From <@UBVM.CC.BUFFALO.EDU:owner-LISTSERV@UBVM.CC.BUFFALO.EDU> Mon Feb 6 17:00:09 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 RAA29335 for ; Mon, 6 Feb 1995 17:00:09 -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 QAA04309 for ; Mon, 6 Feb 1995 16:59:34 -0500 Message-Id: <199502062159.QAA04309@netaxs.com> Received: from UBVM.CC.BUFFALO.EDU by UBVM.cc.buffalo.edu (IBM VM SMTP V2R2) with BSMTP id 3003; Mon, 06 Feb 95 16:58:32 EST Received: from UBVM.CC.BUFFALO.EDU (NJE origin LISTSERV@UBVM) by UBVM.CC.BUFFALO.EDU (LMail V1.2a/1.8a) with BSMTP id 9074; Mon, 6 Feb 1995 12:40:06 -0500 Date: Mon, 6 Feb 1995 12:39:41 -0500 From: "L-Soft list server at UBVM (1.8a)" Subject: File: "GEODESIC LOG9407" To: "Christopher J. Fearnley" Status: RO ========================================================================= Date: Sat, 2 Jul 1994 06:42:56 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Chris Fearnley Organization: Critical Path Project Subject: Re: Developing an operational pi In article <01HE4HR1T7JC9AN51F@delphi.com> "" writes: [Stuff deleted] > > >To say that something is an approximation is to indicate that some perfectly >accurate representation exists. Everything I know contradicts this; >everything is inexact - mutating and changing constanty, subtly, and even >extensively. That which is not does not appear to be measurable. > > > >Still, I can't shake the feeling that when someone wants a rational way of >determining the height of an equilateral triangle made from physical struts, >there is no rational way of calculating the value. This of course is the heart of the matter: Is it easier to get a rational perspective or must we continue to "round off" from our irrational and transcendental XYZ perspective? I think looking at a system from the rational fuller-like perspective is very useful, but when I want to measure or build something I reach for my computer and get an answer rather quickly. Perhaps when we build our synergetic visioning systems we will see ways to calculate what hitherto had been easier to do by calculator. The trial-by-error method does offer a rational way to calculate your height of the equilateral triangle - but it's too slow for an impatient world. And nature doesn't care about its height - she just "expands" the consequences of gravity and radiation. Also it seems possible that mathematics was developed specifically to solve arbitrary problems. Maybe synergetics is more for understanding Nature and solving anticipatory design science problems. In sum I think that mathematics may always be the better tool for arbitrary calculations (the height of some figure such as the equilateral triangle). > >----------------------------------------------------------------------- >Mitch C. Amiano >amiano@delphi.com -- Christopher J. Fearnley UNIX SIG Leader at PACS cfearnl@cpp.pha.pa.us (Philadelphia Area Computer Society) cfearnl@pacs.pha.pa.us Design Science Revolutionary fearnlcj@duvm.ocs.drexel.edu Explorer in Universe ========================================================================= Date: Sat, 2 Jul 1994 14:07:24 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Matthew V. J. Whalen" Subject: Re: Dome magazine In-Reply-To: Your message of "Thu, 30 Jun 1994 16:29:08 EST." <199407010040.UAA06783@tis.telos.com> >I'm reading old posts. Did anyone ever provide the Dome magazine >information. I'm a subscriber so I can provide it if it is still >needed. please do - I've been looking for it for about a year now... thanks -matthew ========================================================================= Date: Mon, 4 Jul 1994 08:22:56 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Anton BAKKER Organization: Schlumberger RPS - France Subject: Lookinh for file: HEDRA.ZIP 9 April 1994. An article was posted in this newsgroup mentioning Hedra.zip on the net. This program would create a wide veriety of polyhedra forms for 3D Studio. were or how do I find this program on the net? Thank in advance for any help! ========================================================================= Date: Tue, 5 Jul 1994 18:30:06 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Vincent J. Matsko" Organization: Sponsored account, Mathematics, Carnegie Mellon, Pittsburgh, PA Subject: Hello! and pi (and somewhat long). Hi! I'm new to this newsgroup. Re: An operational pi: As regards Mitch Amiano's approximation to pi using trigonometry and the response that knowledge of pi was needed to calculate the trig. functions, it is possible, beginning with a hexagon, to perform the "irrational" approximations by doubling the number of sides each time (rather than increasing by one) so that the appropriate half-angle formulae may be used to calculate sines and cosines without any knowledge of pi. Re: Kirby Urner's "table of trig function that always `came out' to some rational number": The only rational multiples of pi whose sine and cosine are both rational are 0, 1/2, 1, 3/2, etc. There are many angles whose sine and cosine are rational (take any Pythagorean triple as sides of a right triangle), but they don't occur that often in the usual geometric exploits. Re: The practicality of pi: I have done quite a bit of solid geometry (Fuller was inspirational for me), and I have never had occasion to use pi. I think the natural choice of "unit" for angles to be "revolutions", thus the range 0-360 degrees is just the range 0-1. Now these numbers are "dimensionless", being interpreted as the fraction of the area of a circle that the sector cut out by the angle occupies. Now let's take the discussion to three dimensions. We wish to have a measure of solid angle so that we may discuss spacefilling ideas. So define the measure of a solid angle to be that fraction of a sphere (centered at the vertex of the angle) cut out be the solid angle. If A, B, and C are the measures of the dihedral angles of a solid angle, and the measure of the angle is 1/2(A + B + C - 1/2). Example: Take the corner of a cube. Each dihedral angle has measure 1/4 (i.e., 90 degrees). So the measure of the solid angle determined by a corner is 1/2(1/4 + 1/4 + 1/4 - 1/2) = 1/8. Now in a cubic packing of space, 8 corners of the cube meet at a point, so it makes sense that each corner should occupy "one-eighth of the space" about the vertex of that corner. Thus, we may talk of solid angles in 3D without needing to bring in pi, the results being, I believe, more geometrically intuitive. And, being that Fuller's rather discrete geometry really never concerns itself with circles or spheres (I suppose excepting sphere packings), pi is not really needed. Re: The irrationality of the height of an equilateral triangle (a la Chris Fearnley): Again, I think it a matter of perspective. One may take an easy way out (I often do) and say that the square of the ratio of the height of an equilateral triangle to its edge is 3/4. Voila, a rational number! Or alternatively, sometimes an expression involving square roots may be described as the solution of a quadratic equation with integer coefficients (i.e., the golden ratio is a root of x^2 = x + 1). Now on another level, this is unsatisfactory, and I can't offer a good answer. Allow me a suggestion: change the comparison. For example, what is the ratio of the volume of a regular tetrahedron to the volume of a cube when both have the same edge length? Answer: irrational and irrelevant! Look through Fuller, and you never see (as least not to my recollection) two such figures. One only encounters a tetrahedron and the circumscribing cube. In this case, the ratio of the volumes is 1/3. Thus, not every "ratio" is rational; it depends upon what one takes the ratio of. And setting a standard is not simple. I believe, for example, that Williams in his book about structure gives data for the Archimedean solids relative to an edge length of one, which I find wholly unsatisfactory. Here is my choice for the "basic" Platonic solids: Begin with a tetrahedron, circumscribe a cube, and for the octahedron, take the dual to the cube (in the sense that the edge of duals perpendicularly bisect each other). Now the cube may in turn be inscribed in a dodecahedron, whose dual is an icosahedron. Now compute ratios of volumes of these figures, not those with edge length of one! For it is these figures which "naturally" occur in concert with each other. For those ratios involving icosahedra/dodecahedra, one must be satisfied with rationals and the golden ratio as well. In fact, I am inclined to submit that the golden ratio be given honorary "rational" status! Well, that's enough for now, I see that I have been long-winded. I hope that I haven't been too vague. Vince Matsko ========================================================================= Date: Tue, 5 