program. 3) A third one comes from the theory of complex multiplication of elliptic curves, and was discovered by S. Ramanujan. This gives a number of beautiful formulas, but the most useful was missed by Ramanujan and discovered by the Chudnovsky's. It is the following (slightly modified for ease of programming): Set k1=545140134;k2=13591409;k3=640320;k4=100100025;k5=327843840;k6=53360; Then in AmsTeX notation $\pi=\frac{k6\sqrt(k3)}{S}$, where $$S=\sum_{n=0}^\infty (-1)^n\frac{(6n)!(k2+nk1)}{n!^3(3n)!(8k4k5)^n}$$ The great advantages of this formula are that 1) It converges linearly, but very fast (more than 14 decimal digits per term). 2) The way it is written, all operations to compute S can be programmed very simply since it only involves multiplication/division by single precision numbers. This is why the constant 8k4k5 appearing in the denominator has been written this way instead of 262537412640768000. This is how the Chudnovsky's have computed several billion decimals. Question: how can I get a C program which computes pi? Answer: if you are too lazy to use the hints above, you can use the following 160 character C program (who is the author of this?) which computes pi to 800 decimal digits. If you want more, it is easy to modify, but you have to understand how it works first. int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5; for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a, f[b]=d%--g,d/=g--,--b;d*=b);} References : (This is a short version for a more comprehensive list contact Juhana Kouhia at jk87377@cc.tut.fi) J. M. Borwein, P. B. Borwein, and D. H. Bailey, "Ramanujan, Modular Equations, and Approximations to Pi", American Mathematical Monthly, vol. 96, no. 3 (March 1989), p. 201 - 220. P. Beckman A history of pi Golem Press, CO, 1971 (fourth edition 1977) J.M. Borwein and P.B. Borwein The arithmetic-geometric mean and fast computation of elementary functions SIAM Review, Vol. 26, 1984, pp. 351-366 J.M. Borwein and P.B. Borwein More quadratically converging algorithms for pi Mathematics of Computation, Vol. 46, 1986, pp. 247-253 J.M. Borwein and P.B. Borwein Pi and the AGM - a study in analytic number theory and computational complexity Wiley, New York, 1987 Shlomo Breuer and Gideon Zwas Mathematical-educational aspects of the computation of pi Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2, 1984, pp. 231-244 David Chudnovsky and Gregory Chudnovsky The computation of classical constants, Columbia University, Proc. Natl. Acad. Sci. USA, Vol. 86, 1989. Y. Kanada and Y. Tamura Calculation of pi to 10,013,395 decimal places based on the Gauss-Legendre algorithm and Gauss arctangent relation Computer Centre, University of Tokyo, 1983 Morris Newman and Daniel Shanks On a sequence arising in series for pi Mathematics of computation, Vol. 42, No. 165, Jan 1984, pp. 199-217 E. Salamin Computation of pi using arithmetic-geometric mean Mathematics of Computation, Vol. 30, 1976, pp. 565-570 D. Shanks and J.W. Wrench, Jr. Calculation of pi to 100,000 decimals Mathematics of Computation, Vol. 16, 1962, pp. 76-99 Daniel Shanks Dihedral quartic approximations and series for pi J. Number Theory, Vol. 14, 1982, pp.397-423 David Singmaster The legal values of pi The Mathematical Intelligencer, Vol. 7, No. 2, 1985 Stan Wagon Is pi normal? The Mathematical Intelligencer, Vol. 7, No. 3, 1985 J.W. Wrench, Jr. The evolution of extended decimal approximations to pi The Mathematics Teacher, Vol. 53, 1960, pp. 644-650 [...] ========================================================================= Date: Tue, 9 Aug 1994 09:39:50 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: PI to 20 or 30 sig-figs ?? Steve Mather wrote: LW>>In the first iteration, my hypotenuse is simply the square LW>>root of 2. And it is used to derive the new hypotenuse. LW>> LW>>newH = sqrt( pow(h/2,2) + pow(1-sqrt(1-pow(h/2,2)),2) ); SM> I don't think I understand how one is used to determine SM> the next. Could you explain? (I hate ASCII characters =) After trying several times to draw a circle, incsribed square, etc. with ASCII characters, I gave up and drew it on a "CAD" program. (MacDraw II for you Mac fans) If you like, I'd be glad to FAX you the three page output. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Tue, 9 Aug 1994 16:59:15 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "David Q. Spitzley" Organization: Illuminati Online (on pentagon) Subject: World Game Does anyone out there have any details on how to get a copy of the rules for the World Game, or any comments from experience on running it "off-campus", i.e. outside it's normal residence? ========================================================================= Date: Tue, 9 Aug 1994 13:14:53 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: World Game "David Q. Spitzley" writes > Does anyone out there have any details on how to get a copy of the rules > for the World Game, or any comments from experience on running it > "off-campus", i.e. outside it's normal residence? > I have to check for the address of the World Game institute in Penn. (unless someone beats me to it). The institute was at GWU two years ago to run a Game. About 30 people participated and it was an education as well as fun. I will locate my old post that described the experience. Have you seen the Hypercard stack from the World Game Institute? I've seen it offered from BFI but haven't gotten a copy myself."Just call me Tri mtab" RBF _______ / \ BDAMICO@GWUVM.GWU.EDU ___________/__________\____________ \ Trimtab: A tiny gear / |\ which moves the rudder / |__\ that turns great ships / --------------------- ========================================================================= Date: Tue, 9 Aug 1994 14:10:06 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: Kirby In Article Chris Fearnley writes: >On Sat, 6 Aug 1994 14:38:00 GMT said: >>Hey, whatever happened to Kirby? >He posted that he was going to Africa with his family for a few weeks(?) >He'll be back soon - judging from history he'll reply to all of the >messages he has "missed." On that note, where would I find the archives? Steve Mather ========================================================================= Date: Tue, 9 Aug 1994 23:56:56 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: World Game In-Reply-To: Message of Tue, 9 Aug 1994 16:59:15 GMT from On Tue, 9 Aug 1994 16:59:15 GMT David Q. Spitzley said: >Does anyone out there have any details on how to get a copy of the rules >for the World Game, or any comments from experience on running it >"off-campus", i.e. outside it's normal residence? We all play the world game every day :) But there is an organization devoted to Bucky concept of the World Game. The World Game Institute 3215 Race Street Philadelphia, PA 19104-2597 Phone: (215)387-0220 Fax: (215)387-3009 The basic rule is to make the world work for 100% of humanity in the shortest period of time without ecological offense nor the disadvantage of anyone (rough paraphrase of Bucky there) -- Christopher J. Fearnley | UNIX SIG Leader at PACS cfearnl@pacs.pha.pa.us | (Philadelphia Area Computer Society) fearnlcj@duvm.bitnet | Design Science Revolutionary fearnlcj@duvm.ocs.drexel.edu | Explorer in Universe 503 S 44th ST | Linux Advocate Philadelphia PA 1914-3907 | (215)349-9681 ========================================================================= Date: Wed, 10 Aug 1994 21:35:04 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Dane Winberg Organization: Drexel University Subject: Re: World Game Chris Fearnley writes: >On Tue, 9 Aug 1994 16:59:15 GMT David Q. Spitzley said: >>Does anyone out there have any details on how to get a copy of the rules >>for the World Game, or any comments from experience on running it >>"off-campus", i.e. outside it's normal residence? >We all play the world game every day :) But there is an organization >devoted to Bucky concept of the World Game. >The World Game Institute >3215 Race Street >Philadelphia, PA 19104-2597 >Phone: (215)387-0220 Fax: (215)387-3009 >The basic rule is to make the world work for 100% of humanity in the shortest >period of time without ecological offense nor the disadvantage of anyone >(rough paraphrase of Bucky there) World Game Institute produces a (typically 3-4 hour) workshop which serves as an introduction to the concepts of World Game, as described by Bucky. In the last 20 years, 90,000 people have participated in workshops held for universities, corporations, high schools, churches, government agencies, and others all over the world. We also do World Game Workshops with a focus (diversity, environment). We're (finally) putting together an excerpt from an upcoming brochure that I can send via e-mail. Also, I have brief descriptions of other products and programs we provide (atlas and database software, playground maps, etc.) If you're interested in that information, our e-mail address is xtm00002@duvm.ocs.drexel.edu. For further information, please call or write (see above). As for comments and experiences of past participants at the workshops, I look forward to seeing them posted here. I've worked for WGI for 11 years and have been repeatedly amazed by the positive responses of participants and hosts. We currently have posted on our bulletin board a letter from a woman whose father-in-law expressed a desire before he died that his grandchildren participate in a grade school version of the workshop he had observed. Dane Winberg World Game Institute ========================================================================= Date: Wed, 10 Aug 1994 19:53:06 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "by way of woodh@sfu.ca Lee Wood" Subject: Bad Address for a DCA User The Internet address you are using for Darren Cromer at DCA is no longer valid, Please discontinue its use. Thanks, Robert C. Covington Director, MIS Operations Digital Communications Associates Voice: (404)442-4887 FAX: (404)442-4361 X.400: c=us; ad=attmail; pr=dca; su=covington; gi=robert Internet: itrcc@dcatla.com ========================================================================= Date: Thu, 11 Aug 1994 07:37:00 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: Bad Address for a DCA User I've been receiving "bounce" warnings each time I post something to this list. And eventually I got the following. This looks like a job for the list owner. ___________________________________________________________________ > The Internet address you are using for Darren Cromer at DCA is >no longer valid, Please discontinue its use. > >Thanks, > >Robert C. Covington >Director, MIS Operations >Digital Communications Associates >Voice: (404)442-4887 >FAX: (404)442-4361 >X.400: c=us; ad=attmail; pr=dca; su=covington; gi=robert >Internet: itrcc@dcatla.com ========================================================================= Date: Thu, 11 Aug 1994 23:53:58 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: Bad Address for a DCA User In-Reply-To: Message of Thu, 11 Aug 1994 07:37:00 -0700 from On Thu, 11 Aug 1994 07:37:00 -0700 Lee Wood said: >I've been receiving "bounce" warnings each time I post something >to this list. And eventually I got the following. > Yes, the problem should be solved soon. I e-mailed the administrator of the bad account telling him how to fix it. And our list owner might beat him to the punch :) > ========================================================================= Date: Fri, 12 Aug 1994 20:14:21 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "" Subject: Re: PI to 20 or 30 sig-figs ?? Ah, the thread awakens... It reminds me of something I read recently: "... Formalism was quite secondary to the to the informal reasoning that is carried out when mathematicians have a first idea about the sort of theorem that they would like to prove; they attempt to do this, and discover that it can't be done because there are unusual counterexamples to their potential theorem. Their response is to first revise and refine their goal and then to try to prove a new theorem. Lakatos showed how some geometrical problems had evolved in this way through a chain of reformulations, counterexamples, and partial proofs until something related to - but quite distinct from - the original proposal was stated and proved. ..." - John D. Barrow, in "PI in the Sky," referencing Imre Lakatos' "Proofs and Refutations" Incidentally, my original effort was aimed at developing a rational approximation of pi as a function of the frequency of a regular polygon, without trig. Leaving out the later restriction, I came up with: /----------------------------------------- pi' = n * \/ (R - R*cos(360/n) )^2 + ( R*sin(360/n) )^2 ---------------------------------------------------- 2*R Where n = frequency of subdivision and R = radius of circumscribing circle. --------------------------------------------------------------------------- Mitch C. Amiano amiano@delphi.com ========================================================================= Date: Sat, 13 Aug 1994 02:24:21 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Shannon A Snider Subject: AIP Fullerene News For those of you on GEODESIC who may be interested in the ongoing research research into Fullerenes (or Buckyballs). A morsel of information distributed by the American Institute of Physics... - Shannon A. Snider Forwarded message: > PHYSICS NEWS UPDATE > A digest of physics news items by Phillip F. Schewe, American > Institute of Physics > Number 189 August 9, 1994 physnews@aip.org > > THAT METAL ATOMS CAN SIT INSIDE BUCKYBALLS has now > been proved by an IBM Almaden-Caltech-Virginia Polytechnic > collaboration. Using transmission electron microscopy, the physicists > showed that the lattice spacing for a pure crystals of C-84 molecules > was the same (11.2 angstroms) as that for a crystal of Sc2@C84 > molecules, the first such pure metallofullerene crystal to be prepared. > They assert that the Sc2@C84 molecules are truly endohedral; that is, > the metal atoms reside within and not alongside the fullerenes. (R. > Beyers et al., Nature, 21 July.) ========================================================================= Date: Mon, 15 Aug 1994 03:09:07 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: CPofLA Organization: America Online, Inc. (1-800-827-6364) Subject: Bucky Ball Kit For a flyer featuring a very cool (well that's what we say) bucky ball kit, contact mondo@mondo.com or call 1 415 455 9330. Chris Paine ========================================================================= Date: Wed, 17 Aug 1994 09:33:58 PDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: AMMSSC@TEVM2.NSC.COM Subject: NO SUBJECT X-To: geodesic%ubvm.bitnet@mitvma.mit.edu FROM: Mike Slama SUBJECT: subscribe geodesic Mike Slama ========================================================================= Date: Fri, 19 Aug 1994 01:44:34 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kevin Sahr Organization: Forest Sciences Laboratory Subject: Q: Uniquely identifying dome segments For a project we are working on we need to uniquely identify each of the faces of a subdivided spherical icosahedron (more specifically, we need some addressing system for dealing with the faces on a computer). A colleague came-up with an interesting line of investigation: does anyone know how the individual faces of a geodesic dome are identified/ marked so that they can be correctly assembled? Thanks, Kevin ========================================================================= Date: Fri, 19 Aug 1994 14:49:53 EST Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "j.marang-acacia-dsn-student-91093738" Subject: Fog Gun anyone? Hi! I just popped onto this list in the hope that someone here could give me some information on an invention of Buckminster's known as a 'Fog Gun'. A lecturer ofmine told me about this invention and he used that name, so I hope it's right. It's been impossible so far to find any information on the invention, because all of the books I've found so far only deal with his major inventions, like thegeodesic domes. Buckminster (so my lecturer tells me) came up with the idea after observing some fishermen coming back after fishing in a fog. Their hands, which should have been oily and dirty, were completely clean. He hypothesised that the combination of fog and high wind had an efficient cleaning effect. He invented a device that forced water through a nozzle to create a fine mist, and a heating device was located at the junction between the hose and nozzle. This is all I know, and I don't know how acurate it is. Can anyone supply me with more information? Thanks, Jennifer jlmarang@acacia.itd.uts.edu.au ========================================================================= Date: Fri, 19 Aug 1994 15:25:07 EST Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "j.marang-acacia-dsn-student-91093738" Subject: Fog Gun anyone? Forwarded message: >From jlmarang Fri Aug 19 14:50:06 1994 From: jlmarang (j.marang-acacia-dsn-student-91093738) Message-Id: <9408190449.AA10052@acacia.itd.uts.EDU.AU> Subject: Fog Gun anyone? To: geodesic@ubvm.bitnet Date: Fri, 19 Aug 94 14:49:53 EST Cc: jlmarang X-Mailer: ELM [version 2.4dev PL17] Hi! I just popped onto this list in the hope that someone here could give me some information on an invention of Buckminster's known as a 'Fog Gun'. A lecturer ofmine told me about this invention and he used that name, so I hope it's right. It's been impossible so far to find any information on the invention, because all of the books I've found so far only deal with his major inventions, like thegeodesic domes. Buckminster (so my lecturer tells me) came up with the idea after observing some fishermen coming back after fishing in a fog. Their hands, which should have been oily and dirty, were completely clean. He hypothesised that the combination of fog and high wind had an efficient cleaning effect. He invented a device that forced water through a nozzle to create a fine mist, and a heating device was located at the junction between the hose and nozzle. This is all I know, and I don't know how acurate it is. Can anyone supply me with more information? Thanks, Jennifer jlmarang@acacia.itd.uts.edu.au ========================================================================= Date: Fri, 19 Aug 1994 15:52:40 EST Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "j.marang-acacia-dsn-student-91093738" Subject: Fog Gun anyone? Forwarded message: >From jlmarang Fri Aug 19 14:50:06 1994 From: jlmarang (j.marang-acacia-dsn-student-91093738) Message-Id: <9408190449.AA10052@acacia.itd.uts.EDU.AU> Subject: Fog Gun anyone? To: geodesic@ubvm.bitnet Date: Fri, 19 Aug 94 14:49:53 EST Cc: jlmarang X-Mailer: ELM [version 2.4dev PL17] Hi! I just popped onto this list in the hope that someone here could give me some information on an invention of Buckminster's known as a 'Fog Gun'. A lecturer of mine told me about this invention and he used that name, so I hope it's right. It's been impossible so far to find any information on the invention, because all of the books I've found so far only deal with his major inventions, like the geodesic domes. Buckminster (so my lecturer tells me) came up with the idea after observing some fishermen coming back after fishing in a fog. Their hands, which should have been oily and dirty, were completely clean. He hypothesised that the combination of fog and high wind had an efficient cleaning effect. He invented a device that forced water through a nozzle to create a fine mist, and a heating device was located at the junction between the hose and nozzle. This is all I know, and I don't know how acurate it is. Can anyone supply me with more information? Thanks, Jennifer jlmarang@acacia.itd.uts.edu.au ========================================================================= Date: Fri, 19 Aug 1994 02:10:42 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: Fog Gun anyone? In-Reply-To: Message of Fri, 19 Aug 1994 14:49:53 EST from On Fri, 19 Aug 1994 14:49:53 EST j.marang-acacia-dsn-student-91093738 said: >Hi! > >I just popped onto this list in the hope that someone here could give me some > information on an invention of Buckminster's known as a 'Fog Gun'. A lecturer > ofmine told me about this invention and he used that name, so I hope it's > right. Here is what I've compiled on the fog gun from the FAQ: Pat Salsbury originally posted the following: The ``fog gun'' was an invention