Jul 1994 22:46:59 -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: Developing an operational pi >>So are we agreed that what Bucky was advocating was a 'grainy-pi' >>using a super scheherezade number with tons of primes folded in? >>Like, the Babylonians chose 360 because of its easy divisibility. >>People came up with "Grads" (on most calculators) dividing >>the circle into 100 degrees -- for the true die-hard decimal-heads. >>So Bucky, in true Babylonian fashion, but acknowledging the new level >>of computing power we've attained, suggested replacing 360 with a >>number with a great many more primes worked in. The idea would be >>to then generate a table of trig functions that always "came out" >>to some rational number. The whole set-up would be "grainy" but >>I think it was Bucky's contention that we would find such a system >>to be sufficient to cover nature's "scalables" -- i.e. we would have >>a rational trigonometry of enough accuracy to do subatomics, >>architecture etc. >> It is an unproven hypothesis to be sure, but worth investigating. The Babylonians didn't just magically come up with 360 either. If I recall correctly, 360 seems to have been one of a series of of counting milestones in their 20 & 60 based numeric system. They didn't use a power-of-ten place-holder system for representing numbers, but an arithmatic (60 plus 20 plus 3 ....) method. The representation of degrees/minutes/seconds further suggests that there may be a way to 'block' an irrational operation from proceeding further, by shifting to another base (the next in a chain of scheherazade factors?) after integrally computing the previous denomination. eg Decimally, 101 = 1*10^2 + 0*10^1 + 1*10^0 but could also be 7*10^1 + 3*10^1 + 1*10^0 = 101 I could imagine what the symbolic form would be, but don't have time right now. >>I have no clear understanding of what it would look like to carry >>this out in practice. Sounds like a job for a computer language. >>My question here though is: does anyone have a different understanding >>of what Bucky meant? And, yes, what about the phenomenal utility of >>such numbers as e, \/2 etc? Although here, again, my earlier >>assertion is relevant: our computers only carry out our symbols >>to a finite number of numbers for crunching purposes in any case, >>so the question "can we get along without computing with non- >>terminating irrationals?" is moot in any case -- we get along fine >>right now. >> >>-- Kirby Such numbers are also good for symbolic usages. ------------------------------------------------------------------------- Mitch C. Amiano amiano@delphi.com ========================================================================= Date: Tue, 5 Jul 1994 22:46:22 -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: Earthquake-Proof >>"JR> of non-wood dome superstructure would be the U.S. Pavillion at >>Canada's >> "JR> Expo67 in Montreal. I believe steel was used for that >> "JR> dome's huge skeleton. >>> "JR> HJ.ROSEN@SRS.GOV< >>>Actually, it was made of aluminum extrusions.< >>>Jeff Weiner< >>>jeff@ican.com< >>My information agrees with the steel construction for the EXPO 67 US >>Pavillion. It was constructed of steel pipes slotted at the ends to join >>onto cast multi-fingered connectors. Once in place the pipe slots were >>welded onto the connector fingers which fit exactly into the slots. I read >>somewhere that one of the problems that this construction system had was >>the fact that it was too rigid and due to thermal stresses tended to be >>susceptible to cracks requiring re-welding. I've seen the joint >>construction up close (years after the fire that destroyed the skin). The >>dome was an incredible accomplishment considering they evidently used log >>tables for calculations, took 2 years and were hoping that there weren't >>errors in the tables. No flame intended, but that's pretty sad. Computer techs call work like that a hack (like we have room to speak...). >>Thermal stress is often referred to as causing problems in dome >>construction. I believe that the way to minimize this is to design an >>insulation and weather break system which goes on the outside of the >>structure so that the structure doesn't have to take the bulk of the >>expansion and contraction. >>kbrown@atc.edmonton.ab.ca >> Isn't this because the domes themselves are too loose? Could one solution be to make them more rigid by increasing the tension in the structure? I suppose this would be easier with a tensegrity or double-layered octetruss construction. -------------------------------------------------------------------- Mitch C. Amiano amiano@delphi.com ========================================================================= Date: Wed, 6 Jul 1994 00:05:33 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: Hello! and pi (and somewhat long). In-Reply-To: Message of Tue, 5 Jul 1994 18:30:06 -0400 from On Tue, 5 Jul 1994 18:30:06 -0400 Vincent J. Matsko said: >Hi! I'm new to this newsgroup. Welcome! And don't worry about those two bounced messages you received everyone who posts seems to get two bounced messages. > [Stuff deleted] > >Re: The practicality of pi: I have done quite a bit of solid geometry >(Fuller was inspirational for me), and I have never had occasion to use >pi. I think the natural choice of "unit" for angles to be >"revolutions", thus the range 0-360 degrees is just the range 0-1. Now >these numbers are "dimensionless", being interpreted as the fraction of >the area of a circle that the sector cut out by the angle occupies. > >Now let's take the discussion to three dimensions. We wish to have a >measure of solid angle so that we may discuss spacefilling ideas. So >define the measure of a solid angle to be that fraction of a sphere >(centered at the vertex of the angle) cut out be the solid angle. If A, >B, and C are the measures of the dihedral angles of a solid angle, and >the measure of the angle is 1/2(A + B + C - 1/2). Example: Take the >corner of a cube. Each dihedral angle has measure 1/4 (i.e., 90 >degrees). So the measure of the solid angle determined by a corner is >1/2(1/4 + 1/4 + 1/4 - 1/2) = 1/8. Now in a cubic packing of space, 8 >corners of the cube meet at a point, so it makes sense that each corner >should occupy "one-eighth of the space" about the vertex of that corner. This is great! Does this formula have a name? Did you derive it yourself? Suddenly everything seems rational again :) Is there a simple way to verify your formula in the general case? >Thus, we may talk of solid angles in 3D without needing to bring in pi, >the results being, I believe, more geometrically intuitive. And, being >that Fuller's rather discrete geometry really never concerns itself with >circles or spheres (I suppose excepting sphere packings), pi is not >really needed. > >Re: The irrationality of the height of an equilateral triangle (a la >Chris Fearnley): Again, I think it a matter of perspective. One may >take an easy way out (I often do) and say that the square of the ratio >of the height of an equilateral triangle to its edge is 3/4. Voila, a >rational number! Or alternatively, sometimes an expression involving >square roots may be described as the solution of a quadratic equation >with integer coefficients (i.e., the golden ratio is a root of x^2 = x + >1). This reminds me of the "canonical form" problem in mathematics. I realized in college that the notion of canonical form is ridiculous. Who cares if you have a sqrt() in the denominator of a fraction? Isn't a fraction of fractions still a fraction? If expressions not in canonical form are "bad" then doesn't that taint the whole derivation? Of course NOT! For synergetics calculations we are defining a new asthetic for canonical form. One that is more geometrically intuitive and hence explainable to young children. From this perspective, it doesn't matter if we need to do "ugly" calculations to get some result - just put it in canonical (synergetic) form at the end. And by trying to do whole derivations entirely in synergetics (canonical) we accomplish the dual objectives of getting a clear geometrical representation and it's the one Nature is actually using in her transformings and intertransformings. >Now on another level, this is unsatisfactory, and I can't offer a good >answer. Allow me a suggestion: change the comparison. For example, >what is the ratio of the volume of a regular tetrahedron to the volume >of a cube when both have the same edge length? Answer: irrational and >irrelevant! Look through Fuller, and you never see (as least not to my >recollection) two such figures. One only encounters a tetrahedron and >the circumscribing cube. In this case, the ratio of the volumes is 1/3. > >Thus, not every "ratio" is rational; it depends upon what one takes