Bucky developed as a water saving alternative to the wastefulness of showers. While Bucky was in the navy, he noted that, while standing on the deck of a ship, in the spray and mist of the sea, nothing seems to stay on your skin for very long. Not even grease. He reasoned that it must have something to do with the abrasive action of the tiny water droplets, so he developed a device that atomized the water (like a perfume bottle with the little bulb that you squeeze to get perfume mist) and ejected it at high speed. He dubbed this the ``fog gun'' and found that it worked very well for cleaning a person off without soap (I'm not sure how he did hair, though) and without wasting a lot of water. (The ``gun'' could clean a family of four with *1 PINT* of water!) > >It's been impossible so far to find any information on the invention, because > all of the books I've found so far only deal with his major inventions, like > thegeodesic domes. Buckminster (so my lecturer tells me) came up with the idea > after observing some fishermen coming back after fishing in a fog. Their > hands, which should have been oily and dirty, were completely clean. He > hypothesised that the combination of fog and high wind had an efficient > cleaning effect. Actually it was Fuller himself who made the obsevation. I don't know of any technical specs for the device, but it's mentioned three times in the index to Fuller's _Critical_Path_. > >He invented a device that forced water through a nozzle to create a fine mist, > and a heating device was located at the junction between the hose and nozzle. Is this REALLY how he atomized the water? Has your instructor seen drawings? Has anyone seen drawings? I'm sure someone here has "The artifacts of R. Buckminster Fuller : a comprehensive collection of his designs and drawings" which may have details on this. I don't have this book :( > >This is all I know, and I don't know how acurate it is. Can anyone supply me > with more information? I hope this helps. > >Thanks, > >Jennifer > >jlmarang@acacia.itd.uts.edu.au -- Christopher J. Fearnley | UNIX SIG Leader at PACS cfearnl@pacs.pha.pa.us | (Philadelphia Area Computer Society) fearnlcj@duvm.bitnet | Design Science Revolutionary fearnlcj@duvm.ocs.drexel.edu | Explorer in Universe 503 S 44th ST | Linux Advocate Philadelphia PA 1914-3907 | (215)349-9681 ========================================================================= Date: Thu, 18 Aug 1994 23:21:51 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Subject: Re: Kirby >In Article >Chris Fearnley writes: >>On Sat, 6 Aug 1994 14:38:00 GMT said: >>>Hey, whatever happened to Kirby? >>He posted that he was going to Africa with his family for a few weeks(?) >>He'll be back soon - judging from history he'll reply to all of the >>messages he has "missed." > >On that note, where would I find the archives? > Steve Mather > > Yeah, well, I'll definitely reply to this one at least. Yes, back from Africa -- Lesotho to be precise, where I read in the London Times this morning (still jet lagged) a coup has just taken place. So now I'm wondering about dad (mom's at a conference in New Mexico). I'm about to email the US Embassy there. Been looking over the PI thread. My software wimps out at 30 iterations but I'm confidant my algorithm was correct. Using pythagoras and the unit circle was all the rage for awhile, but in later decades, other methods have superceded. That quote from Kiyoshi awhile back about someone in the New York Times questioning the "actuality" of PI beyond a certain number of decimals (because nature is finite, not infinite), seems the way to connect back more directly to synergetics. So, yes, I'm out of Africa. Glad to be back with y'all. -- Kirby ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) ========================================================================= Date: Fri, 19 Aug 1994 02:47:40 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: Kirby In-Reply-To: Message of Thu, 18 Aug 1994 23:21:51 -0700 from On Thu, 18 Aug 1994 23:21:51 -0700 Kirby Urner said: > >So, yes, I'm out of Africa. Glad to be back with y'all. Welcome Back. > >-- Kirby > >------------------------------------------------ >Kirby T. Urner pdx4d@teleport.com >4D Solutions (teleport.com is a public access node) ========================================================================= Date: Fri, 19 Aug 1994 22:14:33 EST Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "j.marang-acacia-dsn-student-91093738" Subject: I just went through those books today! In-Reply-To: <9408190625.AA18768@acacia.itd.uts.EDU.AU>; from "Chris Fearnley" at Aug 19, 94 02:10:42 am Hi again, Thanks Chris for the info! I had actually looked through the four volumes of the collection 'The artifacts of Buckminster Fuller;...' and they were very comprehensive - but only on his more reknowned inventions - with no mention of the fog gun. They are very good books though, because they are almost totally made up of his drawings, both engineering and sketchy concepts, of his inventions. Unfortunately, they seem very hard to get a hold of: at the library of my Uni, the books were located in the basement, and were not normally allowed for student viewing. Don't know why... Anyway, thanks again, and if there's anything else anyone knows I'm still interested! Jennifer jlmarang@acacia.itd.uts.edu.au ========================================================================= Date: Fri, 19 Aug 1994 10:10:27 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: Fog Gun anyone? > On Fri, 19 Aug 1994 14:49:53 EST j.marang-acacia-dsn-student-91093738 said: > >Hi! > > > >I just popped onto this list in the hope that someone here could give me some > > information on an invention of Buckminster's known as a 'Fog Gun'. A lecture r > > ofmine told me about this invention and he used that name, so I hope it's > > right. > [Chris] > Here is what I've compiled on the fog gun from the FAQ: > Pat Salsbury originally posted the following: > The ``fog gun'' was an invention Bucky developed as a water saving > alternative to the wastefulness of showers. While Bucky was in the > navy, he noted that, while standing on the deck of a ship, in the spray > and mist of the sea, nothing seems to stay on your skin for very long. > Not even grease. He reasoned that it must have something to do with the > abrasive action of the tiny water droplets, so he developed a device > that atomized the water (like a perfume bottle with the little bulb that > you squeeze to get perfume mist) and ejected it at high speed. He dubbed > this the ``fog gun'' and found that it worked very well for cleaning a > person off without soap (I'm not sure how he did hair, though) and > without wasting a lot of water. (The ``gun'' could clean a family of four > with *1 PINT* of water!) > > > >It's been impossible so far to find any information on the invention, because > > all of the books I've found so far only deal with his major inventions, like > > thegeodesic domes. Buckminster (so my lecturer tells me) came up with the id ea > > after observing some fishermen coming back after fishing in a fog. Their > > hands, which should have been oily and dirty, were completely clean. He > > hypothesised that the combination of fog and high wind had an efficient > > cleaning effect. > > Actually it was Fuller himself who made the obsevation. I don't know of any > technical specs for the device, but it's mentioned three times in the index > to Fuller's _Critical_Path_. > > > >He invented a device that forced water through a nozzle to create a fine mist , > > and a heating device was located at the junction between the hose and nozzle . > > Is this REALLY how he atomized the water? Has your instructor seen drawings? > Has anyone seen drawings? I'm sure someone here has "The artifacts of R. > Buckminster Fuller : a comprehensive collection of his designs and drawings" > which may have details on this. I don't have this book :( > > I'll ask Ed to let me look at his copy of "artifacts". I intend to install these in my retirement home and I will be working with prototypes at my current home. It seems to me that the device (which Bucky did prototype) synergized the water pressure with an air compressor. A mixing valve would cause the air to draw the water through a (or several?) pinhole exit point(s). This would cause the fog to occur. The compressed air would be analogous to the bulb of air in the atomizer. As Fuller might say, that's "Everything I Know" about the fog gun. When I learn more, I will share it. "Just call me Trimtab" {~~~| R. Buckminster Fuller ~~~| _______ | / \ | BDAMICO@GWUVM.GWU.EDU ___________/__________\______|_____ \ Trimtab: A tiny gear / Blaine A. D'Amico |\ which moves the rudder / Systems Specialist ~~~~~~ |__\ that turns great ships / ~~~~~ Design Science Revolutionary --------------------- Comprehensive Generalist ========================================================================= Date: Fri, 19 Aug 1994 10:12:11 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: Kirby > On Thu, 18 Aug 1994 23:21:51 -0700 Kirby Urner said: > > > >So, yes, I'm out of Africa. Glad to be back with y'all. > > Welcome Back. > > > >-- Kirby > > > >------------------------------------------------ > >Kirby T. Urner pdx4d@teleport.com > >4D Solutions (teleport.com is a public access node) > I'll be in Portland the week of the 29th. Can we meet on Wednesday afternoon? Would you like to come by our property and help plan our retirement dome? "Just call me Trimtab" {~~~| R. Buckminster Fuller ~~~| _______ | / \ | BDAMICO@GWUVM.GWU.EDU ___________/__________\______|_____ \ Trimtab: A tiny gear / Blaine A. D'Amico |\ which moves the rudder / Systems Specialist ~~~~~~ |__\ that turns great ships / ~~~~~ Design Science Revolutionary --------------------- Comprehensive Generalist ========================================================================= Date: Fri, 19 Aug 1994 22:01:33 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kevin Sahr Organization: Forest Sciences Laboratory Subject: quantum physics and tetrahedrons (x-post frm sci.math) I saw this on sci.math and thought I'd pass it along... Article: 75090 of sci.math Newsgroups: sci.physics.research,sci.math,sci.physics From: baez@ucrmath.ucr.edu (John Baez) Subject: This Week's Finds in Mathematical Physics (Week 38) Message-ID: <1994Aug19.204415.14387@galois.mit.edu> Originator: jbaez@euclid X-Sender: usenet@galois.mit.edu Nntp-Posting-Host: euclid Organization: MIT, Department of Mathematics Date: Fri, 19 Aug 94 20:44:15 GMT Approved: jbaez@math.mit.edu Lines: 303 This Week's Finds in Mathematical Physics (Week 38) John Baez I've been busy, and papers have been piling up; there are lots of interesting ones that I really should describe in detail, but I had better be terse and list them now, rather than waiting for the mythical day when I will have time to do them justice. So: 1) Topological quantum field theories from generalized 6j-symbols, B. Durhuus, H. P. Jakobsen and R. Nest, Reviews in Math. Physics 5 (1993), 1-67. In "week16" I explained a paper by Fukuma, Hosono and Kawai in which they obtained topological quantum field theories in 2 dimensions starting with a triangulation of a 2d surface. The theories were "topological" in the sense that the final answers one computed didn't depend on the triangulation. One can get between any two triangulations of a surface by using a sequence of the following two moves (and their inverses), called the (2,2) move: O O /|\ / \ / | \ / \ / | \ / \ O | O <----> O-------O \ | / \ / \ | / \ / \|/ \ / O O and the (3,1) move: O O /|\ / \ / | \ / \ / | \ / \ / | \ / \ / _O_ \ <----> / \ / _/ \_ \ / \ / _/ \_ \ / \ /_/ \_\ / \ O-----------------O O-----------------O Note that in either case these moves amount to replacing one part of the surface of a tetrahedron with the other part! In fact, similar moves work in any dimension, and they are often called the Pachner moves. The really *wonderful* thing is that these moves are also very significant from the point of view of algebra... and especially what I call "higher-dimensional algebra" (following Ronnie Brown), in which the distinction between algebra and topology is largely erased, or, one might say, revealed for the sham it always was. For example, as explained more carefully in "week16", the (2,2) move is really just the same as the *associative* law for multiplication. The idea is that we are in a 2-dimensional spacetime, and a triangle represents multiplication: two "incoming states" go in two sides and their product, the "outgoing state", pops out the third side: O / \ / \ / \ A B / \ / \ / \ / \ O--------AB-------O Then the (2,2) move represents associativity: O O /|\ / \ A | (AB)C A A(BC) / | \ / \ O AB O <----> O--BC---O \ | / \ / B | C B C \|/ \ / O O Of course, the distinction between "incoming" and "outgoing" sides of the triangle is conventional, and the more detailed explanation in "week16" shows how that fits into the formalism. Roughly speaking, what we have is not just any old algebra, but an algebra that, thought of as a vector space, is equipped with an isomorphism between it and its dual. This isomorphism allows us to forget whether we are coming or going, so to speak. Hmm, and here I was planning on being terse! Anyway, the still *more* interesting point is that when we think about 3-dimensional topology and "3-dimensional algebra," we should no longer think of O O /|\ / \ / | \ / \ / | \ / \ O | O and O-------O \ | / \ / \ | / \ / \|/ \ / O O as representing *equal* operations (the 3-fold multiplication of A, B, and C); instead, we should think of them as merely *isomorphic*, with the tetrahedron of which they are the front and back being the isomorphism. The basic philosophy is that in higher-dimensional algebra, as one ascends the ladder of dimensions, certain things which had been regarded as *equal* are revealed to be merely isomorphic. This gets tricky, since certain *isomorphisms* that were regarded as equal at one level are revealed to be merely isomorphic at the next level... leading us into a subtle world of isomorphisms between isomorphisms between isomorphisms... which the theory of n-categories attempts to systematize. (I should note, however, that in the particular case of associativity this business was worked out by Jim Stasheff quite a while back: it's the homotopy theorists who were the ones with the guts to deal with such issues first.) Now, it turns out that in 3-dimensional algebra, the isomorphism corresponding to the (2,2) move is not something marvelously obscure. It is in fact precisely what physicists call the "6j symbol", a gadget they've been using to study angular momentum in quantum mechanics for a long time! In quantum mechanics, the study of angular momentum is just the study of representations of the group SU(2), and if one has representations A, B, and C of this group (or any other), the tensor products (A tensor B) tensor C and A tensor (B tensor C) are not *equal*, but merely *isomorphic*. It should come as no surprise that this isomorphism is represented by physicists as a big gadget with 6 indices dangling on it, the "6j symbol". Quite a while back, Regge and Ponzano tried to cook up a theory of quantum gravity in 3 dimensions using the 6j symbols for SU(2). More recently, Turaev and Viro built a 3-dimensional topological quantum field theory using the 6j-symbols of the *quantum group* SU_q(2), and this led to lots of work, which the above article explains in a distilled sort of way. The original Ponzano-Regge and Turaev-Viro papers, and various other ones clarifying the relation of the Turaev/Viro theory to quantum gravity in spacetimes of dimension 3, are listed in "week16". It's also worth checking out the paper by Barrett and Foxon listed in "week24", as well as the following paper, for which I'll just quote the abstract: ...other not-related paper abstracts deleted... ========================================================================= Date: Thu, 18 Aug 1994 19:03:00 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: William Cringan Organization: CRS Online (Toronto, Ontario) Subject: flyer Please send me a flyer with the buckeyball kit. My address is: William Cringan, 2267 Westman Rd. Mississauga, Ontario. L5K 1M7 Thanks Kindly. --- * OLX 2.1 TD * All wiyht. Rho sritched mg kegtops awound? ========================================================================= Date: Sat, 20 Aug 1994 06:41:25 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Subject: Re: flyer >Please send me a flyer with the buckeyball kit. >My address is: > >William Cringan, >2267 Westman Rd. >Mississauga, Ontario. >L5K 1M7 > >Thanks Kindly. > >--- > * OLX 2.1 TD * All wiyht. Rho sritched mg kegtops awound? > > The science toys company which produced a buckyball kit (some of which were sold through the Buckminster Fuller Institute) is: Mondotronics 524 San Anselmo Ave #107 San Anselmo, CA 94960 Phone 414-455-9330 Fax 414-455-9333 email mondo@holonet.net Its a small company run by Roger G. Gilbertson. Their most popular products feature "muscle wires" -- highly processed strands of nickel-titanium (nitinol) which stretch at room temperature but harden into their unstretched shapes when conducting an electrical current. The buckyball kit consists of four sheets of flexible plastic each containing 3 pentagons and 5 hexagons. You pop these out and snap them together edgewise to assemble a buckyball that's a bit smaller than a soccer ball. No glue or other materials required. The kit also comes with a booklet giving background information on buckyballs and even some synergetics. I was the author of the booklet (Roger and Celene designed and edited it). The information is still relevant, if a bit dated (hottest area of chemical research is going to obsolete its info quickly of course). I used a buckyball kit with 6th graders in Lesotho, Africa just about a week ago. We also assembled a vector flexor. The tensegritoy takes more time and will have to wait 'til another time. -- Kirby ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) ========================================================================= Date: Sat, 20 Aug 1994 23:50:01 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Grego10067 Organization: America Online, Inc. (1-800-827-6364) Subject: Geodesic Dome Software Can anyone tell me if there is any software out there to design geodesic domes with, and create different 2, 3, 4, 6, 8 frequency designs ?????????? please E-Mail me on America On Line at Grego10067, or on this forum, I will offer a reword leading to the purchase of this software, ========================================================================= Date: Sun, 21 Aug 1994 20:32:10 -0800 Reply-To: Curtis@dbug.org Sender: List for the discussion of Buckminster Fuller's works From: SimulatedSnow Organization: Baloney Research Institute Subject: Re: World Game In article <32bh68$k2u@dunx1.ocs.drexel.edu>, XTM00003@DUVM.OCS.DREXEL.EDU (Dane Winberg) wrote: | As for comments and experiences of past participants at the workshops, I look | forward to seeing them posted here. I've worked for WGI for 11 years... I attended a WG in the mid 70's & consider it one of the highlights of my life. The perspective stays w/me. I would like for all candidates for public office to be exposed to this...top to bottom. Any word on plans for next year ? | Dane Winberg | World Game Institute -Mahalo for the 11 years & the future ! -- -Planning for years, plant rice -Planning for decades, plant trees -Planning for centuries, educate people ========================================================================= Date: Mon, 22 Aug 1994 06:21:05 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Ross Keatinge Organization: Public Access Internet, Auckland New Zealand Subject: Re: Fog Gun anyone? >From what I have, in Bucky's book "Critical Path", your description is fairly accurate. As a sidenote. We could sure do with some of Bucky's bathroom designs here in Auckland, New Zealand. We are in the grips of a water shortage and are all trying to save water. Now that I'm conscious of it, it has become obvious what a ridiculous amount of water each household uses. Regards Ross Keatinge icosa@iconz.co.nz ========================================================================= Date: Mon, 22 Aug 1994 08:38:56 -0600 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Bob