the >ratio of. And setting a standard is not simple. I believe, for >example, that Williams in his book about structure gives data for the >Archimedean solids relative to an edge length of one, which I find >wholly unsatisfactory. Here is my choice for the "basic" Platonic >solids: Begin with a tetrahedron, circumscribe a cube, and for the >octahedron, take the dual to the cube (in the sense that the edge of >duals perpendicularly bisect each other). Now the cube may in turn be >inscribed in a dodecahedron, whose dual is an icosahedron. Now compute >ratios of volumes of these figures, not those with edge length of one! >For it is these figures which "naturally" occur in concert with each >other. When reading synergetics it struck me that perhaps there are two (or three) basic phases in the Universe - tetra (octa) and icosa. Clearly the tetra and icosa are a bit out of phase. Well, this is really the A, B, and T quanta modules (any others to add to the list). If I remember correctly (haven't checked into this) the A, B, and T quanta modules are not geometrically interderivable. So your regular polyhedra heirarchy may be just another way of looking at the three fundamental geometrical forms in Universe - my so-called canonical (rational) forms. >For those ratios involving icosahedra/dodecahedra, one must be satisfied >with rationals and the golden ratio as well. In fact, I am inclined to >submit that the golden ratio be given honorary "rational" status! > >Well, that's enough for now, I see that I have been long-winded. I hope >that I haven't been too vague. Well, I had to read it twice, but you were pretty clear. Thanks for the intellectual work-out! Do Enjoy! > >Vince Matsko ========================================================================= Date: Wed, 6 Jul 1994 11:13:00 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "H. Jeffrey Rosen" Subject: Re: Earthquake-Proof >>"JR> of non-wood dome superstructure would be the U.S. Pavillion at >>Canada's >> "JR> Expo67 in Montreal. I believe steel was used for that >> "JR> dome's huge skeleton. >>> "JR> HJ.ROSEN@SRS.GOV< >>>Actually, it was made of aluminum extrusions.< >>>Jeff Weiner< >>>jeff@ican.com< >>Thermal stress is often referred to as causing problems in dome >>construction. I believe that the way to minimize this is to design an >>insulation and weather break system which goes on the outside of the >>structure so that the structure doesn't have to take the bulk of the >>expansion and contraction. >>kbrown@atc.edmonton.ab.ca >> Isn't this because the domes themselves are too loose? Could one solution be to make them more rigid by increasing the tension in the structure? I suppose this would be easier with a tensegrity or double-layered octetruss construction. ---------------- Mitch C. Amiano amiano@delphi.com I suggest considering a spherical, pseudo-jitterbug articulated structure which could grow/shrink in harmony with thermal effects on the proposed dome's outer layers. Thermic sensors would feed a central scaling column that would adjust mechanically to keep the dome surface taut. There is a small scale model of this structure in the atrium of the Liberty Science Center in Jersey City, NJ - It's motorized to show the shrink/grow capability to maximum effect. It may be a brainchild of geodesic design. ========================================================================= Date: Wed, 6 Jul 1994 17:58:37 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: scimatec5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: Earthquake-Proof In article <01HED407S78Y9AOEY7@delphi.com>, "" writes: >>>My information agrees with the steel construction for the EXPO 67 US >>>Pavillion. It was constructed of steel pipes slotted at the ends to join >>>onto cast multi-fingered connectors. Once in place the pipe slots were >>>welded onto the connector fingers which fit exactly into the slots. I read >>>somewhere that one of the problems that this construction system had was >>>the fact that it was too rigid and due to thermal stresses tended to be >>>susceptible to cracks requiring re-welding. I've seen the joint >>>construction up close (years after the fire that destroyed the skin). The >>>dome was an incredible accomplishment considering they evidently used log >>>tables for calculations, took 2 years and were hoping that there weren't >>>errors in the tables. > > No flame intended, but that's pretty sad. Computer techs call work like > that a hack (like we have room to speak...). > >>>Thermal stress is often referred to as causing problems in dome >>>construction. I believe that the way to minimize this is to design an >>>insulation and weather break system which goes on the outside of the >>>structure so that the structure doesn't have to take the bulk of the >>>expansion and contraction. >>>kbrown@atc.edmonton.ab.ca >>> > > Isn't this because the domes themselves are too loose? Could one > solution be to make them more rigid by increasing the tension in the > structure? I suppose this would be easier with a tensegrity or > double-layered octetruss construction. Well, perhaps the domes could be more dynamic. Say, make those slots movable clips, rather than welded ones, so that the structure is able to expand and contract with the weather. The the skin could be a one piece (to make weather tight) and somehow dynamic as well (how? I'm not sure =) Steve Mather ========================================================================= Date: Wed, 6 Jul 1994 17:32:15 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Vincent J. Matsko" Organization: Sponsored account, Mathematics, Carnegie Mellon, Pittsburgh, PA Subject: Re: Hello! and pi (and somewhat long). In-Reply-To: About the formula for a solid angle (i.e., 1/2(A + B + C - 1/2)): No, I cannot claim originality for this formula. It is derived from a standard result (in the CRC, e.g.) for the area of a spherical triangle by changing the units to revolutions and by dividing by the surface area of the sphere in question (so as to yield a ratio rather than an "absolute" area). As far as a generalization goes: (1) for an n-hedral angle with dihedral angles A1,...,An, the formula for the measure of the solid angle is 1/2(A1 + A2 + ... + An + 1 - n/2), which reduces to the above for n = 3. (2) However, in higher dimensions, there is no simple formula. Coxeter addresses the issue briefly in his "Regular Polytopes", where he includes a formidable 4D formula derived by Schlafli. It's really rather nasty looking, if I must say. Re: "Synergistic calculations/results": I noticed the use of such terms in your (Chris Fearnley's) post. Could you elaborate on what you mean by these terms? Re: Tetra/Octa/Icosa: Chris Fearnley remarks that the tetra and icosa are "out of phase". Again, I am unsure how to interpret such a phrase. My feeling is that these three polyhedra are quite intimately related. I offer as justification several polyhedral models from Magnus Wenninger's book "Polyhedron Models" - search the stellations of the icosahedron for (1) the compound of five tetrahedra, (2) the compound of ten tetrahedra, and (3) the compound of five octahedra. (Anyone else out there interested in stellations of polyhedra?) For those interested in group theory, from a group theoretical perspective, we can view the symmetry groups of the tetrahedron and the octhahedron as subgroups of the symmetry group of the icosahedron (with reflections included) - so that, in a sense, the tetrahedron and octahedron are "children" of the icosahedron. However, I know of no nice decomposition of the icosahedron into "modules" such as the quanta modules of Fuller (mentioned in Chris Fearnley's post). Does anyone know of such a decomposition? Well, enough for today. P.S. Anyone interested in the geodesic decompositions described in Wenninger's "Spherical Models"? Vince ========================================================================= Date: Thu, 7 Jul 1994 06:20:08 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Wei Chen Organization: Computer Science Department, Stanford University. Subject: newsletter Does anyone know any electronic newsletter on the net about news in geodesic area? -- Wei Chen Dept of Computer Science Stanford University ========================================================================= Date: Thu, 7 Jul 1994 05:02:29 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: Hello! and pi (and somewhat long). In-Reply-To: Message of Wed, 6 Jul 1994 17:32:15 -0400 from On Wed, 6 Jul 1994 17:32:15 -0400 Vincent J. Matsko said: [Stuff deleted] > >Re: "Synergistic calculations/results": I noticed the use of such terms >in your (Chris Fearnley's) post. Could you elaborate on what you mean >by these terms? Without the context it's difficult. I