Bayn, mgr. Academic Services, Computer Center" Subject: Re: Fog Gun anyone? >From: Ross Keatinge >As a sidenote. We could sure do with some of Bucky's bathroom designs here >in Auckland, New Zealand. We are in the grips of a water shortage and are >all trying to save water. Now that I'm conscious of it, it has become >obvious what a ridiculous amount of water each household uses. Utah's cronic water shortage is also acute this year. It's made me think that my dream of a geodesic dome home probably ought to include some facility for greywater capture and reuse (probably for irrigation). What the heck, my preliminary investigations suggest that planners, inspectors and lenders are all going to be hard sells for my efforts, I may as well hit 'em with non-conforming plumbing, too. Unfortunately, I haven't a clue as to where to find info about workable greywater collection systems in new construction. I don't believe that Bucky did any design work on this topic. Anybody know of any specs, layouts or designs that would complement Bucky's dome? -- Robert Bayn http://happy.usu.edu/~bob/index.html o bob@cc.usu.edu I just can't picture having \__^\=* Logan, Utah, USA fun by burning gas. (O)""""o ========================================================================= Date: Mon, 22 Aug 1994 10:41:35 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Tom Buckley Organization: NB*Net Subject: stress analysis Can anyone tell me how to get some software or otherwise so that I can determine the stresses on dome members? ********************************************************* * TERRAIN ENGINEERING LTD. * *P.O. Box 2703,Saint John, New Brunswick,CANADA E2L 4Z1* * * * voice, fax: 506-832-4929 * * 14400 or less, N81, Zmodem (ask) * ********************************************************* ========================================================================= Date: Mon, 22 Aug 1994 11:09:47 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Bob Cook Subject: Solar Village Questionaire Tucson Solar Village Thank you for your help in planning Phase 3 of the Tucson Solar Village Program. We were pleased with the quality and thoughtfulness of the responses to the first cycle. In this second cycle Delphi process we will review responses from the first cycle, try to identify the most important issues and examine them further. This second questionnaire is intended to be somewhat more specific than the first. We encourage you to add ideas and suggestions to those listed. The final questionnaire will incorporate responses from this and the first one and will become more detailed and specific. It is important that you mail or fax your responses in time to be received by Monday, August 29, 1994 to: Robert Cook, Solar Village Committee Chair Tucson-Pima County Metropolitan Energy Commission BCOOK@Pimacc.Pima.Edu FAX (602) 748-4754 Phone (602) 748-4745 Thank you again for your participation. In this second cycle of the Delphi process, we complete responses to the first cycle for each of the main issues. Now, would you please review these lists and help make them as complete as possible by adding important items that have not been mentioned so far. Then, we ask that you select the 3-5 most important items and explain briefly why they are important. The explanations are important to the process, so please give them careful consideration. Finally, we ask for your creative help with two questions that are of particular concern at this time. Thank you again for your help. 1. Following is a list of important features in Civano cited by respondents. Please add any others you think should be listed, then select 3-5 features by letter reference that you feel are most important, and explain why. a. Passive solar design b. Photovoltaic generation system c. Pedestrian path system d. Resource conservation area e. Bicycle path system f. Connection to public transportation g. Neighborhoods to support design flexibility h. Diverse socio-economic inclusion i. Sense of neighborhood & place j. Community & on-site water harvesting k. Neighborhood social & recreation centers l. Clustered building layout with open space m. Access to regional trails system n. Proximity to on-site jobs o. Walkable village center p. Community education program q. Recycling options r. Community gardens, orchards s. Reclaimed effluent system t. Community tram system u. Other: v. Other: Letter Why important 2. Following is a list of activities cited by respondents as important to assure the successful development of Civano. Please add any others you think should be listed, then select 3-5 activities which you think are most important, and explain why. a. Offer competitive home value b. Involve associations, groups & individuals c. Provide incentives for developers d. Integrate environmental education e. Build demonstration program now f. Build office work stations in each home g. Use best minds of designers & engineers h. Attract a sympathetic developer i. Establish a Community Resource Center j Form management/overseeing body now k. Prepare focused market analysis/strategy l. Conduct public info/relations program m. Integrate with community revitalization n. Define photovoltaic program with TEP o. Redefine land sale to include all values p. Committed people on design review q. Prepare strong agency commitments r. Structural efficiencies built into all phases s. Include range of uses in all phases t. Seek additional resources to assure goals u. Build village center & plaza early v. Screen buyers for "Civano" values w. Other: x. Other: Letter Why important 3. Following is a list of Civano aspects cited by respondents as important to implement in the community at large. Please add any others you think should be listed, then select 3-5 activities/projects which you think are most important, and explain why. a. Develop Civano concepts as infill project b. Solar: water heat, passive design, solar thermal, pv systems c. Preserve wash habitat corridors d. Target areas for ped-friendly design e. Education re sustain. design principles f. Ped access to neighborhood nodes g. Focus on getting Civano built as demo h. Awards program for exceeding targets i. Apply Civano performance targets to all j. More affordable housing options k. More trees & shade l. Provide bicycle-friendly design features m. Pedestrian emphasis throughout region n. Energy conserv'n equipment, techniques o. Water conserv'n equipment, techniques p. Recycling to the max q. Document existing local examples r. Build infill demonstration housing s. Alternative transportation corridors t. Other: u. Other Letter Why important 4. Following is a list of ways cited by respondents to bring the best people with the right resources together to get the job done. Please add any others you think should be listed, then select 3-5 ways which you feel are most important, and explain why. a. Let a committed developer manage it b. Secure interest of Fortune 500 company c. Form public-private-nonprofit partnership d. Continue getting the word out e. Network with inclusive stakeholder groups f. Develop a management association g. Advertise nationally for "right" developer h. Aggressive leadership & management i. Leverage funding: seek additional grants j. Task forces to deal with important aspects k. Continue doing what MEC is now doing. l. Support full time staff to ensure continuity m. Consider each retailer, employer, resident as stockholder n. Let the successful land purchaser manage it o. Convince larger public of economic viability p. Other: q. Other: Letter Why important 5. Many of the respondents to the first questionnaire suggested that marketing techniques and economic feasibility analyses of specific features in Civano are very important in assuring its successful development. If you were the enlightened sales manager for the Civano developer, what would be the elements of your marketing strategy? 6. Many respondents to the first questionnaire felt that leadership and developing partnerships and cooperation between builders, developers, public agencies, civic organizations and utility/service providers is critical to developing the Solar Village Program. What would be the best way(s) to form such public/private partnership(s) or cooperation? 7. Many respondents to the first questionnaire felt that there should be a set of developer incentives to encourage conformance with the Civano goals and objectives. What kinds of incentives do you think would be appropriate to assure inclusion of the following features. FEATURES INCENTIVES Clustered housing with open space around Village commercial center to begin with Village & neighborhood social centers Separate bicycle and pedestrian paths Natural open space & drainage preservation Jobs creation within Civano Create personal office setup in home Photovoltaic generating system Community education program Community gardens, orchards Co-housing options "Sweat equity" housing options Low-income housing options Connection to public transit alternatives Recycling of materials 8. Please feel free to add any other comments/ideas/suggestions which you think would further the implementation of Civano and encourage the development of related projects throughout the community, or comments on the questionnaires thus far. Robert Cook, Solar Village Committee Chair Tucson-Pima County Metropolitan Energy Commission BCOOK@Pimacc.Pima.Edu FAX (602) 748-4754 Phone (602) 748-4745 Again, thank you for your participation. ========================================================================= Date: Mon, 22 Aug 1994 14:29:56 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Re: Fog Gun anyone? > > >From: Ross Keatinge > >As a sidenote. We could sure do with some of Bucky's bathroom designs here > >in Auckland, New Zealand. We are in the grips of a water shortage and are > >all trying to save water. Now that I'm conscious of it, it has become > >obvious what a ridiculous amount of water each household uses. > > Utah's cronic water shortage is also acute this year. It's made me > think that my dream of a geodesic dome home probably ought to include > some facility for greywater capture and reuse (probably for irrigation). > What the heck, my preliminary investigations suggest that planners, > inspectors and lenders are all going to be hard sells for my efforts, > I may as well hit 'em with non-conforming plumbing, too. Unfortunately, > I haven't a clue as to where to find info about workable greywater > collection systems in new construction. I don't believe that Bucky > did any design work on this topic. Anybody know of any specs, layouts > or designs that would complement Bucky's dome? > -- > Robert Bayn http://happy.usu.edu/~bob/index.html o > bob@cc.usu.edu I just can't picture having \__^\=* > Logan, Utah, USA fun by burning gas. (O)""""o > Actually, Bucky focused on not creating as much grey water to start with. Along with the fog gun was the dry toilet which Bucky proposed. Bucky wanted us to stop taking the waste products that nature has so carefully seperated into liquid and solid and mix them up again in order to dispose of them. He proposed a toilet that plastic wrapped dry waste in hermetically sealed packages. Solid waste could then be converted into energy at the homesite or trucked away to third party energy plants. Dont' know about any gery water projects that Bucky did. "Just call me Trimtab" {~~~| R. Buckminster Fuller ~~~| _______ | / \ | BDAMICO@GWUVM.GWU.EDU ___________/__________\______|_____ \ Trimtab: A tiny gear / Blaine A. D'Amico |\ which moves the rudder / Systems Specialist ~~~~~~ |__\ that turns great ships / ~~~~~ Design Science Revolutionary --------------------- Comprehensive Generalist ========================================================================= Date: Mon, 22 Aug 1994 15:40:50 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: Fog Gun anyone? Robert Bayn wrote: >Utah's cronic water shortage is also acute this year. It's made me >think that my dream of a geodesic dome home probably ought to include >some facility for greywater capture and reuse (probably for irrigation). >[snip] >Anybody know of any specs, layouts or designs that would complement >Bucky's dome? I was considering the problem recently and decided that a solar powered evaporator/condenser would be worth looking into. Such a device would produce recycled water of sufficient quality for drinking. I don't have any plans, but I'll describe what I had in mind: - a greywater tank within a passive, solar collector box; - pipes carry the water vapor from the tank through a trough of cold water, where it condenses back into liquid. The trough is chilled by the solid state cooling devices they use on those new portable ice-chests you plug into your car's cigarette lighter [sorry, I've forgotten their name]) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Mon, 22 Aug 1994 23:29:27 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: stress analysis In-Reply-To: Message of Mon, 22 Aug 1994 10:41:35 GMT from On Mon, 22 Aug 1994 10:41:35 GMT Tom Buckley said: >Can anyone tell me how to get some software or otherwise so that I can >determine the stresses on dome members? FElt 2.0 is availble for Unix systems (binary releases for Linux, HP-UX and SUNOS). It is a finite element analysis package. Although I haven't played with it yet, it seems like a good package for simple structural analysis. Robert L. Read reported on this group awhile ago "It allows you to input a geometry of a structure, assign material types to various components, add on continuous forces like roof loads and specific forces like a 10,000 pound weight at a certain point, and then compute the forces in each member. It can handle lots of stuff (hence "Finite Element") but of particular interest to fans of Fuller is that it easily handles "tensegrity" structures, where a tensegrity structures is one in which force is transmitted only axially along members (cables or struts)." It is copylefted software (GPL) and available for anonymous ftp at cs.ucsd.edu:/pub/felt. For Linux users a binary is on sunsite.unc.edu in /pub/Linux/apps/math. >********************************************************* >* TERRAIN ENGINEERING LTD. * >*P.O. Box 2703,Saint John, New Brunswick,CANADA E2L 4Z1* >* * >* voice, fax: 506-832-4929 * >* 14400 or less, N81, Zmodem (ask) * >********************************************************* -- Christopher J. Fearnley | UNIX SIG Leader at PACS cfearnl@pacs.pha.pa.us | (Philadelphia Area Computer Society) fearnlcj@duvm.bitnet | Design Science Revolutionary fearnlcj@duvm.ocs.drexel.edu | Explorer in Universe 503 S 44th ST | Linux Advocate Philadelphia PA 1914-3907 | (215)349-9681 ========================================================================= Date: Tue, 23 Aug 1994 02:34:28 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Postmaster Subject: Re: Fog Gun anyone? Sorry for jumping in it so abruptly. I think those solid sate cooling devices consumes a lot of energy. It may be a problem with a PV system. Also, they are based on a principle named the "Pelletier effect". I find your project very interesting. Don't give up! Marc Tremblay, marc@harfang.login.qc.ca ========================================================================= Date: Tue, 23 Aug 1994 07:43:33 -0500 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kiyoshi Kuromiya Subject: Re: stress analysis X-cc: rich@cpp.pha.pa.us In-Reply-To: from "Chris Fearnley" at Aug 22, 94 11:29:27 pm Chris: Even if only axial stresses are measured in a pure tensegrity structure rather than a dome with multiple nodes, the stress analysis of dome members is still far too complex to compute with any existing software. Fuller compared such an analysis with analyzing the movement of all the air molecules in a basketball simultaneously. For that reason, structurl engineers were forced by this to test geodesic structures by loading them to failure. Because of this fact, geodesic domes were often greatly overbuilt because of the need for conservativism, since the mathematical tools were not and still are not available. --Kiyoshi Quoted message begins here: > > On Mon, 22 Aug 1994 10:41:35 GMT Tom Buckley said: > >Can anyone tell me how to get some software or otherwise so that I can > >determine the stresses on dome members? > > FElt 2.0 is availble for Unix systems (binary releases for Linux, HP-UX and > SUNOS). It is a finite element analysis package. Although I haven't > played with it yet, it seems like a good package for simple structural > analysis. Robert L. Read reported on this group awhile ago > > "It allows you to input a geometry of a structure, assign material types > to various components, add on continuous forces like roof loads and > specific forces like a 10,000 pound weight at a certain point, and then > compute the forces in each member. It can handle lots of stuff (hence > "Finite Element") but of particular interest to fans of Fuller is that > it easily handles "tensegrity" structures, where a tensegrity structures > is one in which force is transmitted only axially along members (cables > or struts)." > > It is copylefted software (GPL) and available for anonymous ftp at > cs.ucsd.edu:/pub/felt. For Linux users a binary is on sunsite.unc.edu > in /pub/Linux/apps/math. > > >********************************************************* > >* TERRAIN ENGINEERING LTD. * > >*P.O. Box 2703,Saint John, New Brunswick,CANADA E2L 4Z1* > >* * > >* voice, fax: 506-832-4929 * > >* 14400 or less, N81, Zmodem (ask) * > >********************************************************* > -- > Christopher J. Fearnley | UNIX SIG Leader at PACS > cfearnl@pacs.pha.pa.us | (Philadelphia Area Computer Society) > fearnlcj@duvm.bitnet | Design Science Revolutionary > fearnlcj@duvm.ocs.drexel.edu | Explorer in Universe > 503 S 44th ST | Linux Advocate > Philadelphia PA 1914-3907 | (215)349-9681 > ========================================================================= Date: Tue, 23 Aug 1994 08:17:40 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Greywater Recycling (was: Fog Gun anyone?) Marc Tremblay wrote: > I think those solid sate cooling devices >consumes a lot of energy.[snip] Also, they >are based on a principle named the "Pelletier effect". Right, "Pelletier Effect Devices", thank you. As for the power consumption question, I will plead ignorance. It was not one of my design concerns. In fact, I had only two main design considerations: 1 - I started from the premise that water was an extremely valuable resource and that water recycling would be mandatory, irrespective of cost. 2 - I was designing for a solar power source, so once installed, the electricity would "free". On the plus side, when based on a photo-voltaic/battery electrical supply, the entire system would have NO moving parts. So, compared to a refrigerator, which has MANY moving parts, AS WELL AS high power consumption, this system *should* have a longer life and be essentially maintenance free. Two final notes: 1 - Evaporative cooling was not an option, since it wastes water. 2 - The CAPS are not intended to indicate "shouting" i.e. a FLAME response. They are merely for emphasis. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Tue, 23 Aug 1994 12:39:36 -0500 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Bill Long Organization: SUNY at Plattsburgh, New York, USA Subject: Dome for Taj Mahal? Hello, I caught an article in our local paper over the weekend saying that a lawyer in the Agra region of India has begun circulating a petition to get over 8000 factories moved out of the area as a means of preserving the Taj Mahal from the effects of pollution. This made me wonder if building a large geodesic structure to enclose the building and then controlling the atmosphere within the dome wouldn't be easier (and cheaper and a more permanent solution) than trying to displace that much industry. Think a structure that large could handle the stress of its own weight? Anyone on this forum who lives in that area that would be interested in forwarding this idea to local planners? To me a dome of this size would be almost as incredible a feat as the construction of the Taj Mahal itself. Anyone know about how large such a dome would have to be? Enough questions...thanx for your time. Bill Long >-- StarGazer ========================================================================= Date: Tue, 23 Aug 1994 12:48:18 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Mark Phillips Subject: Re: Greywater Recycling (was: Fog Gun anyone?) In-Reply-To: <9408231517.AA17624@rain.org> I am also looking into using these thermoelectric cooling devices (Peltier device) in conjunction with a solar electric source. On the surface, these devices seem to be pretty good, no moving parts, 12 VDC operation, low power (ie. 3 A @ 12VDC for 36 watts), no freon or other working fluid, quiet. But I've talked to several people, including one from a peltier device manufacturer that have tried to tell that they are not good for refrig/freezers. Part of the problems are: - they usually do need two fans (like in the back of your computer) to transfer the heat to/from the cold/heat sinks. - There are compressors out there optimized for 12VDC operation that are efficient with power consumption. - It seems that a single device only brings the internal temperature down to 41 deg F. Well above freezing. And that temperature is dependent on what the ambient temperature is. With all that, I am not daunted but and am a little more wary and not quite as excited or optimistic. You can cascade (use 2 devices back to back) for a larger temperature drop. And I still think the devices might be more reliable and have a longer life. It is just going to be a larger engineering design challenge. These companies manufacture Peltier devices and if you contact them, they will send you there literature complete with applications guides, product desc. and price lists: MELCOR 1040 Spruce St Trenton NJ 08648 609-393-4178 MARLOW INDUSTRIES 10451 Vista Park Rd Dallas, TX 75238 214-340-4900 SUPERCOOL US PO DRAWER 9066 San Rafeal, CA 94912 415-459-3777 Also check the Thomas register under Cooling, Thermoelectric devices or Thermoelectric cooling devices. I hope this helps you and doesn't discourage you. If it was easy, it would have been done by now, but that doesn't mean it's impossible. Good luck and good designing. ========================================================================= Date: Wed, 24 Aug 1994 09:15:45 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Alpha, Beta, Hexagonal closest packing I have a question that no one I know can answer. Why is the alpha state of crystals usually weaker than the beta state? Is it because the beta state makes a more efficient use of materials (i.e. is less dense,) or what? Steve Mather ========================================================================= Date: Wed, 24 Aug 1994 17:51:45 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: The "PI without Trig" Monster Returns SUBJECT: The "PI without Trig" Monster Returns I've stumbled upon a mystery which I hope someone here can help me solve. Unfortunately, it's quite detailed, so I hope you'll bear with me during the explanation. Here goes: As you may recall, a few weeks ago we discussed an algorithm for determining PI without using trig. [The algorithm divided a circle (radius=1) into smaller and smaller right triangles, each of whose hypotenuse lay progressively closer to the circumference of the circle. The hypotenuse was of a known length, and was an integer fraction of the circumference of the circle, so we could then calculate a good approximation for the length of the actual circumference. Knowing the radius and circumference, we could then determine PI to a fair degree of accuracy.] Well, I decided to apply a similar technique to finding the volume of a sphere (also of radius 1). My algorithm filled the sphere with flat cylinders (more like hockey pucks, actually), all contained (mostly) within the surface of the sphere. I then calculated the volume for each cylinder, and obtained their sum. NB actually, I filled only the upper hemisphere with cylinders, and multiplied their volumes by 2. The following diagram is supposed to resemble the hemisphere filled with 4 cylinders. My program used 10,000 cylinders, but drawing that was out of the question. As someone said earlier, "I hate ascii". __.__.__ _.|________|._ .|______________|. .|__________________|. .|____________________|. The resulting sum of volumes agreed quite closely to the valued given if you use the standard formula for the volume of a sphere: Vsphere = (4/3)PI*Radius*Radius*Radius = 4.1887902 (my program said Vsphere was equal to 4.1887901) Encouraged by this result, I modified the program to calculate the area of each cylinder wall. The sum of all 10,000, I thought, should be a good approximation of the surface area of the sphere. (of some interest to people who design domes, yes?) The equation for the Surface Area of a Sphere said: SAsphere = 4*PI*Radius*Radius = 12.5664 But my program said SAsphere = 9.8696 The error is more than 20% !! This was surprising, considering the accuracy I had achieved with my sphere-volume program. So I thought up a *new* algorithm. ALGORITHM #2 - Start with a hemisphere (radius=1). - Draw a line from the north pole to the equator. - Move a very small distance along the equator, then draw a second line from the north pole to this new point on the equator. We have just drawn a long, skinny isosceles triangle. It's base is on the equator, and it's apex is at the north pole. Now we cover the entire surface of the hemisphere with triangles identical to the one just described. (Before beginning, we cleverly ensured that the base of the triangle would be some integer fraction of the distance around the equator, say 1/1000th.) Each apex lies at the north pole, and each base lies along the equator. The triangles do not overlap, and they cover the hemisphere completely. We can easily determine the area of a triangle: Area = Base*Height/2 So we multiply that area by the number of triangles covering the hemisphere, then double that quantity to obtain the surface area of the entire sphere. DONE! But before programming this new algorithm, I first worked it out on paper. I allowed the number of triangles to grow very large and discovered that the Height approached a value equal to the circumference around the sphere (a great circle) divided by four. This simplified the algebra *significantly*. Almost everything canceled out, and I was left with: SAsphere = PI*PI*Radius*Radius (i.e. PI squared times the Radius squared) But since the Radius=1, it reduces to: SAsphere = PI*PI Being of an inquisitive nature, I took out my calculator and punched in PI*PI. To my surprise and consternation, it told me that PI squared was equal to 9.8696 !! This is the EXACT value my "Summation of Cylinder Walls" program gave me. So that's my mystery. The odds against two *completely* different algorithms yielding identical (but wrong) values is astronomical. I'm forced to conclude, therefore, that: EITHER 1 - I made some *profoundly* fundamental error (like forgetting how to multiply) OR 2 - this very old, very respected formula for determining the surface area of a sphere is very wrong! NB In order to verify the accuracy of my programs, I specifically went out and purchased a small book called _The_Essentials_of_Geometry_II_. It was here that I found the Volume and Surface Area formulas. The book cost me only $7 CDN, so suppose I won't mind too much if this mystery proves that Pythagoras was wrong and I win a Nobel Prize in Mathematics. ;) Can anyone solve this mystery? =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Wed, 24 Aug 1994 23:06:59 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Subject: Re: Alpha, Beta, Hexagonal closest packing In-Reply-To: Message of Wed, 24 Aug 1994 09:15:45 GMT from On Wed, 24 Aug 1994 09:15:45 GMT said: >I have a question that no one I know can answer. > >Why is the alpha state of crystals usually weaker >than the beta state? Hmm, Linus Pauling's 1955 College Chemisty doesn't mention what alpha and beta states of crystals are - maybe you could expalin what these are? Of course, I could get an up-to-date chemistry reference :) > >Is it because the beta state makes a more efficient >use of materials (i.e. is less dense,) or what? > Steve Mather -- Christopher J. Fearnley | UNIX SIG Leader at PACS cfearnl@pacs.pha.pa.us | (Philadelphia Area Computer Society) fearnlcj@duvm.bitnet | Design Science Revolutionary fearnlcj@duvm.ocs.drexel.edu | Explorer in Universe 503 S 44th ST | Linux Advocate Philadelphia PA 1914-3907 | (215)349-9681 ========================================================================= Date: Wed, 24 Aug 1994 20:43:38 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Subject: Re: Dome for Taj Mahal? Having visited to Taj, I'd say awfully big. A couple miles in diameter maybe. The Taj design is about points of view some distance away, with that reflecting pool ala the Lincoln monument in DC being part of the picture. I think putting domes over those factories and piping their pollution somewhere else would be more practical. Or just treating the pollution -- I bet scrubbers and such would do the trick. >Hello, > >I caught an article in our local paper over the weekend saying that a >lawyer in the Agra region of India has begun circulating a petition to get >over 8000 factories moved out of the area as a means of preserving the Taj >Mahal from the effects of pollution. This made me wonder if building a >large geodesic structure to enclose the building and then controlling the >atmosphere within the dome wouldn't be easier (and cheaper and a more >permanent solution) than trying to displace that much industry. Think a >structure that large could handle the stress of its own weight? Anyone on >this forum who lives in that area that would be interested in forwarding this >idea to local planners? To me a dome of this size would be almost as >incredible a feat as the construction of the Taj Mahal itself. Anyone know >about how large such a dome would have to be? > >Enough questions...thanx for your time. > >Bill Long >-- StarGazer > > ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) ========================================================================= Date: Wed, 24 Aug 1994 00:59:07 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Postmaster Subject: Re: Greywater Recycling (was: Fog Gun anyone?) On Tue, 23 Aug 1994 08:17:40 -0700, "Lee Wood" wrote: > As for the power consumption question, I will plead ignorance. > It was not one of my design concerns. In fact, I had only two > main design considerations: My high power consumption considerations about the Pelletier devices were based on practical usage. I do not have any real data. > 1 - I started from the premise that water was an extremely > valuable resource and that water recycling would be mandatory, > irrespective of cost. If you could see they amout of drinkable water that we throw, or waste, here in Quebec, you'll be traumatised. With your statement, I agree. > 2 - I was designing for a solar power source, so once > installed, the electricity would "free". Yah! Free? > On the plus side, when based on a photo-voltaic/battery > electrical supply, the entire system would have NO > moving parts. So, compared to a refrigerator, which has > MANY moving parts, AS WELL AS high power consumption, > this system *should* have a longer life and be > essentially maintenance free. No moving parts: It's lovely. I must admit it: I do also like those Pelletier effect devices. I was told that if electricity polarity is reversed, they produces heat. Is it right? Is it Pelletier or Peletier? PV panels are usually warranted for 10 years but they can last for 20-25 years easily. Would it be possible to build the system in the center of a stellite dish so that the dish, at night when unused, would be put in horizontal position, by a timer or a computer, in order to collect dew? To improve your system's performance? Marc Tremblay, marc@harfang.login.qc.ca "Hier, j'ai appris que les services secrets Russe avaient capture des contrebandiers et trois kilos d'Uranium 238. Aujourd'hui, j'ai appris que les services secrets Russe avaient capture des contrebandiers et dix kilos d'Uranium 238. Bonne journee and have a nice day." ========================================================================= Date: Thu, 25 Aug 1994 11:40:37 EST/EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: DAMICO@GELMAN.CIRC.GWU.EDU Subject: Recent Fuller articles Anyone seen these? TITLE: Hey Bucky, can you spare a paradigm? Reflections on synergetic education. AUTHOR(S): Wagschal, Peter H. SOURCE: NASSP Bulletin Mar 1994, v78n560, p. 51-61 SUBJECT DESCRIPTORS: Education Future SPECIAL FEATURES: p. 51-61 Includes references. ABSTRACT: Unlike any other sector of US life, education has managed to hold on to its basic structure throughout the 20th century. The paradigms through which public schools currently approach their role(s) in US society and how the structure of schools will inevitably change are examined. NOTE: Availability: UMIACHo2206.00 . Article Length: Long (31+ col inches). Article Type: Feature. TITLE: Books : Cosmography by R. Buckminster Fuller. AUTHOR(S): Wavering, Michael J. SOURCE: Science Teacher Feb 1993, v60n2, p. 68 (1 pages) SUBJECT DESCRIPTORS: Nonfiction Geometry Technology Universe SPECIAL FEATURES: p. 68W NOTE: Availability: UMIACHo1597.00 . Article Length: Medium (10-30 col inches). Article Type: Book Review-Favorable. "Just call me Trimtab" {~~~| R. Buckminster Fuller ~~~| _______ | / \ | BDAMICO@GWUVM.GWU.EDU ___________/__________\______|_____ \ Trimtab: A tiny gear / Blaine A. D'Amico |\ which moves the rudder / Systems Specialist ~~~~~~ |__\ that turns great ships / ~~~~~ Design Science Revolutionary --------------------- Comprehensive Generalist ========================================================================= Date: Thu, 25 Aug 1994 19:03:35 -0500 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Robert L. Read" Organization: CS Dept, University of Texas at Austin Subject: Re: stress analysis The following recently-posted statement about the analyzability of domes is totally wrong, IMHO. >From kiyoshi@CPP.PHA.PA.US Thu Aug 25 18:33:58 CDT 1994: >Chris: > Even if only axial stresses are measured in a pure tensegrity structure >rather than a dome with multiple nodes, the stress analysis of dome >members is still far too complex to compute with any existing software. >Fuller compared such an analysis with analyzing the movement of all the >air molecules in a basketball simultaneously. For that reason, structurl >engineers were forced by this to test geodesic structures by loading them to >failure. Because of this fact, geodesic domes were often greatly overbuilt >because of the need for conservativism, since the mathematical tools were >not and still are not available. > >--Kiyoshi The fact that Fuller compared X to Y is regrettably not a proof that X and Y have anything to do with each other. I submit as proof that domes are analyzable the following arguments: 0) I've used FElt to analyze a design for a geodesic gazebo that is not quite a "dome" in the sense that it only has 6 roof triangles and 12 wall triangles, but is nevertheless a rigid omnitriangulated tensegrity. It works. Adding more trinagles would slow it down, but I don't think it would make it choke. 1) I heard about FElt from its authors, who responded to my request for software that can analyze a dome by saying --- "Geodesic domes? All the force axial in the members, right? Piece of cake! FElt can do it!" 2) Part of the confusion arises in that geodesic domes are not highly, highly, "statically indeterminate". This means that the the forces in the various members under a given load cannot be determined without appealing to the geometry of the structure and the material properties of its members. This is characteristic of structures that have redundant members. Geodesic domes in general are maximally redundant, and that redundancy means that forces are quickly spread throughout the structure, which is the key to its excellent structural properties. However, introductory statics textbooks often only explain how to analyze "statically determinate" structures. However, methods exist for analyzing statically indeterminate structures, and the most popular is the "stiffness method" which involves the construction of a very large matrix which is relatively easy for a computer to do and very hard to do as a MechE 101 homework assignment. However, the stiffness method can handle ANY skeletal structure built out of rods and cables. The book "Computer Analysis of Structural Frameworks", by James A.D. Balfour, Second Edition, 1992, Oxford University Press, 200 Madison Avenue, NY, NY 10016 is a good book on structural analysis and the stiffness method and computer implementation thereof, although, unfortunately, it does not include a geodesic dome as an example. * * * This does however, raise an interesting issue: Had the stiffness method not been invented when Bucky made the statement alledged above? Was he simply wrong, or ignorant? Or was he somehow referring to something else, such as analyzing a dome at a greater level of detail that, for instance, engineers find practical for the steel grillages in skyscrapers? I would like to know. Quoted message begins here: > > On Mon, 22 Aug 1994 10:41:35 GMT Tom Buckley said: > >Can anyone tell me how to get some software or otherwise so that I can > >determine the stresses on dome members? > > FElt 2.0 is availble for Unix systems (binary releases for Linux, HP-UX and > SUNOS). It is a finite element analysis package. Although I haven't > played with it yet, it seems like a good package for simple structural > analysis. Robert L. Read reported on this group awhile ago > > "It allows you to input a geometry of a structure, assign material types > to various components, add on continuous forces like roof loads and > specific forces like a 10,000 pound weight at a certain point, and then > compute the forces in each member. It can handle lots of stuff (hence > "Finite Element") but of particular interest to fans of Fuller is that > it easily handles "tensegrity" structures, where a tensegrity structures > is one in which force is transmitted only axially along members (cables > or struts)." > > It is copylefted software (GPL) and available for anonymous ftp at > cs.ucsd.edu:/pub/felt. For Linux users a binary is on sunsite.unc.edu > in /pub/Linux/apps/math. > > >********************************************************* > >* TERRAIN ENGINEERING LTD. * > >*P.O. Box 2703,Saint John, New Brunswick,CANADA E2L 4Z1* > >* * > >* voice, fax: 506-832-4929 * > >* 14400 or less, N81, Zmodem (ask) * > >********************************************************* > -- > Christopher J. Fearnley | UNIX SIG Leader at PACS > cfearnl@pacs.pha.pa.us | (Philadelphia Area Computer Society) > fearnlcj@duvm.bitnet | Design Science Revolutionary > fearnlcj@duvm.ocs.drexel.edu | Explorer in Universe > 503 S 44th ST | Linux Advocate > Philadelphia PA 1914-3907 | (215)349-9681 > -- Robert L. Read, Member of the League for Programming Freedom ========================================================================= Date: Thu, 25 Aug 1994 15:57:39 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: Alpha, Beta, Hexagonal closest packing In Article Chris Fearnley writes: >On Wed, 24 Aug 1994 09:15:45 GMT said: >>I have a question that no one I know can answer. >> >>Why is the alpha state of crystals usually weaker >>than the beta state? > >Hmm, Linus Pauling's 1955 College Chemisty doesn't mention what alpha and >beta states of crystals are - maybe you could expalin what these are? >Of course, I could get an up-to-date chemistry reference :) I think it's been known about for some time. I ran across it a while back while reading about high strength and temperature materials. Titanium, in particular, is much stronger in its beta state. >>Is it because the beta state makes a more efficient >>use of materials (i.e. is less dense,) or what? >> Alpha (generally speaking about metal crystal structures, but also substances like silicon carbide, etc.) structure is hexagonal closest packing. Beta is center-bodied cubic (I think. =) Alpha (this is based on memory, i.e. not absolute) is usually found at higher temperatures. Steve Mather ========================================================================= Date: Thu, 25 Aug 1994 16:24:01 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: The "PI without Trig" Monster Returns In Article <199408250051.RAA18337@whistler.sfu.ca> Lee Wood writes: >SUBJECT: The "PI without Trig" Monster Returns Damn, I'd just stopped checking under my bed too. >I've stumbled upon a mystery which I hope someone here can >help me solve. Unfortunately, it's quite detailed, so I hope >you'll bear with me during the explanation. Here goes: I hope so too, 'cause now you've got me wondering. >Well, I decided to apply a similar technique to finding the >volume of a sphere (also of radius 1). My algorithm filled >the sphere with flat cylinders (more like hockey pucks, actually), >all contained (mostly) within the surface of the sphere. I then >calculated the volume for each cylinder, and obtained their sum. > >NB actually, I filled only the upper hemisphere with cylinders, >and multiplied their volumes by 2. The following diagram is >supposed to resemble the hemisphere filled with 4 cylinders. >My program used 10,000 cylinders, but drawing that was out >of the question. As someone said earlier, "I hate ascii". > > __.__.__ > _.|________|._ > .|______________|. > .|__________________|. >..|____________________|. Could you explain better how these "pucks" are packed? I don't know if it'll help any, but it couldn't hurt since I can't offer any explaination yet. For example, could you show it from the side (the end of the cylinders?) >The resulting sum of volumes agreed quite closely to the valued >given if you use the standard formula for the volume of a sphere: > >Vsphere = (4/3)PI*Radius*Radius*Radius = 4.1887902 > (my program said Vsphere was equal to 4.1887901) > > > >Encouraged by this result, I modified the program to calculate >the area of each cylinder wall. The sum of all 10,000, I thought, >should be a good approximation of the surface area of the sphere. >(of some interest to people who design domes, yes?) > >The equation for the Surface Area of a Sphere said: >SAsphere = 4*PI*Radius*Radius = 12.5664 > >But my program said SAsphere = 9.8696 > >The error is more than 20% !! This was surprising, considering >the accuracy I had achieved with my sphere-volume program. >So I thought up a *new* algorithm. > >ALGORITHM #2 >- Start with a hemisphere (radius=1). >- Draw a line from the north pole to the equator. >- Move a very small distance along the equator, then draw a second > line from the north pole to this new point on the equator. > >We have just drawn a long, skinny isosceles triangle. It's >base is on the equator, and it's apex is at the north pole. > >Now we cover the entire surface of the hemisphere with triangles >identical to the one just described. (Before beginning, we >cleverly ensured that the base of the triangle would be some >integer fraction of the distance around the equator, say 1/1000th.) >Each apex lies at the north pole, and each base lies along the >equator. The triangles do not overlap, and they cover the >hemisphere completely. > >We can easily determine the area of a triangle: > Area = Base*Height/2 Does this work for spherical triangles as well? If not, what you're describing is a cone. I don't know if this corresponds to the equation you reach though. >So we multiply that area by the number of triangles covering >the hemisphere, then double that quantity to obtain the surface >area of the entire sphere. > >DONE! Or so we hope =) >But before programming this new algorithm, I first worked it >out on paper. I allowed the number of triangles to grow very >large and discovered that the Height approached a value equal >to the circumference around the sphere (a great circle) divided >by four. This simplified the algebra *significantly*. >Almost everything canceled out, and I was left with: > >SAsphere = PI*PI*Radius*Radius > (i.e. PI squared times the Radius squared) > >But since the Radius=1, it reduces to: > >SAsphere = PI*PI > >Being of an inquisitive nature, I took out my calculator and punched >in PI*PI. To my surprise and consternation, it told me that >PI squared was equal to 9.8696 !! This is the EXACT value my >"Summation of Cylinder Walls" program gave me. > > >So that's my mystery. The odds against two *completely* >different algorithms yielding identical (but wrong) values >is astronomical. I'm forced to conclude, therefore, that: > >EITHER > >1 - I made some *profoundly* fundamental error > (like forgetting how to multiply) > >OR > >2 - this very old, very respected formula for > determining the surface area of a sphere is > very wrong! OR you set up the same problem twice in entirely different ways. I'm not sure though. >NB In order to verify the accuracy of my programs, I specifically >went out and purchased a small book called >_The_Essentials_of_Geometry_II_. It was here that I found the >Volume and Surface Area formulas. > >The book cost me only $7 CDN, so suppose I won't mind too much >if this mystery proves that Pythagoras was wrong and I win a >Nobel Prize in Mathematics. ;) > >Can anyone solve this mystery? Elementary my dear Wa-- uh, no luck so far. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | >Lee_Wood@sfu.ca | INTJ spoken here. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Thu, 25 Aug 1994 16:39:27 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: Greywater Recycling (was: Fog Gun anyone?) In Article <199408231517.IAA10811@whistler.sfu.ca> Lee Wood writes: >Marc Tremblay wrote: >> I think those solid sate cooling devices >>consumes a lot of energy.[snip] Also, they >>are based on a principle named the "Pelletier effect". > > >Right, "Pelletier Effect Devices", thank you. > > > >As for the power consumption question, I will plead ignorance. >It was not one of my design concerns. In fact, I had only two >main design considerations: > >1 - I started from the premise that water was an extremely >valuable resource and that water recycling would be mandatory, >irrespective of cost. ^^^^^^^^^^^^ There are cheap ways to do it. >1 - Evaporative cooling was not an option, since >it wastes water. Speaking of evaporation, perhaps a clear dome or clear parabolic structure would work (the second's probably the better.) The waste water could flow into a dish in the center and evaporate from the warmth. The clear ceiling would collect the water and drive it to the edges where it could be collected and pumped back into the house. The limitation is size, however. If a great deal of waste water were to run through it, it couldn't process all of it. I s'pose the use of Bucky's inventions could help here. Another possiblity is a small enclosed gravel/cattail (wetlands) treatment area. The problem with this is that the water wouldn't be fit to be used in the house afterwards, though it could be discharged, or used in watering the lawn or garden. Also, zoning regulations usually prevent the implementation of these. Steve Mather >2 - The CAPS are not intended to indicate "shouting" >i.e. a FLAME response. They are merely for emphasis. ALRIGHT! =) >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | >Lee_Wood@sfu.ca | INTJ spoken here. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Sat, 9 Feb 1980 00:35:13 MST Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Neil MacMullen Subject: Re: The "PI without Trig" Monster Returns On Wed, 24 Aug 1994 17:51:45 -0700, "Lee Wood" wrote: > SUBJECT: The "PI without Trig" Monster Returns > > I've stumbled upon a mystery which I hope someone here can > help me solve. Unfortunately, it's quite detailed, so I hope > you'll bear with me during the explanation. Here goes: > > NB actually, I filled only the upper hemisphere with cylinders, > and multiplied their volumes by 2. The following diagram is > supposed to resemble the hemisphere filled with 4 cylinders. > My program used 10,000 cylinders, but drawing that was out > of the question. As someone said earlier, "I hate ascii". > > __.__.__ > _.|________|._ > .|______________|. > .|__________________|. > ..|____________________|. > > Encouraged by this result, I modified the program to calculate > the area of each cylinder wall. The sum of all 10,000, I thought, > should be a good approximation of the surface area of the sphere. > (of some interest to people who design domes, yes?) At a cursory glance, I think the mistake you have made is to assume that the limit of the sum of the cylinder surface areas is equal to the surface area of the sphere. Really, you should be looking at the slanting 'rind' which connects the corners of each cylinder. As an analogy, imagine trying to find the length of a diagonal line by dividing it into a series of smaller and smaller horizontal and vertical segments ... no matter how small you make the segments, you will come out with the same answer - root 2 times the true length of the line. Approximation Diagonal Line _| / _| / _| / > ALGORITHM #2 > - Start with a hemisphere (radius=1). > - Draw a line from the north pole to the equator. > - Move a very small distance along the equator, then draw a second > line from the north pole to this new point on the equator. > > We have just drawn a long, skinny isosceles triangle. It's > base is on the equator, and it's apex is at the north pole. > > Now we cover the entire surface of the hemisphere with triangles > identical to the one just described. (Before beginning, we > cleverly ensured that the base of the triangle would be some > integer fraction of the distance around the equator, say 1/1000th.) > Each apex lies at the north pole, and each base lies along the > equator. The triangles do not overlap, and they cover the > hemisphere completely. Perhaps I have misunderstood you here but it sounds like you are calculating the surface area of a cone ! :-) Neil -- Neil MacMullen Vc 1-403-264-2338 US Cybernetics Vc 289-0252 neil@powerstor.cuc.ab.ca ========================================================================= Date: Fri, 26 Aug 1994 06:57:01 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Grego10067 Organization: America Online, Inc. (1-800-827-6364) Subject: Computer Software I am still looking for any software out there somewhere that will help one to design geodesic domes, if anyone knows of anything along these lines, or can help steer me in that direction, please post me. Grego10067 America On Line, or on the Compuserve network Greg Matherly 75024,1260 or post your message here..... Thank You ========================================================================= Date: Fri, 26 Aug 1994 14:01:33 +0100 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Andy Wardley Subject: Re: Computer Software In-Reply-To: <9408261252.AA14870@Q.icl.co.uk> from "Grego10067" at Aug 26, 94 06:57:01 am Grego10067 writes: > >I am still looking for any software out there somewhere that will help one >to design geodesic domes, if anyone knows of anything along these lines, >or can help steer me in that direction, please post me. Grego10067 America >On Line, or on the Compuserve network Greg Matherly 75024,1260 or post >your message here..... > Thank You > I don't think it's quite what you want, but I wrote a utility to generate dome data for ray tracing systems. It's primitive as it currently only tessellates octahedrons, but the new version (due out RSN) should do much more. However, all it will do it give you a list of co-ordinates and stuff. If this is any use to you, you can find it at ftp.uwa.edu.au in /pub/povray/utilities. It's called "geodome.zip" and includes source code which may (or may not) help you. I know I've mentioned this before, but what the heck - a picture of mine which I created using the utility is available there in /pub/povray/HALL_OF_FAME. If you like geodomes, check it out. Cheers Andy This Spot Is Allowed whatever K says. Here, have some Andy Wardley chocolate, it's Terry's. No smug bait for Derek. M#0 abw@oasis.icl.co.uk Badgers are your friends. OK, so they don't frink but have you seen them forage? DAMN! I've run out of sp ========================================================================= Date: Fri, 26 Aug 1994 13:24:13 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Rywalt Organization: Prodigy Services Company, White Plains, New York Subject: Re: stress analysis In article <33jbgn$b6q@peaches.cs.utexas.edu>, read@cs.utexas.edu (Robert L. Read) writes: |> However, introductory statics textbooks often only explain how |> to analyze "statically determinate" structures. However, methods |> exist for analyzing statically indeterminate structures, and the |> most popular is the "stiffness method" which involves the construction |> of a very large matrix which is relatively easy for a computer to |> do and very hard to do as a MechE 101 homework assignment. |> However, the stiffness method can handle ANY skeletal structure built out |> of rods and cables. |> |> The book "Computer Analysis of Structural Frameworks", by James A.D. |> Balfour, Second Edition, 1992, Oxford University Press, 200 Madison |> Avenue, NY, NY 10016 is a good book on structural analysis and |> the stiffness method and computer implementation thereof, although, |> unfortunately, it does not include a geodesic dome as an example. I would be very interested in seeing whether or not this yields a proper analysis of the structure. I would tend to doubt it. Why? Because the analysis of any skeletal structure is not the same as the analysis of a geodesic structure. Don't take my word for it: here's a piece from _Geodesic Math_ by Hugh Kenner (written in 1976). Kenner has just finished 40 or so pages of exposition on tensegrity structures and has just gotten to discussing "rigid tensegrities" -- geodesic domes being a special case of rigid tensegrity. He writes about increasing the frequency of the tensegrity, which means correspondingly shorter tendons; if we increase the frequency sufficiently, the tendons will only be as long as the thickness of the struts of the tensegrity, and the struts will touch. He goes on: "Let us suppose that the system is made of wood. We can cut the ends of the upper struts so that they meet squarely, and we can bevel the upper edge of the lower strut so it fits snugly under them....[The tendons] may easily be replaced by a couple of bolts. Tightening the nuts will draw everything together. "Now, note carefully: When we commenced this process, the rest of the system was tending to pull all three components apart. Now that they are touching each other, this is still true. The two upper struts are not in compressive contact: if released, they would pull *apart*. And they are not resting on the lower strut: if released, they would move outward and away from it. "The bolts and nuts are quite literally serving to *draw the elements together.* They are performing the function of the tension members in the original tensegrity.... "Having developed this structure [a dome] from Tensegrity principles, we know about the tensile forces that are holding it together. Traceable throught the bolts and the wooden beams, they form a tensional continuity, resembling the surface tension of a liquid sphere in being concentrated on the outer surface of the polyhedron. "But a casual observer of the structure would soon commence studying the joints, and seeing what is diagrammed in Diagram 6.3 [unfortunately not ASCII-izable] he would suppose that the bottom member was bearing the weight of the upper ones. He would suppose that the structure was in *compression,* like a frame house, and that the bolts functioned merely as do carpenter's nails, to prevent lateral slippage. He would therefore form a wholly erroneous idea of the system's dynamics.... "However variant geometries and connecting systems may modify this account, it remains the first principle of geodesics." |> This does however, raise an interesting issue: Had the stiffness method |> not been invented when Bucky made the statement alledged above? |> Was he simply wrong, or ignorant? Or was he somehow referring to |> something else, such as analyzing a dome at a greater level |> of detail that, for instance, engineers find practical for the |> steel grillages in skyscrapers? I would like to know. I imagine that, using this "stiffness method" -- which I know nothing about -- you would be analyzing the structure in much the same was as Kenner describes, and thus get erroneous results. If you don't start with tensegrity, you can't end with proper analysis. From what I know of engineering -- a passing knowledge from Stevens Tech, where I was studying Computer Science -- I doubt very much that they'd be taking tension into account on such a scale. Chris. crywalt@tinman.dev.prodigy.com ========================================================================= Date: Fri, 26 Aug 1994 15:10:37 -0500 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Robert L. Read" Organization: CS Dept, University of Texas at Austin Subject: Re: stress analysis >From crywalt@tinman.dev.prodigy.com Fri Aug 26 14:16:49 CDT 1994 Rob wrote: |> However, introductory statics textbooks often only explain how |> to analyze "statically determinate" structures. However, methods |> exist for analyzing statically indeterminate structures, and the |> most popular is the "stiffness method" which involves the construction |> of a very large matrix which is relatively easy for a computer to |> do and very hard to do as a MechE 101 homework assignment. |> However, the stiffness method can handle ANY skeletal structure built out |> of rods and cables. |> |> The book "Computer Analysis of Structural Frameworks", by James A.D. |> Balfour, Second Edition, 1992, Oxford University Press, 200 Madison |> Avenue, NY, NY 10016 is a good book on structural analysis and |> the stiffness method and computer implementation thereof, although, |> unfortunately, it does not include a geodesic dome as an example. crywalt@tinman.dev.prodigy.com writes: >I would be very interested in seeing whether or not this yields a proper > analysis of the structure. I would tend to doubt it. Why? > Because the analysis of any skeletal structure is not the same as the > analysis of a geodesic structure. Rob writes: Ah, but I believe that the analysis of a geodesic structure is exactly the same as the analysis of a skeletal structure. [.. A long excerpt from "Geodesic Math" is deleted here] >I imagine that, using this "stiffness method" -- which I know nothing about -- > you would be analyzing the structure in much the same was as Kenner > describes, and thus get erroneous results. If you don't start with > tensegrity, you can't end with proper analysis. From what I know > of engineering -- a passing knowledge from Stevens Tech, where I > was studying Computer Science -- I doubt very much that they'd be > taking tension into account on such a scale. Why should you have to start with tensegrity? An artifact is an artifact, no matter what mental processes are used in its conception and realization. Unfortunately, I have zero idea why your excerpt from "Geodesic Math", which I have read, suggests that a structural analysis would lead to an erroneous result. In addition to arguing about this issue, I'll try to put my money where my mouth is. I will obtain the coordinate data for a fairly large dome, analyze it with FElt, post the input and output files here. Since FElt is freely available, anyone will be able to duplicate this experiment if they so desire. The output of "FElt" will be the force and deformation in every strut in the dome. This is the critical data, because if we manufacture the connectors so that they are stronger than the struts, the force required to break a strut is the failure point of the dome. I claim that this will be a proof that geodesic domes are analyzable, and analyzable with existing, freely available software. I furthermore promise to do this by October 1, 1994. Unfortunately, your statement "I doubt very much that they'd be taking tension into account on such a scale" suggests that you might believe that the results would not truly represent a proper analysis of a dome. To counter this point, I ask you to consider the following two arguments. A geodesic dome constructed out of rods with connectors that do not constrain the angles of the rods (corresponding to a "pin" joint in structural engineering) is simply a structure like any other. The effect of its design is "magical" in the sense that it has some excellent properties relative to rectilinear structures. It has nothing that prevents normal structural analysis from providing correct answers. The fact that it is similar to a degenerate tensegrity in which the the cables