quickly jumped to the newsreader and came back, but I'm doubtfull that I can remeber the context. I think in the broadest sense "synergetics calculations/results" are those geometric problems one raises when working from Bucky's perspective. So one is trying to keep to the "rational" if at all possible. > >Re: Tetra/Octa/Icosa: Chris Fearnley remarks that the tetra and icosa >are "out of phase". Again, I am unsure how to interpret such a phrase. > My feeling is that these three polyhedra are quite intimately related. >I offer as justification several polyhedral models from Magnus >Wenninger's book "Polyhedron Models" - search the stellations of the >icosahedron for (1) the compound of five tetrahedra, (2) the compound of >ten tetrahedra, and (3) the compound of five octahedra. (Anyone else >out there interested in stellations of polyhedra?) "out of phase" could mean incommensurable in terms of volume relationship and symmetrical (5-fold vs. 3-fold). Of course they are related in many ways too. The volume incommensurability issue I think is the main worry for me about how to keep "rational" perspective." > >For those interested in group theory, from a group theoretical >perspective, we can view the symmetry groups of the tetrahedron and the >octhahedron as subgroups of the symmetry group of the icosahedron (with >reflections included) - so that, in a sense, the tetrahedron and >octahedron are "children" of the icosahedron. However, I know of no >nice decomposition of the icosahedron into "modules" such as the quanta >modules of Fuller (mentioned in Chris Fearnley's post). Does anyone >know of such a decomposition? Was I making unwarrented assumptions or is my intuition correct: The T module is based on the rhombic triacontahedron. Is it not intimately connected with the icosa? Perhaps with a rational volume ratio? Now that you got me curious - I wonder what type of "orthoscheme" breakdowns of the icosa would prove fruitful? Neat stuff about the symmetry groups - two bad I hadn't fully read your post before replying above :) > >Well, enough for today. > >P.S. Anyone interested in the geodesic decompositions described in >Wenninger's "Spherical Models"? > >Vince ========================================================================= Date: Thu, 7 Jul 1994 23:44:23 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: scimatec5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: Earthquake-Proof In article "H. Jeffrey Rosen" writes: > I suggest considering a spherical, pseudo-jitterbug articulated structure which > could grow/shrink in harmony with thermal effects on the proposed dome's > outer layers. Thermic sensors would feed a central scaling column that would > adjust mechanically to keep the dome surface taut. > > There is a small scale model of this structure in the atrium of the > Liberty Science Center in Jersey City, NJ - It's motorized to show the > shrink/grow capability to maximum effect. It may be a brainchild of > geodesic design. I once thought of making a similar structure, but failed to see any use. I didn't think of thermal expansion needs, they usually don't matter in a conventional structure. The idea uses center-bodied octet truss tensegrity components (octahedral) that are pneumatic. They can extend or contract when needed, in this case according to thermal stresses. With this, no central column is required, because the structure retains its rigidness through this system. As the trusses expand, the tensile components slide along the ends of these, causing the structure to shrink in surface area and volume (and vise-versa) according to thermal stresses. The entire structure can be covered by a sort of "scale- like" (as in reptile scales) covering. The covering (roof/walls)will be able to expand and contract with the structure to provide weather protection. Steve Mather ========================================================================= Date: Fri, 8 Jul 1994 02:47:29 -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: Deriving PI without trig RE: Deriving PI w/o trigonometry I've come up with an algorithm for deriving pi that uses no trig, just pythagoras. Involves filling a unit circle with a fractal pattern of similar triangles, thereby approaching pi as an *area* (vs circumference). The algorithm is easiest expressed as a short computer program: ---------------------------------------------------- pi=2 hypot=2^0.5 FOR n=1 TO 30 height=1-(1-(hypot/2)^2)^0.5 newhypot=(height^2+(hypot/2)^2)^0.5 newarea=1/2*height*hypot pi=pi+2^(n+1)*newarea ? pi hypot=newhypot ENDFOR ---------------------------------------------------- The output of which (from the line reading '? pi') reads: First 5 terms: 2.828427124746190000 3.061467458920718000 3.121445152258052000 3.136548490545939000 3.140331156954753000 Last 7 terms: 3.141592653589789000 3.141592653589793000 3.141592653589793000 3.141592653589793000 3.141592653589793000 3.141592653589793000 3.141592653589793000 As you can see, I reach the limits of my computer's accuracy (using this particular programming language) at about 25 iterations. There's some specific geometric reasoning that led to this algorithm of course, which involves starting with an inscribed square (2 triangles) and successively bisecting outer edges (hypotenuses) to create a series of smaller and smaller similar triangles pushing into the unfilled arc regions. As the triangles get smaller, their numbers multiply exponentially, hence the fractal-like (self-similar) nature of the algorithm. This method may have already been published many times, but I derived it from scratch I'm proud to say. -- Kirby Urner ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) Portland, Oregon ========================================================================= Date: Fri, 8 Jul 1994 08:53:00 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "H. Jeffrey Rosen" Subject: Re: Earthquake-Proof In article "H. Jeffrey Rosen" writes: > I suggest considering a spherical, pseudo-jitterbug articulated structure which > could grow/shrink in harmony with thermal effects on the proposed dome's > outer layers. Thermic sensors would feed a central scaling column that would > adjust mechanically to keep the dome surface taut. > > There is a small scale model of this structure in the atrium of the > Liberty Science Center in Jersey City, NJ - It's motorized to show the > shrink/grow capability to maximum effect. It may be a brainchild of > geodesic design. I once thought of making a similar structure, but failed to see any use. I didn't think of thermal expansion needs, they usually don't matter in a conventional structure. The idea uses center-bodied octet truss tensegrity components (octahedral) that are pneumatic. They can extend or contract when needed, in this case according to thermal stresses. With this, no central column is required, because the structure retains its rigidness through this system. As the trusses expand, the tensile components slide along the ends of these, causing the structure to shrink in surface area and volume (and vise-versa) according to thermal stresses. The entire structure can be covered by a sort of "scale- like" (as in reptile scales) covering. The covering (roof/walls)will be able to expand and contract with the structure to provide weather protection. Steve Mather The overlapping scales concept has some inherent physical liabilities. One example is that scale-to-scale contact admits many mechanical failure and corrosion points. Another is the need to accomodate three dimensional expansion/contraction physics due to intrusion of water during freeze/thaw temperature cycles. The Montreal dome applied an inside surface solution to this puzzle by "pulling the windowshades" on certain translucent skin panels receiving direct sunlight. Perhaps this technique can be paired with a suitable photo-optical phase change or dichroic material to minimize the greenhouse heating (and subsequent lift!) within large domes. Jeff Rosen ========================================================================= Date: Fri, 8 Jul 1994 13:41:35 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Mike Kohl Subject: Residential Domes I am looking for manufacturers of residential size geodesic domes. Any help would be appreciated. *************************************************************************** Michael P. Kohl, P.E. Phone (404) 453-7455 Michael_Kohl@INS.com Fax (404) 740-1506 International Network Services Pager (800) 710-0104 "There are only two mistakes one can make in business - > acting when you shouldn't and not acting when you should". > Dr. W. Edwards Deming Consultant in Statistical Studies *************************************************************************** ========================================================================= Date: Fri, 8 Jul 1994 15:15:59 EDT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Jack Goodman Organization: The University of Kentucky Subject: Re: Residential Domes In article <9407081741.AA02228@provider.ins.com> Mike Kohl writes: >I am looking for manufacturers of residential size geodesic domes. ... One that I'm aware of (the source of the shell components we bought) is Oregon Dome, Inc. 3215 Meadow Lane Eugene, OR 97402 (503) 689-3443 -jg *** pha146.ukcc.uky.edu The Odyssey Farm, Bourbon County, Kentucky *** ========================================================================= Date: Fri, 8 Jul 1994 18:19:04 -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: Developing an operational pi (part of post deleted) Chris writes... >The trial-by-error method does offer a rational way to calculate your >height of the equilateral triangle - but it's too slow for an impatient >world. And nature doesn't care about its height - she just "expands" >the consequences of gravity and radiation. In a narrowly defined category of problems, this is true. In other contexts, it is irrelevant; but then Synergetics isn't concerned with other contexts - or is it? (I don't like the apparently arbitrary casting-off of entire problem domains that, well, most Synergetic-accounting discussions tend to take on. That's not a flame but an observation. If someone asks, 'What is the length of a right-triangle's hypotenuese," the synergeticist' response would be 'It has no structural meaning, and is therefore irrelevant.' But this is not valid line of reasoning, because clearly these relationships do have significance at times.) > Also it seems possible that >mathematics was developed specifically to solve arbitrary problems. >Maybe synergetics is more for understanding Nature and solving >anticipatory design science problems. In sum I think that mathematics >may always be the better tool for arbitrary calculations (the height >of some figure such as the equilateral triangle). Not all problems are rationally tractable, so I think this is true. But mathematics was not 'developed specifically' (no intonation intended, just using your words) so I think it could not have been 'developed specifically to solve arbitrary problems'. Mathematics is an outgrowth of the human mind; the human mind is not entirely rational, therefore mathematics is not entirely rational. ---------------------------------------------------------------------- Mitch C. Amiano amiano@delphi.com ========================================================================= Date: Fri, 8 Jul 1994 19:04:50 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Matthew V. J. Whalen" Subject: Re: Residential Domes In-Reply-To: Your message of "Fri, 08 Jul 1994 15:15:59 EDT." <199407081958.PAA13611@tis.telos.com> >In article <9407081741.AA02228@provider.ins.com> >Mike Kohl writes: > >>I am looking for manufacturers of residential size geodesic domes. ... > > One that I'm aware of (the source of the shell components we bought) is > > Oregon Dome, Inc. > 3215 Meadow Lane > Eugene, OR 97402 > (503) 689-3443 You can also try: Timberline Geodesics 2015 Blake St. Berkeley, California 94704 (800) DOME-HOME -matthew ========================================================================= Date: Fri, 8 Jul 1994 19:41:43 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: scimatec5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Thermal expansion problems, was Re: Earthquake-Proof In article <01HEGIZX09QS001Q9K@mr.srs.gov>, "H. Jeffrey Rosen" writes: > In article "H. Jeffrey Rosen" writes: > >> I suggest considering a spherical, pseudo-jitterbug articulated structure >> which >> could grow/shrink in harmony with thermal effects on the proposed dome's >> outer layers. Thermic sensors would feed a central scaling column that would >> adjust mechanically to keep the dome surface taut. (some deleted) >> I once thought of making a similar structure, but >> failed to see any use. I didn't think of thermal >> expansion needs, they usually don't matter in a >> conventional structure. >> >> The idea uses center-bodied octet truss tensegrity >> components (octahedral) that are pneumatic. They >> can extend or contract when needed, in this case >> according to thermal stresses. With this, no central >> column is required, because the structure retains its >> rigidness through this system. As the trusses expand, >> the tensile components slide along the ends of these, >> causing the structure to shrink in surface area and >> volume (and vise-versa) according to thermal stresses. >> The entire structure can be covered by a sort of "scale- >> like" (as in reptile scales) covering. The covering >> (roof/walls)will be able to expand and contract with >> the structure to provide weather protection. >> >> Steve Mather > > The overlapping scales concept has some inherent physical liabilities. > One example is that scale-to-scale contact admits many mechanical failure > and corrosion points. Another is the need to accomodate three dimensional > expansion/contraction physics due to intrusion of water during freeze/thaw > temperature cycles. > > The Montreal dome applied an inside surface solution to this puzzle by > "pulling > the windowshades" on certain translucent skin panels receiving direct > sunlight. > Perhaps this technique can be paired with a suitable photo-optical phase > change> or dichroic material to minimize the greenhouse heating (and > subsequent lift!) > within large domes. There is a gel of some sort I once read about that does this (I'll have to look it up.) It can be sandwiched between two windows. It turns opaque, or translucent (I can't remember) when the temperature inside gets above what's desirable. The scale idea does have its problems. Perhaps some sort of sliding rubber buffer (much like the sealing around conventional doors, but moving) could take the abrasion and prevent leakage of air or water. The scales would overlap in all directions, much like shingles, thus accomadating (I know I spelled that one wrong) three dimentional expansion. I'm working on a model right now to work out any problems involved. If any of you notice any now, please post. Steve Mather ========================================================================= Date: Fri, 8 Jul 1994 19:51:17 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: scimatec5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: Deriving PI without trig In article <199407080948.CAA11846@teleport.com>, Kirby Urner writes: > RE: Deriving PI w/o trigonometry > > I've come up with an algorithm for deriving pi that uses > no trig, just pythagoras----- Much deleted > 3.141592653589793000 > > As you can see, I reach the limits of my computer's > accuracy (using this particular programming language) at > about 25 iterations. > > There's some specific geometric reasoning that led to this > algorithm of course, which involves starting with an > inscribed square (2 triangles) and successively bisecting > outer edges (hypotenuses) to create a series of smaller > and smaller similar triangles pushing into the unfilled > arc regions. As the triangles get smaller, their numbers > multiply exponentially, hence the fractal-like > (self-similar) nature of the algorithm. > > This method may have already been published many times, > but I derived it from scratch I'm proud to say. Correct me if I'm wrong (I haven't fully understood this discussion =) but isn't this different from the other calculations in that it's two dimensional rather than three? This I understand. The three-dimentional I do not. Perhaps I've been flirting with Plato too long. =) Steve Mather ========================================================================= Date: Sat, 9 Jul 1994 12:31:11 -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: Deriving PI without trig >In article <199407080948.CAA11846@teleport.com>, Kirby Urner > writes: >> RE: Deriving PI w/o trigonometry >> >> I've come up with an algorithm for deriving pi that uses >> no trig, just pythagoras----- > >Much deleted > >> 3.141592653589793000 >> > >Correct me if I'm wrong (I haven't fully understood this >discussion =) but isn't this different from the other >calculations in that it's two dimensional rather than three? > >This I understand. The three-dimentional I do not. >Perhaps I've been flirting with Plato too long. =) > > Steve Mather > > As I recall, the thread began with a polygon (not hedron) inscribed in a circle, with a question about how to use its increasing number of edges to generate pi, but without using trig functions. I haven't seen any "solid" geometric derivations proposed --it'd be overkill I think, just to get pi. My solution is to tile a circle with smaller and smaller