are very short is irrelevant to the fact that it is a structure composed of rods, and hence analyzable. Furthermore, the idea that domes are unanalyzable leads to a contradiction. Surely, we can agree that a tetrahedron constructed out of steel rods is "analyzable". As per "Geodesic Math", one method of constructing a dome is to tesselate the faces of one of the regular polyhedra (like the tetrahedron) with symmetric triangles and the "bulge them out" until the intersection points of all triangles lie on the surface of the circumscribing sphere by pushing them outward along the line from the center of the tetrahedron through the intersection point. Suppose that we do this for just one fact, with just the simplest equilateral tesselation of a triangle : * / \ * - * / \ / \ *---*---* Now, this 12 strut system is surely analyzable, because the tesselated face can be analyzed by itself and then substituted for the untesselated face, since the "stiffness" or "elasticity" of a structure is just the superposition of the stiffnesses and elasticities of its components. Now we can tesselate all faces, and the same argument still holds, and we have a very simple geodesic dome (although as Kenner points out the tetrahedron is so non-spherical that people usually use the octahedron or the isocahadron as a starting point instead.) And of course this can be carried out for higher frequency divisions of a face, and for any start polyhedron. Sorry for such a long post. Perhaps there is a lurker on the group who is a structural engineer who can shed some light on this subject? I will post my analysis by Oct. 1, but I cannot make you believe that it is correct. Please be careful, as uninformed opinion can badly muddle issues as complicated as these. P.S. -- I know a little bit about engineering myself. >Chris. > crywalt@tinman.dev.prodigy.com -- Robert L. Read, Member of the League for Programming Freedom ========================================================================= Date: Fri, 26 Aug 1994 15:51:46 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: The Mystery Egg-On-Face Retraction: After speaking with someone from the Computer Science Dept. at one of the local universities last night, I've abandoned the "divide the surface of a hemisphere into a gazillion little triangles" approach. He convinced me to return to the "divide the interior of a hemisphere into a gazillion little hockey pucks" approach. Except, where before I allowed the hockey pucks to have vertical walls (i.e. cylinders), a more accurate approximation could be achieved by beveling the sides to parallel the slope of the sphere at that location. So instead of stacked cylinders: ---------- | | -------------- | | -------------- we have stacked frustums: ------ / \ ------------ / \ -------------- So I summed the surface areas of the 10,000 frustum walls (actually my program did) and achieved a value for the surface area of the sphere equal to what the equation from the book said it should be: 4*PI*Radius*Radius = 12.566370+ As for the mystery, I still don't see why the tiny-triangle approach should yield PI squared. But I see now that in the hockey puck approach, measuring only the vertical components of the cylinders, will always be equal to the radius. Do they have a Nobel Prize category for "close, but no cigar"? :) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Fri, 26 Aug 1994 15:32:11 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: smtc5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Non-synergetics I have two questions unrelated to synergetics: (At least directly) Why does water drain counter-clockwise? and What does IMHO mean? Steve Mather ========================================================================= Date: Fri, 26 Aug 1994 22:56:02 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Hawku Organization: America Online, Inc. (1-800-827-6364) Subject: Re: Computer Software In article <33khpt$dkn@search01.news.aol.com>, grego10067@aol.com (Grego10067) writes: >I am still looking.... I found the following post in comp.soft-sys.wavefront >From: Francisco X DeJesus , xx xDate: Wed, 29 Jun 1994 16:06:00 -0700 xMessage-ID: x<199406292306.QAA13365@archimedes.chinalake.navy.mil> x> I have found geodesic spheres to be useful in certain modeling x> situations in Wavefront. So, I wrote a program to create them. x> x> If anyone is interested, I can email a uuencoded copy of the x> program compiled under IRIX 5.2. xThe uuencoded binary is now on avalon.chinalake.navy.mil, xfilename: x/pub/utils/misc/geodesic_obj.uue xHaven't had a chance to try it yet... x-- xFrancisco X DeJesus \-------------------\ xdejesus@archimedes.chinalake.navy.mil xScience Applications International Corp. \--------------\ xdejesus@c3ot.saic.com ----------------------- Headers ----------------------- xPath: xsearch01.news.aol.com!newstf01.cr1.aol.com!news.ans.net!meaddatxa!babbage .ece.uc.edu!news.kei.com!MathWorks.Com!zombie.ncsc.mil!pxaladin.american.ed u!auvm!ARCHIMEDES.CHINALAKE.NAVY.MIL!dejesus xComments: Gated by NETNEWS@AUVM.AMERICAN.EDU xNewsgroups: comp.soft-sys.wavefront xX-Mailer: ELM [version 2.4 PL21] xMIME-Version: 1.0 xContent-Type: text/plain; charset=US-ASCII xContent-Transfer-Encoding: 7bit xContent-Length: 538 xMessage-ID: x<199406292306.QAA13365@archimedes.chinalake.navy.mil> xDate: Wed, 29 Jun 1994 16:06:00 -0700 xSender: WaveFront Software xFrom: Francisco X DeJesus x xSubject: Re: Program to create geodesic spheres xIn-Reply-To: x<199406281554.IAA03792@archimedes.chinalake.navy.mil> from "Terrence Pong" at Jun 28, 94 10:54:13 am xLines: 13 I hope this helps. You can Email my "reword". ========================================================================= Date: Fri, 26 Aug 1994 23:01:04 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Hawku Organization: America Online, Inc. (1-800-827-6364) Subject: Re: Non-synergetics In article , smtc5@uoft02.utoledo.edu writes: >Why does water drain counter-clockwise? and What does IMHO mean? 1) IMHO, water drains counterclockwise because it moves from the future to the past. 2) IMHO, IMHO means "in my humble opinion" ========================================================================= Date: Sat, 27 Aug 1994 11:03:21 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: The "PI without Trig" Monster Returns smtc5@UOFT02.UTOLEDO.EDU writes: >Could you explain better how these "pucks" are packed? >I don't know if it'll help any, but it couldn't hurt >since I can't offer any explaination yet. For example, >could you show it from the side (the end of the cylinders?) They weren't "packed", so much as "stacked". They were of varying diameters, stacked one on top of another, largest on the bottom. i.e. __.__.__ _.|________|._ .|______________|. .|__________________|. .|____________________|. This IS a side view. Think of them as pancakes heaped under a glass dome. --==o==-- And Neil MacMullen wrote: > I think the mistake you have made is to assume > that the limit of the sum of the cylinder surface areas is equal to > the urface area of the sphere. Really, you should be looking at the > slanting 'rind' which connects the corners of each cylinder [snip] > Approximation Diagonal Line > _| / > _| / > _| / Yep, you got it. When I changed the program to produce a slop-sided cylinder (i.e. frustum -- I had to look up that word in my geometry book.) instead of a vertical sided cylinder, it worked properly. And, responding to ALGORITHM #2, Neil wrote: > Perhaps I have misunderstood you here but it sounds like you are > calculating the surface area of a cone ! :-) Right again. I assumed (incorrectly) that, as the number of little triangles approached infinity, the shape would approximate a sphere. But you're right. It's just a cone. Unfortunately, my calculus and solid geometry skill are not what they once were, so I cannot see how to go about correcting the algorithm. If anyone has a suggestion, I'd be glad to hear about it. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Sun, 28 Aug 1994 00:54:18 MDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Ken G. Brown" Organization: BEST Online Subject: Re: Fog Gun anyone? >Forwarded message: >From jlmarang Fri Aug 19 14:50:06 1994 >I just popped onto this list in the hope that someone here could give me some > information on an invention of Buckminster's known as a 'Fog Gun' >Can anyone supply me with more information? >Thanks, >Jennifer >jlmarang@acacia.itd.uts.edu.au I too have been interested in the fog gun shower but haven't yet followed up. I did however note the following info in the Jan '89 Trimtab I believe. A book ($4) and a thesis were referred to, available from: Minimum Cost Housing Group School of Architecture McGill University m3480 University Street Montreal, Quebec, Canada H3A 2A7 Book: Water Conservation and the Mist Experience- Alex Morse, Vikram Bhatt, Witold Rybozynski (or maybe Rybczynski) Master's Thesis in Architecture Thesis: The use of Atomization for Washing & Showering to Conserve Water- Alex Moss Another book: Stop the 5 gallon Flush Please let me know if you manage to obtain these as I would like to know if they are still available. -Ken- (kbrown@atc.edmonton.ab.ca) ========================================================================= Date: Sat, 27 Aug 1994 21:21:13 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Subject: Geodesic sphere algorithm The way I come at geodesic spheres is through the Descarte's discovery, elaborated upon in Synergetics, that a simple concave/convex polyhedron of n vertices has 720 degrees all told, summing the angles converging at those vertices, than 360*n. In other words, the difference between a network with no curvature, and one which connects around to itself to provide a concave inside and concave outside, is 720. Bucky was pleased to identify the tetrahedron, a 720 degree construct, with this difference. In other words, subtract a tetrahedron's worth of degrees from a flat surface, and get... a tetrahedron. Or a cube. Or an octahedron. Shape in general. Somethingness is begotten by zero spitting out a negative and positive tetrahedron. In pure principle, they intercancel, but in time-worn Universe, we have giraffes. OK, now moving to geodesic spheres, we think of a super high frequency icosasphere. Like we're standing on one the size of the earth. What we see are triangles arranged in hexagons, stretching to the horizon. We see a mosaic of triangular tiles. We're told that in twelve and only twelve places on the surface of our earth, 5 triangular tiles surround a vertex, instead of 6. We might spend our whole lives searching for a such a pentagonal convergence. The other 11 will be equally spaced, around the surface of the earth, from the first we find, on the plan of a spherical icosahedron. The important thing to remember is that each triangle is locally flat. At each vertex of the gazillion vertices on this earth-sized mosaic of 1 inch triangles, 6 triangles meet (except in 12 sacred spots, as mentioned). To you and I, the tiling seems flat. Detecting curvature is difficult, given our scale relative to the earth. But the way this geodesic sphere was constructed was by subtracting exactly the same fraction of a degree from each vertex. So whereas it *appears* that each vertex is surrounded by 6 equilateral triangles, giving 360 degrees, in actual fact the triangles are ever so slightly non-equilateral, and each vertex is surrounded by more like 359.9999999 degrees. The sacred 12 are likewise surrounded by 359.9999999 degrees. The missing .0000001 degree from each vertex contributes to the making of a tetrahedron of 720 degrees, erected somewhere on our earth to symbolize its inverse. So this brings me to the algorithm (conceptual only): I give you a very large number. Not just any number. It has to be the number of vertices in a super high frequency icosanet. Actually, Fuller gives us a formula: 10F*F+2. So I give you F, lets say. Your job (you being the algorithm) is to create a geodesic sphere with the goal of distributing the "missing" 720 degees among all vertices equally. In other words, every vertex should be surrounded by 360-k degrees where n*k=720 and n is the number of vertices given by 10*F*F+2. What you have to play with is the lengths of the spokes coming out from each vertex. Defining the 5 spokes from a sacred spot (one of 12) as unity (arbitrary), every other spoke in the scheme will be expressed relative to 1 (e.g. 1.00003234). Does this algorithm give us the closest thing to the ideal geodesic sphere we can imagine? Are the stipulations constraining enough to provide a true numerical recipe. How many different, unique edge lengths will such an algorithm generate? A much lower frequency earth-net (not inch triangles, but kilometer ones, say), begins to show how we can transfer the real earth's bumpiness to scaled triangles on a corresponding tiny sphere (in the palm of my hand). If the edges between all the triangles in our super high frequency geosphere are hinges, and if we cut along certain edges to create the opening sinuses, will the flat triangles open up to lie flat, giving the Dymaxion Map? They ought to. ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) ========================================================================= Date: Sun, 28 Aug 1994 15:01:00 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: William Cringan Organization: CRS Online (Toronto, Ontario) Subject: The "PI without Trig" Mon WO>I've stumbled upon a mystery which I hope someone here can WO>help me solve. Unfortunately, it's quite detailed, so I hope WO>you'll bear with me during the explanation. Here goes: [snip] WO> __.__.__ <--- the top edge of one disc, WO> _.|________|._ <-- to the top edge of the other disc WO> .|______________|. is the hypotenuse of the right triangle WO> .|__________________|. formed by the edge of the first disc and WO>.|____________________|. the increment in size of the second over the first disc. This measurement is overlooked in your figures. The sum of the surfaces of the edges of the discs will come up short because they do not account for the fact that surface of the sphere is not flush with the edges of the discs. A corrigated surface is much more in area than a flat or curved surface covering the same expanse. You have only included one edge of the corrigation and not the other. what you need to do is sum the surface area of hypotenuse (spelling?) Take a while and let it sink in and you'll see what I mean. regards, Bill. --- * OLX 2.1 TD * hAS ANYONE SEEN MY cAPSLOCK KEY? ========================================================================= Date: Mon, 29 Aug 1994 08:00:47 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Organization: 4D Solutions Subject: Geodesic Sphere (continued) Actually, the algorithm for going from a high frequency, triangle-mosaiced earth model to a flat dymaxion map is not quite so simple as imagining hinges between all the triangles I don't think. The crux of the Dymaxion Map algorithm is to place a regular, flat-faced icosahedron inside our geosphere. Radii from the spherical center will intersect the surface of the icosa on their way out to the vertices of our earth model. To each triangle on the earth model there will correspond a triangle on the icosahedron. Transferring the data from the earth model to the icosa (i.e. the picture of the earth -- rivers, mountains and what not), happens first. Then we unfold the icosa into a Dymaxion Map. Questions: We can subdivide the equilateral faces of the icosa into F*F subtriangles, also equilateral, using the 3-way crosshatching of lines parallel to the 3 sides. Will radii through the vertices so derived, out to the surface of a sphere, define vertices which, when connected by chords, will each be surrounded by the same number of degrees? Put another way: given my algorithm for subtracting the same fraction of a degree from each vertex to produce smooth curvature (with chords approximating sphericity), are we defining an orthonormal projection of equilateral triangles subdividing an icosahedron's 20 faces onto the concave surface of the sphere in which it is inscribed? If not, then what would be a numerical recipe for extending radii through the vertices of a subdivided icosa of frequency x to the surface of a sphere? The solution need only address any one of the 20 triangular sectors. Fuller worked on a similar problem using great circles of spin defined by rotating an icosa on axes through its faces/edges/corners. ========================================================================= Date: Mon, 29 Aug 1994 17:00:24 GMT Reply-To: wherrett@mech.ubc.ca Sender: List for the discussion of Buckminster Fuller's works From: Geoffrey Wherrett Organization: Department of Mechanical Engineering, UBC Subject: Re: Non-synergetics In article , smtc5@uoft02.utoledo.edu writes: >Why does water drain counter-clockwise? and What does IMHO mean? Water drains counterclockwise due to Coriolis effects. In the Southern Hemisphere, water drains in the opposite direction. ========================================================================= Date: Mon, 29 Aug 1994 10:43:58 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: Geodesic sphere algorithm On August 27 Kirby Urner wrote: >So this brings me to the algorithm (conceptual only): I give you >a very large number. Not just any number. It has to be the number >of vertices in a super high frequency icosanet. Actually, Fuller >gives us a formula: 10F*F+2. So I give you F, lets say. Your job >(you being the algorithm) is to create a geodesic sphere with the >goal of distributing the "missing" 720 degees among all vertices >equally. In other words, every vertex should be surrounded by >360-k degrees where n*k=720 and n is the number of vertices given >by 10*F*F+2. What you have to play with is the lengths of the >spokes coming out from each vertex. Defining the 5 spokes from a >sacred spot (one of 12) as unity (arbitrary), every other spoke >in the scheme will be expressed relative to 1 (e.g. 1.00003234). > >Does this algorithm give us the closest thing to the ideal geodesic >sphere we can imagine? Are the stipulations constraining enough >to provide a true numerical recipe. How many different, unique >edge lengths will such an algorithm generate? The algorithm's overall goal is to describe a geodesic sphere, however, I believe it would run much faster if we could define some sub-goals for it to achieve along the way. e.g. When choosing a spoke length, what would make one length "better" then another? Also, is there a fixed set of spoke lengths from which to choose? =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Mon, 29 Aug 1994 16:35:09 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: mail problem Does anyone know why I get the following error msg, whenever I post to this list? Not-Delivered-To: mhs!dca/G=Darren/S=Cromer/O=CCGATE/OU1=DCAICCMK/RECIPNUM=1/MTA-BASIC@attmail .com due to code 01 "Invalid Addres Specification"; arrived Mon Aug 29 18:11:16 GMT 1994; Or, more to the point; does anyone know how to fix it? =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Mon, 29 Aug 1994 19:58:10 GMT Reply-To: garym@[199.71.70.30] Sender: List for the discussion of Buckminster Fuller's works From: Gary Lawrence Murphy Organization: Ontario Science Centre Research & Exhibit Planning Subject: Re: Geodesic Sphere (continued) In-Reply-To: pdx4d@teleport.com's message of Mon, 29 Aug 1994 02:59:34 yes, Gary's back (almost) although maya.isis.org is long gone ;-) My news access (via Hannover's NNTP service) had unfortunately removed the first message before I was able to read it :-( but this Dymaxion-projection method stuff is _very_ intriguing. I have an application for a world-map which is tailor-made for a Dymaxion projection and I'm hoping someone in net-land might lend some expertise to a good cause ;-) ... I am building a major, permanent exhibit on the "Information Highway" for the Ontario Science Centre. Part of the 2200 sq.ft exhibit area in the Hall of Technology will have public access internet workstations. I would like (via a proxy httpd server) to trap the calls from the exhibit machines, match the IP numbers to a geographic database, and then to project their travels on a large video-projected worldmap display, along with other net-gathered global information such as internet traffic or even global temperatures &c. (ie an electronic geo-scope!). In their innocence ;-) the Science Centre has many displays which depend heavily on Fuller (such as all their exhibit space-frames) yet, prior to my arrival, his name was largely unknown. I hope to change this and since the Dymaxion projection is such an iconic 'trademark' shape (Bucky's coke-bottle?) I'd love to put it on a large, promenent display wall. Unfortunately, I can't spare the manhours to do it right and I may be constrained by budget and schedules to settle for a simple cylindrical projection (yuch!) If possible, I would at least like to get a 'reprint' of that earlier message and any other tips or references anyone may have handy on mapping between Dymaxion and geographic coordinates. I'd also be very interested in hearing from any and everyone interested in this exhibit plan and of course, I'm always most eager to talk to potential volunteers :-) -- Gary Lawrence Murphy --------------------------- garym@[199.71.70.30] Sr.Scientist, Media Technology --------- Research & Exhibits Planning Ontario Science Centre ------------------- voice: (416) 429-4100x2215 770 Don Mills Road, Don Mills, Ontario M3C 1T3 -------- fax: 696-3181 -------------------------------------- nothing surpasses the ordinary ========================================================================= Date: Mon, 29 Aug 1994 21:44:44 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Subject: Re: Geodesic sphere algorithm >>What you have to play with is the lengths of the >>spokes coming out from each vertex. Defining the 5 spokes from a >>sacred spot (one of 12) as unity (arbitrary), every other spoke >>in the scheme will be expressed relative to 1 (e.g. 1.00003234). >> >>Does this algorithm give us the closest thing to the ideal geodesic >>sphere we can imagine? Are the stipulations constraining enough >>to provide a true numerical recipe. How many different, unique >>edge lengths will such an algorithm generate? > > >The algorithm's overall goal is to describe a geodesic sphere, >however, I believe it would run much faster if we could define >some sub-goals for it to achieve along the way. > >e.g. When choosing a spoke length, what would make one length >"better" then another? Also, is there a fixed set of spoke lengths >from which to choose? >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | What's true is that this algorithm will not give a single spoke length for use throughout the sphere. If all the edges were equal, then the 6 triangles around a vertex would be trully equilateral, and therefore perfectly flat -- no curvature. The goal is to 'distort' the edges by a hair, longer or shorter, to cause each vertex to be surrounded by exactly k degrees less than 360 (k is konstant for all vertices). Furthermore, k times the number of total vertices will equal 360. I assume that the 5 spokes of a pentagonal convergence will be the same length, and arbitrarily make this 1. I'm wondering if these contraints will now automatically (once the numerical algorithm is in place) determine the unique chord lengths for the rest of the sphere. Note: computation need only be done for one of the 20 icosa- sphere's triangles. ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) ========================================================================= Date: Tue, 30 Aug 1994 07:34:55 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: John Kirk Organization: Netaxs Philadelphia local provider Subject: question re: felt applicability Apologies in advance for any clumsiness in posting. This is my first. I'm an inveterate lurker. I've done some mathematical derivation and some calculation regarding a variety of tensegrity structures, and due to recent postings, looked over the documentation for the felt program. The snag I ran into, trying to see how to apply felt, can be simplified to the following question. (It seems to me to be a user interface issue rather than one with the underlying calculation theory) If I give locations A and B in cartesian coordinates, how do I specify that I wish to connect them with a cable (made, perhaps, of steel for instance) whose length when not under tension is only ninety-eight or ninety-five percent of the distance between A and B? The idea is that such a cable would be stretched sufficiently to reach to be connected, and the felt program would at least tell me, based on elasticity of materials specified, what tension resulted in the cable. Even more interestingly, it would tell me what the resulting compression would be in the struts. Robert Read, with your familiarity with felt, can you tell be how to specify pre-stressed elements? I only see a way to say the end point locations and the material for an element. I presume this means that this means the resulting element will be this length when under no tension. Am I missing something in my reading of the documentation for the program? -- John Kirk (215) 382-3040 email: dystan@netaxs.com ========================================================================= Date: Tue, 30 Aug 1994 09:02:16 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: Geodesic sphere algorithm I have a few questions regarding the algorithm. Kirby Urner wrote: >What's true is that this algorithm will not give a single spoke >length for use throughout the sphere. [snip] >The goal is to 'distort' the edges by a hair, longer or shorter, >to cause each vertex to be surrounded by exactly k degrees less >than 360 (k is konstant for all vertices). Furthermore, >k times the number of total vertices will equal 360. shouldn't that be: "k times the number of total vertices will equal 720"? > I assume >that the 5 spokes of a pentagonal convergence will be the >same length, and arbitrarily make this 1. I'm wondering if >these contraints will now automatically (once the numerical >algorithm is in place) determine the unique chord lengths for >the rest of the sphere. >Note: computation need only be done for one of the 20 icosa- >sphere's triangles. Are these the only restraints? 1 - each vertex = 360-k 2 - k times the number of vertices = a known constant [360 or 720] 3 - the entire structure must have "closure" (i.e. actually form a closed geodesic sphere) On the face of it, I would expect there to be more than one solution. (just my gut reaction) Do you have a feel for the number of possible spoke lengths which will satisfy these 3 restraints? And assuming you allow the program to "find" the spoke lengths, (whether by brute force or some sophisticated search technique) what increments would you tell it to use? And finally, if the spokes of the pentagons are all equal, shouldn't the vertex at each pentagon total 360, rather than the 360-k found at each hexagon's vertex? =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Tue, 30 Aug 1994 19:40:38 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kevin Sahr Organization: Forest Sciences Laboratory Subject: Re: Geodesic sphere algorithm In article <199408301602.JAA12619@whistler.sfu.ca> Lee Wood writes: >I have a few questions regarding the algorithm. > >Kirby Urner wrote: >>What's true is that this algorithm will not give a single spoke >>length for use throughout the sphere. >[snip] >>The goal is to 'distort' the edges by a hair, longer or shorter, >>to cause each vertex to be surrounded by exactly k degrees less >>than 360 (k is konstant for all vertices). Furthermore, >>k times the number of total vertices will equal 360. > I've been reading this whole (fascinating! don't stop!!) discussion but somewhere became confused. Where do we begin distorting _from_? From the vertices on a sub-divided icosahedron? (By sub-divided icosahedron I mean: think of each face of the icosahedron as a triangle of closest-packed spheres of the desired edge frequency, then move each sphere out radially until it lies on the surface of the sphere in which the icosahedron is contained. I should mention that one of the things I'm wrestling with personally is the relationship between a "sub-divided icosahedron" as described above and the geodesics used in domes). >shouldn't that be: >"k times the number of total vertices will equal 720"? > >> I assume >>that the 5 spokes of a pentagonal convergence will be the >>same length, and arbitrarily make this 1. Does this necessarily have to be true? (I think it has to be true of a sub-divided icosahedron, but if that isn't what we're looking for here then I'm not sure I see why it would have to be true of the result). For instance, one could begin by stipulating that the spokes on a given 6-vertex all equal 1.... >>I'm wondering if >>these contraints will now automatically (once the numerical >>algorithm is in place) determine the unique chord lengths for >>the rest of the sphere. >>Note: computation need only be done for one of the 20 icosa- >>sphere's triangles. > > >Are these the only restraints? > >1 - each vertex = 360-k > >2 - k times the number of vertices = a known constant [360 or 720] > >3 - the entire structure must have "closure" > (i.e. actually form a closed geodesic sphere) > > >On the face of it, I would expect there to be more than one solution. >(just my gut reaction) Do you have a feel for the number of possible >spoke lengths which will satisfy these 3 restraints? On the face of it, I would expect there to be an _infinite_ number of solutions. But then again, I can't quite figure-out where we're trying to go here (though it's a damn interesting journey!). Maybe I don't see all the implications of constraint (1) above (hell, I'll even drop the "maybe" on that sentence!). I guess it goes back to what's being looked for here; I wish I'd kept the original post. Are we trying to find the geodesic structure with the minimum total edge lengths? The minimum variance in edge lengths? The minimum variance (whatever this means) in the voronoi of the vertices? None of the above? By the way; assuming you can precisely characterize what you're looking for and where you want to start looking this looks like a pretty straight-forward optimization problem (my first guess is that with a reasonable starting-point like the sub-divided icosahedron you probably wouldn't have to worry about local minima/maxima very much, though I've learned that I'm not usually correct until about the third guess...). > >And assuming you allow the program to "find" the spoke lengths, >(whether by brute force or some sophisticated search technique) >what increments would you tell it to use? > >And finally, if the spokes of the pentagons are all equal, >shouldn't the vertex at each pentagon total 360, rather than the >360-k found at each hexagon's vertex? >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | >Lee_Wood@sfu.ca | INTJ spoken here. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Keep it up Kirby! This kind of thing reminds me why I still hang- out here... Kevin ========================================================================= Date: Tue, 30 Aug 1994 20:23:20 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Rywalt Organization: Prodigy Services Company, White Plains, New York Subject: Re: Geodesic sphere algorithm Kirby wrote: > We can subdivide the equilateral faces of the icosa into F*F subtriangles, > also equilateral, using the 3-way crosshatching of lines parallel to the > 3 sides. Will radii through the vertices so derived, out to the surface of > a sphere, define vertices which, when connected by chords, will each > be surrounded by the same number of degrees? No. The construction you're talking about is called, according to Hugh Kenner's _Geodesic Math_, a class I method 1 geodesic. (I could go into his discussion of the several types, but I don't want to type in the whole book.) According to the math Kenner uses and a program I spent today hacking out to use his chord factors, the angles of a three-frequency icosahedral geodesic -- basically, what you're talking about with the triangular faces of the icosa cut into nine (three second-powered) smaller equilangular triangles -- are as follows: [cheesy ASCII graphic follows] /a\ / \ /b____\ /\ c /\ /d \ / ` /___f\e/_. /\ / \ ' ` ' ` (Note that we only need these six angles to define the whole icosa face, since the rest are symmetrical.) a = 70.73 b = 54.63 c = 60.71 d = 60.71 e = 58.58 f = 58.58 Therefore, around the vertex in the center of the hexagon, we have (f*4)+(e*2) degrees, or 351.48 degrees, for a spherical excess of about 9.5 degrees. However, at the center vertex of the pentagons, we have a spherical excess of only about 7.5 degrees. Different classes and methods of geodesics (not to mention using different base polyhedra, like tetrahedra or octahedra) result in different chord factors and thus different angles around each vertex. If we were to demand that the 720 degrees required to close the surface be deleted from each angle equally, I have no idea what would happen. Kenner's book doesn't mention such a possibility. Chris. crywalt@tinman.dev.prodigy.com ========================================================================= Date: Tue, 30 Aug 1994 19:21:08 -0500 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Robert L. Read" Organization: CS Dept, University of Texas at Austin Subject: Re: question re: felt applicability Unfortunately, John Kirk, I think you have not missed anything in the documentation of FElt; my impression is that it does not allow you to analyze a pre-stressed structure. I have, however, taken the liberty of forwarding your question to the authors of FElt, and will post their answer to this group. >From dystan@netaxs.com Tue Aug 30 17:26:28 CDT 1994 >Apologies in advance for any clumsiness in posting. This is my first. >I'm an inveterate lurker. I've done some mathematical derivation and >some calculation regarding a variety of tensegrity structures, and due >to recent postings, looked over the documentation for the felt program. >The snag I ran into, trying to see how to apply felt, can be simplified >to the following question. (It seems to me to be a user interface issue >rather than one with the underlying calculation theory) >If I give locations A and B in cartesian coordinates, how do I specify >that I wish to connect them with a cable (made, perhaps, of steel for >instance) whose length when not under tension is only ninety-eight or >ninety-five percent of the distance between A and B? This question reveals a subtle limitation of FElt and static structural analyis based on the stiffness method -- that one must assume that when loaded a structure does not change its geometry sufficiently enough to change any angles in the system. If you were to construct a structure out of materials that deformed as much as 5% before failing then you could not count on FElt providing a correct answer. Although there are many interesting materials that elastic, steel is not one of them, at least not in axial deformation. Steel cables stretch enough that you have to add turnbuckles, but they don't stretch 5% of their length. (I think.) >The idea is that such a cable would be stretched sufficiently to reach >to be connected, and the felt program would at least tell me, based on >elasticity of materials specified, what tension resulted in the cable. >Even more interestingly, it would tell me what the resulting compression >would be in the struts. Robert Read, with your familiarity with felt, >can you tell be how to specify pre-stressed elements? I only see a way >to say the end point locations and the material for an element. I >presume this means that this means the resulting element will be this >length when under no tension. Am I missing something in my reading of >the documentation for the program? > -- John Kirk (215) 382-3040 email: dystan@netaxs.com An unpleasant way to do part of what you may desire is to use FElt to put the element representing the cable under tension, and then (if you have specified the elasticity data correctly), FElt will tell you the deformation (the amount that that element stretched). This could give you a rough means of relating tension to deformation. However, in my opinion it would not let you answer what is (to me) the interesting question --- How much force can I apply to a pre-stressed tensegrity before one of the cables snaps or a strut buckles? It just occurs to me that in using FElt to analyze a tensegrity with cables, you have the additional problem that FElt does not recognize a material like a cable -- that has strength in tension but zero resistance to compression. Perhaps you have foreseen this problem and tried to get around it by pre-stressing, which would allow a tensegrity to be built in which no external force less that the failure force would cause a cable to "go slack" (i.e. have zero or negative tension.) For some tensegrities this might not be a problem; however, in general the fact that FElt will treat an element that should represent a "cable" as a rod that will transfer a compressive and tensile force would cause FElt to over-estimate the stiffness of the tensegrity and hence could not be trusted, although it might still be interesting. Finally (sorry for rambling on) this is not a problem for domes built out of 2x6s, for instance, and I still believe FElt can be used to analyze such structures. -- Robert L. Read, Member of the League for Programming Freedom ========================================================================= Date: Tue, 30 Aug 1994 23:16:48 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Subject: Re: Geodesic sphere algorithm >I have a few questions regarding the algorithm. > >Kirby Urner wrote: >>The goal is to 'distort' the edges by a hair, longer or shorter, >>to cause each vertex to be surrounded by exactly k degrees less >>than 360 (k is konstant for all vertices). Furthermore, >>k times the number of total vertices will equal 360. > >shouldn't that be: >"k times the number of total vertices will equal 720"? > Yes yes. 720, the number of degrees in any tetrahedron. > > >Are these the only restraints? > >1 - each vertex = 360-k > >2 - k times the number of vertices = a known constant [360 or 720] 720! >3 - the entire structure must have "closure" > (i.e. actually form a closed geodesic sphere) > Instead I'd say 3 - the number of vertices must equal 10F*F+2 (F any integer) and all be equidistant from a common center in the pattern of a spherical icosahedron subdivided into triangles 4 - the triangles should be as close to equilateral as possible without violating rule 1 >On the face of it, I would expect there to be more than one solution. >(just my gut reaction) Do you have a feel for the number of possible >spoke lengths which will satisfy these 3 restraints? I think lots of different edge lenghts, depending on the frequency of the sphere. >And assuming you allow the program to "find" the spoke lengths, >(whether by brute force or some sophisticated search technique) >what increments would you tell it to use? User-defined. Higher frequencies would require high precision. >And finally, if the spokes of the pentagons are all equal, >shouldn't the vertex at each pentagon total 360, rather than the >360-k found at each hexagon's vertex? No, the spokes are just the radial members out to the rim. The chords around the rim would not be of the same length as the spokes, but a tad bit shorter (by a micro-millimeter perhaps), thus betting a very slight bulge -- the beginning of curvature and hence closure. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | >Lee_Wood@sfu.ca | INTJ spoken here. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= > ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) ========================================================================= Date: Tue, 30 Aug 1994 23:30:26 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Subject: Re: Geodesic sphere algorithm >Kirby wrote: >> We can subdivide the equilateral faces of the icosa into F*F subtriangles, >> also equilateral, using the 3-way crosshatching of lines parallel to the >> 3 sides. Will radii through the vertices so derived, out to the surface of >> a sphere, define vertices which, when connected by chords, will each >> be surrounded by the same number of degrees? > >No. Hmmmmm. That's interesting. What bothers me about equating the notion of equal spherical excess around each vertex with perfect curvature is that assumes vertices are equidistant, which they are not, especially in the locus of the 12 pentagons. In other words, the same amount of curvature should be spread over the same distance, and if each vertex represents X amount of curvature, and yet are not distributed evenly as to distance, then the ideal of perfectly smooth curvature is not being approached. Still, I am intrigued by the possibility of using an icosahedral dispersion of vertices with "forces" smooth curvatude by adjusting vertexes relative to one another to approach this ideal of equal spherical excess around each vertex. I am tending towards the view that there is a unique solution, and also to agree with you that this would not be a Class I, II or III pattern. >[cheesy ASCII graphic follows] > > /a\ > / \ > /b____\ > /\ c /\ > /d \ / ` > /___f\e/_. > /\ / \ > ' ` ' ` > Highly useful graphic for the purposes of this discussion. Thanks for taking the time. >(Note that we only need these six angles to define the whole icosa face, since > the rest are symmetrical.) >a = 70.73 >b = 54.63 >c = 60.71 >d = 60.71 >e = 58.58 >f = 58.58 Excellent data. Thank you Hugh Kenner (one of those voted most likely to be a genius in all ages types -- his Pound Era is magnificent too). > > Chris. > crywalt@tinman.dev.prodigy.com > BTW, the existence of Class III molecular dispersions in viruses, and the modification of the 10F*F+2 formula needed to accommodate this case, is what led (I believe) virologists to cut Fuller out of the recognition loop for helping to shed light on the structure of viruses. Instead, Goldberg and a couple other guys get all the credit for a more generic version of 10*F*F+2. What the virologists don't realize is that Fuller's 10*F*F+2 was also a subcase of a more generic formula dealing with other sphere packings than the cuboctahedral/icosahedral. Fuller deserves more credit for his sphere packing and concommitant virological discoveries than he gets. He came upon 10*F*F+2 independently and in ways that are not derivative. ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) ========================================================================= Date: Tue, 30 Aug 1994 23:52:52 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kirby Urner Subject: Re: Geodesic sphere algorithm > >I've been reading this whole (fascinating! don't stop!!) discussion >but somewhere became confused. Where do we begin distorting _from_? From >the vertices on a sub-divided icosahedron? I think I meant distorting from a perfectly flat hexagonally tiled floor, where all the edges are indeed of equal length. Given edge length as the only thing we have to play with, we begin adding a little here, subtracting a little there to cause curvature. We also allow 12 hexagons to transform into pentagons (these don't mathematically have to happen on the pattern of an icosahedron to beget closure -- but there *do* have to be exactly 12 of them -- I'm sticking to the icosahedral pattern in this scenario). >(By sub-divided icosahedron I mean: think of each face of the icosahedron >as a triangle of closest-packed spheres of the desired edge frequency, >then move each sphere out radially until it lies on the surface of the >sphere in which the icosahedron is contained. I should mention that one >of the things I'm wrestling with personally is the relationship between >a "sub-divided icosahedron" as described above and the geodesics used in >domes). I think Chris has shown you are describing a Class I geodesic dome. But apparently the number of degrees around each vertex will not be the same -- meaning my algorithm is striving to approach another ideal (but how different would it look, I wonder?). >I guess it goes back to what's being looked for here; I wish I'd kept >the original post. Are we trying to find the geodesic structure with >the minimum total edge lengths? The minimum variance in edge lengths? >The minimum variance (whatever this means) in the voronoi of the >vertices? None of the above? My thought was to let the number of unique edge lengths go through the roof if necessary. The goal is to distribute curvature as evenly as possible, where curvature is defined as k "missing" degrees around each vertex. The goal is to have the sum of the angles around each vertex equal the same number (e.g. 359.9999) AND to have the vertices be as evenly distributed as the plan of the icosahedron permits. The regular tetrahedron has exactly 120 degrees around each vertex and all are the same distance from the other 3. Your intuitive take on "minimum variance... in the voronoi of the vertices" sounds like a good description. I want a constant K curvature spread evenly, using whatever edge lengths will do the job. > >By the way; assuming you can precisely characterize what you're >looking for and where you want to start looking this looks like a >pretty straight-forward optimization problem (my first guess is that >with a reasonable starting-point like the sub-divided icosahedron >you probably wouldn't have to worry about local minima/maxima very >much, though I've learned that I'm not usually correct until about >the third guess...). Dunno. I can imagine a program that gets into perpetually micro-adjusting vertices, creating an endless ripple pattern that never settles down. > >Keep it up Kirby! This kind of thing reminds me why I still hang- >out here... > >Kevin > > Well thanks. But it takes these response posts to keep it flowing. I personally feel a long ways from a numerical recipe. It would have to start with a low value of F. If F=1, it should return the icosahedron itself. But for higher values of F, we are apparently diverging from the classic Class I sphere. Surely the final creature exists, and looks a lot *like* a Class I sphere?? ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) ========================================================================= Date: Wed, 31 Aug 1994 08:48:57 -0700 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Lee Wood Subject: Re: Geodesic sphere algorithm Kirby and I wrote: LW>>And finally, if the spokes of the pentagons are all equal, LW>>shouldn't the vertex at each pentagon total 360, rather than the LW>>360-k found at each hexagon's vertex? KU>No, the spokes are just the radial members out to the rim. The KU>chords around the rim would not be of the same length as the KU>spokes, but a ad bit shorter (by a micro-millimeter perhaps), KU>thus betting a very slight bulge -- the beginning of curvature KU>and hence closure. Thank you. Yes, I see now. - The pentagons are all identical. - The hexagons are all identical. - And the triangles which make up a pentagon are isosceles, but not *quite* equilateral. Are all the triangles in a pentagon identical? i.e. The spokes are all the same length, but are their chords all the same? And for the triangles which make up hexagons: Are their spokes all the same length? And their chords? If the answer to all the above is "Yes", then I would agree with you; there probably is only one solution. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Wed, 31 Aug 1994 12:27:42 -0500 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Robert L. Read" Organization: CS Dept, University of Texas at Austin Subject: Re: Geodesic sphere algorithm I wonder if it would be possible to construct a Penrose tiling of spheres as per the ideas discussed in this thread? (Roger Penrose invented away of tiling a plane based on two figures, called a "kite" and a "dart", I think, that need never repeat itself -- it is an irregular, but all-plane painting tiling.) This would not be cartographically or structurally interesting, but it is certainly intellectually interesting, and could be considered yet another score for Fuller, or at least the synergetic mind-set. -- Robert L. Read, Member of the League for Programming Freedom ========================================================================= Date: Wed, 31 Aug 1994 19:12:37 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "" Subject: Re: The "PI without Trig" Monster Re >When I changed the program to produce a slop-sided cylinder >(i.e. frustum -- I had to look up that word in my geometry book.) >instead of a vertical sided cylinder, it worked properly. > >Unfortunately, my calculus and solid geometry skill are not what >they once were, so I cannot see how to go about correcting the >algorithm. If anyone has a suggestion, I'd be glad to hear >about it. > A quick look through a technical calculus textbook reveals a formula for calculating the area of a surface of revolution - in this case a hemisphere formed by rotating the curve y= sqrroot( r^2 - x^2 ) about the Y axis; the formula, however, presumes an established constant value for PI, making use of it here merely an exercise in circular reasoning. @-) / b Area = 2PIr^2 = 2PI | sqrrt(r^2 - x^2)*sqrrt( 1 + x^2*(r^2 - x^2) ) dx a/ | r = 2PIrx| = 2PIr^2 | 0 ---------------------------------------------------------------------- Mitch C. Amiano amiano@delphi.com ========================================================================= Date: Wed, 31 Aug 1994 19:12:48 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "" Subject: Re: Non-synergetics >I have two questions unrelated to synergetics: >(At least directly) > >Why does water drain counter-clockwise? > and >What does IMHO mean? > > Steve Mather In My Humble Opinion, water drains clockwise, but we could both be correct! Anyone from down-under care to comment? ---------------------------------------------------------------------------- Mitch C. Amiano amiano@delphi.com ========================================================================= Date: Wed, 31 Aug 1994 19:30:05 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Bob Hiltner Organization: Eskimo North (206) For-Ever Subject: Re: Non-synergetics Geoffrey Wherrett (wherrett@mech.ubc.ca) wrote: : In article , smtc5@uoft02.utoledo.edu : writes: : >Why does water drain counter-clockwise? : Water drains counterclockwise due to Coriolis effects. In the Southern Hemisphere, water drains in the opposite direction. Only in THEORY, really. The Coriolis effect is quite negligible compared to other probable effects (geometry of the draining vessel, initial movement of the water, etc.) I think winds and such larger systems DO show the effect, but will leave that as an excercise for the reader ;-) -- Bob Hiltner "It is not enough to do well (and I hope you do), you must also do good" ^^^^ ========================================================================= Date: Wed, 31 Aug 1994 22:56:33 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Kevin Sahr Organization: Forest Sciences Laboratory Subject: geosphere-related sci.math posts The on-going geosphere discussion here made me think some of you might find the following thread from sci.math interesting... Article: 74443 of sci.math From: cet1@cus.cam.ac.uk (Chris Thompson) Newsgroups: sci.math Subject: Re: Maximal polyhedrons Date: 10 Aug 1994 21:20:17 GMT Organization: University of Cambridge, England Lines: 48 Message-ID: <32bgah$snd@lyra.csx.cam.ac.uk> References: <47Doqc5w165w@bbs.dsnet.com> <323o5l$dh2@gap.cco.caltech.edu> NNTP-Posting-Host: apus.cus.cam.ac.uk In article <47Doqc5w165w@bbs.dsnet.com>, samurai@bbs.dsnet.com (John Basol) asks about the problem of choosing n points on a (say) unit sphere so as to maximise the volume of their convex hull. In article <323o5l$dh2@gap.cco.caltech.edu>, allenk@beat.ugcs.caltech.edu (Allen Knutson) provides the correct terminology and writes inter alia: |> What I wonder first about your very interesting problem is what sort |> of faces one expects to have, for large n. In particular, should they |> all be triangles? I'd be curious to know what the combinatorics are |> of an n=40 example - how many faces, how many edges those faces each have. It is known that all the faces must be triangular (this goes back to Fejes Toth, at least). It is conjectured in [1] that the valencies of the n vertices differ by at most 1 (so that they are the integers on either side of 6-12/n), but this has not been proved in general. The solutions are known with certainty only up to n=8. [Croft/Falconer/Guy "Unsolved Problems in Geometry" section F17, from which I got much of the content of this posting, say it "has been solved for n<=9 by Berman & Hanes" but this seems to be a misprint.] For n=4 it is the regular tetrahedron; for n=5,6,7 it is the triangular/square/pentagonal bipyramid respectively (i.e. for n=6 it is the regular octahedron). The solution for n=8 is more interesting. It was found, as a local maximum, by a computer search in [1], and was proved globally maximal in [2]. By suitable choice of coordinates the points can be taken to be P1 = ( sin 3t, 0, cos 3t) P5 = (0, -sin 3t, -cos 3t) P2 = ( sin t , 0, cos t ) P6 = (0, -sin t . -cos t ) P3 = (-sin t , 0, cos t ) P7 = (0, sin t , -cos t ) P4 = (-sin 3t, 0, cos 3t) P8 = (0, sin 3t, -cos 3t) where t is such that cos^2 t = (15+sqrt(145))/40. [t is about 34.7 degrees.] There are edges joining P1 to P2,P4,P6,P7,P8 and from P2 to P1,P3,P5,P8; and for the other vertices correspondingly. The volume of the polyhedron is sqrt((475+29*sqrt(145))/250) = 1.815716... [1] Donald W. Grace Search for largest polyhedra Math. Comp. 17 (1963) 197-199 [2] Joel D. Berman & Kit Hanes Volumes of polyhedra inscribed in the unit sphere in E^3 Math. Ann. 188 (1970) 78-84 Chris Thompson Internet: cet1@phx.cam.ac.uk JANET: cet1@uk.ac.cam.phx Article: 74446 of sci.math From: cet1@cus.cam.ac.uk (Chris Thompson) Newsgroups: sci.math Subject: Variants of n points on a sphere (was: Re: Maximal polyhedrons) Date: 10 Aug 1994 22:17:06 GMT Organization: University of Cambridge, England Lines: 57 Message-ID: <32bjl2$snd@lyra.csx.cam.ac.uk> References: <47Doqc5w165w@bbs.dsnet.com> <323o5l$dh2@gap.cco.caltech.edu> NNTP-Posting-Host: apus.cus.cam.ac.uk In article <323o5l$dh2@gap.cco.caltech.edu>, allenk@beat.ugcs.caltech.edu (Allen Knutson) writes: |> |> The related problem that people ask about on sci.math every few months, |> for some reason, asks how to place n vertices so as to maximize the |> smallest distance between them (or the average square distance). More often it starts with some very vague question about how to spread n points on a sphere so as to... well, you know, look spread out. Which is why it was so refreshing to see one in which the variant was clearly described, even if the poster didn't know the canonical terminology. And is why I have moved this to a different thread. |> This question is known to have very asymmetric answers even for n=12,20 |> where one would expect to get Platonic solids. (See Conway & Sloane, |> _Sphere Packings, Lattices, and Groups_, near the beginning.) For the packing (Tammes) and covering (dual of Tammes) versions, n=12 does give the regular icosahedron. (For these it is fairly easy to prove that if a regular polyhedron with n vertices and triangular faces exists, then it is the unique solution.) There is a very large number of different problems involving spreading out n points on a sphere. None of them are completely solved, and the solutions to the different problems are not identical. The following list is very far from exhaustive: Tammes' problem: maximise the minimum interpoint distance, or equivalently maximise the size of n equal circles (or spherical caps) that can be placed on the sphere without overlapping. Dual of Tammes: the covering problem. Minimise the maximum distance of any point on the sphere from a point of the set, or equivalently minimise the size of n equal circles (or spherical caps) that can be placed to completely cover the sphere. Maximise the p-th root mean p-th power of the n(n-1)/2 interpoint distances (in R^3). The cases usually concentrated on are p=1 (maximise the sum of the distances) and p=-1 (mimimise the potential energy of n equal point charges). I have often thought that p=0 (maximise the product of the distances) might be worth looking at a bit harder. Maximum convex hull: maximise the volume of the convex hull of the n points. Dual of that: minimise the volume of the polyhedron formed by the tangent planes at the n points. Croft, Falconer & Guy "Unsolved Problems in Geometry" Springer (1991) ISBN 0-387-97506-3 sections D7,D8,F17 has a useful collection of references for all of these. A footnote: you suggest maximising "the average square distance". Maybe you meant the angular distance on the spherical surface, but it would seem a little perverse to square that. If you mean the distance _through_ the sphere, i.e. the 3rd problem above with p=2, the problem has an easy (and maybe unexpected) answer. With that as a hint, I will leave it as an exercise! Chris Thompson Internet: cet1@phx.cam.ac.uk JANET: cet1@uk.ac.cam.phx Article: 74513 of sci.math From: jbuddenh@artsci.wustl.edu (Jim Buddenhagen) Newsgroups: sci.math Subject: Re: Maximal polyhedrons Date: 11 Aug 1994 17:17:29 GMT Organization: College of Arts and Sciences -- Washington University, St. Louis, Missouri, USA Lines: 37 Message-ID: <32dmf9$o2q@bigfoot.wustl.edu> References: <47Doqc5w165w@bbs.dsnet.com> <323o5l$dh2@gap.cco.caltech.edu> Reply-To: jb1556@daditz.sbc.com NNTP-Posting-Host: apricot.wustl.edu X-Newsreader: TIN [version 1.2 PL2] Allen Knutson (allenk@beat.ugcs.caltech.edu) wrote: : samurai@bbs.dsnet.com (John Basol) writes: : > I wrote a computer program that asks for a value for n, picks : >n random points on the sphere, determines the polyhedron "one would get : >from stretching a sheet of rubber around the points(I forget the term)", : "convex hull" : >and then proceeds to find the maximal polyhedron by simulating the : >polyhedron filled with a gas under pressure that pushes the faces : >outward and slides the vertices (points) "wherever they'd like to go," so : >to speak. [much deleted] : The related problem that people ask about on sci.math every few months, : for some reason, asks how to place n vertices so as to maximize the : smallest distance between them (or the average square distance). : This question is known to have very asymmetric answers even for n=12,20 : where one would expect to get Platonic solids. (See Conway & Sloane, : _Sphere Packings, Lattices, and Groups_, near the beginning.) For n=12 the regular icosahedron is best (for maximizing the minimum edge). For n=4 the regular tetrahedron, for n=6 the regular octahedron. The cube and reg. dodecahedron are not optimal. For n=24 the snub-cube is best. No other regular or Archimedian polyhedra are optimal. See D.A. Kottwitz "The Desest Packing of Equal Circles on a Sphere" Acta Cryst. (1991) A47, 158-165 and references therein. As to the problem of maximizing the volume of a polyhedron with n vertices on a sphere (the original poster's query), the best known results (to my knowledge) have been made publically available by Hardin, Sloane et.al. in the form of coordinate files at netlib.att.com in /netlib/att/math/sloane .. They have coordinates for "max the min edge" polyhedra there also, several improving and extending Kottwitz's results. -- Jim Buddenhagen (jb1556@daditz.sbc.com) Article: 74798 of sci.math From: jbuddenh@artsci.wustl.edu (Jim Buddenhagen) Newsgroups: sci.math Subject: Re: Maximal polyhedrons Date: 16 Aug 1994 19:48:17 GMT Organization: College of Arts and Sciences -- Washington University, St. Louis, Missouri, USA Lines: 20 Message-ID: <32r561$ft2@bigfoot.wustl.edu> References: <47Doqc5w165w@bbs.dsnet.com> <323o5l$dh2@gap.cco.caltech.edu> Reply-To: jb1556@daditz.sbc.com NNTP-Posting-Host: apricot.wustl.edu X-Newsreader: TIN [version 1.2 PL2] Allen Knutson (allenk@beat.ugcs.caltech.edu) wrote: : What I wonder first about your very interesting problem is what sort : of faces one expects to have, for large n. In particular, should they : all be triangles? I'd be curious to know what the combinatorics are : of an n=40 example - how many faces, how many edges those faces each have. : Allen K. [problem is to place n points on unit sphere such that the convex hull has maximal volume]. The putative best for n=40 (per the maxvol files of Hardin et al) is a polyhedron with tetrahedral symmetry. Start with a regular tetrahedron: put one point at each of its vertics (for 4 pts), two on each edge (for 12 pts) and 6 on each face (an equilateral triangle surrounded by three isosceles triangles, for 24 more pts). Adjust till volume is max :-). Resulting convex hull has 7 distinct edge lengths. All faces are triangles. -- Jim Buddenhagen (jb1556@daditz.sbc.com)