triangles, like one of those Escher things with angels and devils shrinking ad infinitum towards the outer edge. I can compute the area of each successive generation of triangle, and their number, and add this area to the cummulative total, gradually approaching and area of pi (unit circle area = pi) in that way. Something similar could be done with tetrahedra I suppose. David Koski is a master of the self-similar tetrahedron fractal. He uses the golden ratio (phi -- not pi) to scale T modules. Phi-scaled T-mods of various sizes actually pack together to make cubes, icosahedra and other 5-fold symmetric solids. Even more shapes may be made if variant modules, each assembled from 6 of the 7 unique edges of the golden cuboid* are admitted to the phi-scaled building-block inventory. Yasushi Kajikawa of the Synergetics Institute in Japan has a competing module set for assembling 5-fold symmetric shapes, and a hypercard stack for the Macintosh to show how it works. Kajikawa's work was actually published in Scientific American (Japanese edition only) whereas David Koski's work is as yet unpublished. The literature of module sets, finding a minimum inventory of building blocks for assembling a wide variety of shapes (ala Fuller's Mite, Kyte, Syte discussion) is fairly large. There's that dome architect from Iceland who's into it, and that book on particle physics which tries to model quantum mechanics using polyhedra (Fuller pushed in this direction of course). I lose comprehension when the discussion tries to phase energy into the picture. I'm more comfortable with "pre-frequency" pure principle abstractions. But I digress. None of this is about pi per se, but about phi and the ability of tetrahedra (irregular) to assemble into various 5-fold regular shapes. Plus we have the A and B quanta for the 4-fold shapes: regular tetra, octa, rhombic dodeca, cubocta etc. -- Kirby Urner * Golden cuboid: a phi-scaled brick, with face diagonals and the body diagonal giving additional unique edges ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) Portland, Oregon ========================================================================= Date: Sun, 10 Jul 1994 00:23:23 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: scimatec5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: Deriving PI without trig In article <199407091932.MAA02473@teleport.com>, Kirby Urner writes: >>In article <199407080948.CAA11846@teleport.com>, Kirby Urner >> writes: >>> RE: Deriving PI w/o trigonometry >>> >>> I've come up with an algorithm for deriving pi that uses >>> no trig, just pythagoras----- >> >>Much deleted >> >>> 3.141592653589793000 >>> > >> >>Correct me if I'm wrong (I haven't fully understood this >>discussion =) but isn't this different from the other >>calculations in that it's two dimensional rather than three? >> >>This I understand. The three-dimentional I do not. >>Perhaps I've been flirting with Plato too long. =) >> >> Steve Mather >> >> > > As I recall, the thread began with a polygon (not hedron) > inscribed in a circle, with a question about how to use > its increasing number of edges to generate pi, but without > using trig functions. I haven't seen any "solid" geometric > derivations proposed --it'd be overkill I think, just to get pi. I was more lost than I thought. =) > My solution is to tile a circle with smaller and smaller > triangles, like one of those Escher things with angels and > devils shrinking ad infinitum towards the outer edge. I can > compute the area of each successive generation of triangle, > and their number, and add this area to the cummulative total, > gradually approaching and area of pi (unit circle area = pi) > in that way. The picture that I had of yours (albiet I was guessing =) was of successive triangles based from the center. First you would take the square, get the ratio of its edges to its diameter. Then make an octagon and get its ratio. Then a 16 sided figure, and so on until infinity (or until your computer gives up.) I'll have to go back and read those old posts I guess.... =) I haven't done any programming, so I had no idea what you were doing there. Steve Mather ========================================================================= Date: Sun, 10 Jul 1994 00:39:59 GMT Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: scimatec5@UOFT02.UTOLEDO.EDU Organization: University of Toledo Subject: Re: Developing an operational pi In article <01HEH1JNV9VM9APL78@delphi.com>, "" writes: > (part of post deleted) > > Chris writes... > >>The trial-by-error method does offer a rational way to calculate your >>height of the equilateral triangle - but it's too slow for an impatient >>world. And nature doesn't care about its height - she just "expands" >>the consequences of gravity and radiation. What you're speaking about with height is also called distance. Distance is never wholly "rational" not only because it's dynamic, but also because it doesn't move in whole number incremants. It moves in fluidly from one place to another and back, etc.. > In a narrowly defined category of problems, this is true. In other > contexts, it is irrelevant; but then Synergetics isn't concerned with other > contexts - or is it? (I don't like the apparently arbitrary casting-off of > entire problem domains that, well, most Synergetic-accounting discussions > tend to take on. That's not a flame but an observation. If someone asks, > 'What is the length of a right-triangle's hypotenuese," the synergeticist' > response would be 'It has no structural meaning, and is therefore > irrelevant.' But this is not valid line of reasoning, because clearly these > relationships do have significance at times.) > >> Also it seems possible that >>mathematics was developed specifically to solve arbitrary problems. >>Maybe synergetics is more for understanding Nature and solving >>anticipatory design science problems. In sum I think that mathematics >>may always be the better tool for arbitrary calculations (the height >>of some figure such as the equilateral triangle). > > Not all problems are rationally tractable, so I think this is true. But > mathematics was not 'developed specifically' (no intonation intended, just > using your words) so I think it could not have been 'developed specifically > to solve arbitrary problems'. Mathematics is an outgrowth of the human > mind; the human mind is not entirely rational, therefore mathematics is not > entirely rational. Agreed. Steve Mather ========================================================================= Date: Sun, 10 Jul 1994 10:25:13 -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: Re: Deriving PI without trig > >The picture that I had of yours (albiet I was guessing =) was >of successive triangles based from the center. First you would >take the square, get the ratio of its edges to its diameter. >Then make an octagon and get its ratio. Then a 16 sided figure, >and so on until infinity (or until your computer gives up.) >I'll have to go back and read those old posts I guess.... =) >I haven't done any programming, so I had no idea what you were >doing there. > > Steve Mather > I didn't give a very complete description since I know it can be frustrating to do geometry in ASCII. I'll give it a whirl though. Inscribe a square in a circle. Now imagine the mid-edges of the square moving out to touch the circle, making 4 triangles using each of the square's edges as a base. Now have the outer mid-edges of those new triangles move out to the circumference again, making more, smaller triangles. Repeat until the computer runs out of significant digits. This is not a picture of a pie with narrower and narrower slices, all converging at the center. It's a pie with big sections at the center and smaller and smaller ones pressing out towards the edge of the circle. I'm going to try distilling my little algorithm down into a simpler summation, using a SIGMA symbol. I'll post it soon. BTW: my wife, teenage step daughter, and 1-month-old infant and myself are flying to Lesotho, Africa in a week. I friend Nick, really into Fuller and a lot of other folk (John Cage, Bohm, Krishnamurti) will be manning the workstation in my absence. If he can figure how to work the email (I've done a training and left detailed instructions), he should be able to participate in the GEODESIC discussions. ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) Portland, Oregon ========================================================================= Date: Sun, 10 Jul 1994 15:25:32 -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: Deriving PI (trigless algorithm) RE: algorithm for pi using + - / * and powering only (no trig functions). I've further simplified, or at least re-expressed, an algorithm for generating pi without using trig functions. Unfortunately, ASCII makes the expressions look more complicated than they are. Those interested should rewrite using conventional notation. [] means subscript. ^ means "raised to power" e.g. 2^.5 means "2 to the one-half" or "2nd root of 2". SIGMA means one of those greek summation symbols (just a fancy symbol for a programmer's DO-loop)... (1) h[0]=2 (2) h[i+1]=(2-(4-h[i]^2)^.5)^.5 (3) pi=SIGMA{(2^i)*h[i]*(1-1/2*(4-h[i]^2)^.5)} where (i=0,1,2...) In other words, you start with h[0]=2, then plug that in to the left side of expression (2) to get h[1], plug h[1] in to get h[2] and so on. Expression (3) is a summation of terms indexed on i where i=0,1,2,3,4... and so on, as long as you want to continue. Note the term 2^i -- a successive doubling with each new term in the series, reflective of the doubling number of smaller and smaller triangles, the area of which is provided my the next two terms (a base*height expression). I've also simplified the computer program a bit: area = 0 hypot=2 FOR n=0 TO 25 height=1-(1-(hypot/2)^2)^0.5 area=area+2^n*height*hypot hypot=(height^2+(hypot/2)^2)^0.5 ENDFOR ? area After 25 iterations, area should = pi to 15 decimals. --------------------------------------------------------- (c) Kirby Urner, 1994 ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) Portland, Oregon ========================================================================= Date: Sun, 10 Jul 1994 19:56:30 -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: Deriving PI without trig X-cc: rich@cpp.pha.pa.us In-Reply-To: from "Kirby Urner" at Jul 9, 94 12:31:11 pm >From Kirby Urner's message of 7/9/94: > > As I recall, the thread began with a polygon (not hedron) > inscribed in a circle, with a question about how to use > its increasing number of edges to generate pi, but without > using trig functions. I haven't seen any "solid" geometric > derivations proposed --it'd be overkill I think, just to get pi. > > My solution is to tile a circle with smaller and smaller > triangles, like one of those Escher things with angels and > devils shrinking ad infinitum towards the outer edge. I can > compute the area of each successive generation of triangle, > and their number, and add this area to the cummulative total, > gradually approaching and area of pi (unit circle area = pi) > in that way. > > Something similar could be done with tetrahedra I suppose. > David Koski is a master of the self-similar tetrahedron fractal. > He uses the golden ratio (phi -- not pi) to scale T modules. > Phi-scaled T-mods of various sizes actually pack together > to make cubes, icosahedra and other 5-fold symmetric solids. > Even more shapes may be made if variant modules, each assembled > from 6 of the 7 unique edges of the golden cuboid* are admitted to > the phi-scaled building-block inventory. > > Yasushi Kajikawa of the Synergetics Institute in Japan has > a competing module set for assembling 5-fold symmetric shapes, and > a hypercard stack for the Macintosh to show how it works. > Kajikawa's work was actually published in Scientific American > (Japanese edition only) whereas David Koski's work is as yet > unpublished. > > The literature of module sets, finding a minimum inventory of > building blocks for assembling a wide variety of shapes (ala > Fuller's Mite, Kyte, Syte discussion) is fairly large. There's > that dome architect from Iceland who's into it, and that book > on particle physics which tries to model quantum mechanics > using polyhedra (Fuller pushed in this direction of course). > I lose comprehension when the discussion tries to phase energy > into the picture. I'm more comfortable with "pre-frequency" > pure principle abstractions. > > But I digress. None of this is about pi per se, but about phi > and the ability of tetrahedra (irregular) to assemble into > various 5-fold regular shapes. Plus we have the A and B quanta > for the 4-fold shapes: regular tetra, octa, rhombic dodeca, > cuboctahedron ... --Kirby Urner Greeting, Kirby-- I thought I would share part of an article ("Cosmic Noise") by George Johnson in today's New York Times (7/9/94): In trying to construct a science of science, people like Dr. Chaitin and Dr. Landauer are questioning some of the deepest assumptions of their craft. Since Newton, scientific laws have been expressed in the form of differential equations, which have exact solutions, and with the so-called real numbers, which can be expressed as infinitely long decimal expansions. Pi equals 3.14159 .... In practice, science inevitably falls short of this ideal of infinite precision. In quantum physics, the simplest atom--hydrogen, with one proton and one electron--can be described precisely. But the equation for the helium atom, with its additional proton--is intractable. We must make do with good approximations. Estimates of the size of the shards of the Shoemaker-Levy comet vary so widely that some scientists predict there will be no measureable impact on Jupiter at all. Science has long operated on the assumption that space is continuous, with infinitely many points between two marks on a line. Mathematicians have calculated pi beyond a billion decimal places. But 61 decimal places are enough to describe a circle girding the visible universe with a deviation of less than a single Planck length--a unit 10-to the twentieth power (1 followed by 20 zeroes) times smaller than a proton. this seems as close to perfectly circular as a real circle can be. Do the rest of the decimal places have any meaning? The mathematician Herman Weyl once said that the belief in an infinite continuum of numbers "taxes the strength of our faith hardly less than the doctrines of the early Fathers of the Church or the Scholastic philosophers of the Middle Ages." Few scientists are ready to abandon differential equations and real numbers for the more realistic mathematics Dr. Chaitin is proposing. but in seeking a foundation for science, everything is up for grabs, including the universality of mathematics. For centuries philosophers have debated whether mathematics is invented or discovered. Taking a middle ground, the 19th-century mathematician Leopold Kronecker declared, "God made the integers; all else is the work of man." Einstein, it seems, went even further. Even the integers, he wrote, are obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences." I found the above excerpt interesting, as is the work of Chaitin. Both Ed Applewhite and myself have attended lectures given by Chaitin in recent years. --Kiyoshi Kuromiya kiyoshi@cpp.pha.pa.us Critical Path Project BBS (215) 463-7160 Fuller Information Xchange expanding to 8 phone lines, all 14.4 k bps 2.6 gigabytes storage ========================================================================= Date: Sun, 10 Jul 1994 21:37:12 -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: LOOKING FOR SYNERGETICS II I have been looking for a copy of Synergetics II for several years. I have tried Booklook and other book search organizations to no avail. Any clues or pointers would be greatly appreciated. Having recently discovered this newsgroup, I am delighted to find an ongoing discussion of Fuller's ideas and hope to make some contribution soon. Richard Hawkins ========================================================================= Date: Sun, 10 Jul 1994 20:24:00 -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: Deriving PI without trig >From Kiyoshi Kuromiya >Science has long operated on the assumption that space is continuous, with >infinitely many points between two marks on a line. Mathematicians have >calculated pi beyond a billion decimal places. But 61 decimal places are >enough to describe a circle girding the visible universe with a deviation >of less than a single Planck length--a unit 10-to the twentieth power >(1 followed by 20 zeroes) times smaller than a proton. this seems as close >to perfectly circular as a real circle can be. Do the rest of the >decimal places have any meaning? > >The mathematician Herman Weyl once said that the belief in an infinite >continuum of numbers "taxes the strength of our faith hardly less than the >doctrines of the early Fathers of the Church or the Scholastic philosophers >of the Middle Ages." > > >I found the above excerpt interesting, as is the work of Chaitin. Both >Ed Applewhite and myself have attended lectures given by Chaitin in recent >years. > >--Kiyoshi Kuromiya > kiyoshi@cpp.pha.pa.us > Critical Path Project BBS (215) 463-7160 > Fuller Information Xchange > expanding to 8 phone lines, all 14.4 k bps > 2.6 gigabytes storage Excellent citation!! Let's get some of this in a next version of the GEODESIC FAQ, how 'bout. Note: I posted my pi algorithm to sci.math. Respondants are saying Archimedes did the same thing, but they're thinking I'm doing the n-gon thing, with narrower and narrower slices all converging at the center. But that's not what I'm doing. ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) Portland, Oregon ========================================================================= Date: Mon, 11 Jul 1994 00:09:40 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: Bucky Fuller FAQ Greetings All, Since last month I have added very, very little new material to the FAQ. I have, however, converted the FAQ to SGML. This means that I can generate ASCII, nroff (actually gtroff), postscript, LaTeX (the best, of course :), and html (!!!!) output. The ASCII form looks OK, but I need to examine the source code and make modification (hoo boy). I haven't got Mosaic up yet (I don't have an internet link yet - though I can run Mosaic under X Windows to take a gander at it). If anyone would like me to e-mail the html formated version, let me know (it may take a week or so to do, I've read the manual but that never seems to be enough :) Does any one want to offer an ftp site for the LaTeX and/or Postscript versions (postscript doesn't work yet, but I think the problem will subcumb to a small effort - LaTeX is GREAT). I've been unable to access Critical Path Project for two weeks (modem trouble and etc. :( ) So I was unable to get the FAQ approved by news.answers. I want to post the FAQ here for any newcomers, but I'm slightly disappointed by my accomplishments on this matter :( Maybe next week? On the bright side I have rendered several tensegrities and will be uuencoding them and posting them soon (In fact, one just finished rendering now - looks rather fantastic - but my tomorrow morning I'll probably be redoing it - that never ending search for perfection - ya' know :) BTW, GIF format seems too dark. I will do some experiments with jpeg. Does everyone have jpeg viewers? I have utilities to get nearly any format you need. Let me know. Until the problems with Critical Path are resolved You should e-mail me at either cfearnl@pacs.pha.pa.us or this address fearnlcj@duvm.ocs.drexel.edu. Do Enjoy! ========================================================================= Date: Sun, 10 Jul 1994 21:39:30 -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: Bucky Fuller FAQ >Greetings All, > >On the bright side I have rendered several tensegrities and will be >uuencoding them and posting them soon (In fact, one just finished >rendering now - looks rather fantastic - but my tomorrow morning I'll >probably be redoing it - that never ending search for perfection - >ya' know :) BTW, GIF format seems too dark. I will do some experiments >with jpeg. Does everyone have jpeg viewers? I have utilities to get >nearly any format you need. Let me know. Got my uudecoder and jpeg viewer dusted off and ready for action. Upload away!! ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) Portland, Oregon ========================================================================= Date: Mon, 11 Jul 1994 04:46:00 -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: Algorithm for PI (continued) Algorithm for PI ...continuation h[0]=2 h[1]=SQRT(2) h[2]=SQRT(2-SQRT(2)) h[3]=SQRT(2-SQRT(2+SQRT(2)) h[4]=SQRT(2-SQRT(2+SQRT(2))) h[5]=SQRT(2-SQRT(2+SQRT(2+SQRT(2)))) ... h[n]=SQRT(2-SQRT(2+SQRT(2+SQRT(2+...)))))... The above succession of terms derives from h[0]=2 h[i+1]=SQRT(2-SQRT(4-h[i]^2)) Another way of expressing the continued radical: i=0...n k[0]=0 h[0]=2 k[i+1]=SQRT(2+k[i]) h[i+1]=SQRT(2-k[i]) This "continued radical" (like a continued fraction) feeds terms into the summation: SIGMA{(2^i)*h[i]*SQRT(1-1/2*(4-h[i]^2))} to give pi. (c) Kirby Urner ========================================================================= Date: Mon, 11 Jul 1994 06:14:05 -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: LOOKING FOR SYNERGETICS II Richard Hawkins writes: >I have been looking for a copy of Synergetics II for several years. I ran across it (and a few other Fuller titles) last week in a used bookstore in Vancouver, Canada. Perhaps they'd be willing to mail it to you... in exchange for you VISA number, ov course. ;) Ashley's Books 3754 W 10th Ave Vancouver, BC Canada (604) 228-1180 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Mon, 11 Jul 1994 06:58:52 -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: Bucky Fuller FAQ WRT Chris Fearnley's FAQ post, "It was Greek to me". >...SGML...nroff...gtroff... LaTeX...html...Mosaic...GIF...jpeg. [Though Postscript and ASCII I've heard of.] Perhaps someone can tell me which, if any, format from this list is appropriate for a Mac user. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ========================================================================= Date: Mon, 11 Jul 1994 09:55:00 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "H. Jeffrey Rosen" Subject: Re: LOOKING FOR SYNERGETICS II Richard Hawkins writes: >I have been looking for a copy of Synergetics II for several years. I ran across it (and a few other Fuller titles) last week in a used bookstore in Vancouver, Canada. Perhaps they'd be willing to mail it to you... in exchange for you VISA number, ov course. ;) Ashley's Books 3754 W 10th Ave Vancouver, BC Canada (604) 228-1180 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Lee Wood | Lee_Wood@sfu.ca | INTJ spoken here. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= You could also inquire of the Fuller Institute in California. They once had a pretty respectable stockpile of Bucky's works, in new condition. Jeff Rosen hj.rosen@srs.gov ========================================================================= Date: Mon, 11 Jul 1994 10:42:28 -0400 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: "Vincent J. Matsko" Organization: Sponsored account, Mathematics, Carnegie Mellon, Pittsburgh, PA Subject: Re: Deriving PI without trig In-Reply-To: <199407091932.MAA02473@teleport.com> With regard to Kirby Urner's post of 9 July - Is there any more that you can offer about Koski's and Kajikawa's set of modules from which solids with icosahedral symmetry may be constructed? In particular, any specific references? Also, is the Japanese Sci. Am. article available in English? -Vince Matsko ========================================================================= Date: Mon, 11 Jul 1994 19:00:35 -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: Deriving PI without trig >With regard to Kirby Urner's post of 9 July - Is there any more that you >can offer about Koski's and Kajikawa's set of modules from which solids >with icosahedral symmetry may be constructed? In particular, any >specific references? Also, is the Japanese Sci. Am. article available >in English? > >-Vince Matsko > > David Koski and I are friends and have worked together on his modules. I attended Kajikawa's Synergetica workshop at UCLA and learned things about his system, but don't know it as well as I know Koski's. I don't know for sure whether the Japanese Sci. Am article is available in English. The issue has dinosaurs on the cover (velociraptors?) -- same as the USA version. ------------------------------------------------ Kirby T. Urner pdx4d@teleport.com 4D Solutions (teleport.com is a public access node) Portland, Oregon ========================================================================= Date: Tue, 12 Jul 1994 01:16:03 -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: curVE I have made a model based on the Vector Equilibrium using quadrants (90 degree arcs) in place of straight lines. Visualize a cube with circular faces. All of the circumferentials (no radials in this model) are equidistant from the center of gravity; facilitating motion. View the model as 4 groups of 6 quadrants each forming "circuits" analogous to the 4 hexagonal components of the VE (cuboctahedron). Locate a rotating armature (straight-line structure) with its' pivot point at the center of gravity and ends at opposing points on one of the 4 "circuits". (I have used 4 different colors to help differentiate these in the model.) Animate the armature to make a complete revolution (keyframes at the beginning of each quadrant) around each "circuit" , alternating continously through the 4 different axes of rotation (4-D). Grouping another armature at 90 degrees centered to the first produced a surprising (to me) result. For each revolution around a "circuit" by the first armature, the grouped armature tracks opposing spherical triangles twice. It bobs and weaves! Sorry if this verbal description is not easy to visualize. I think I could upload the animation if someone could give me formatting and addressing instructions. Questions and comments welcome. Richard Hawkins ========================================================================= Date: Tue, 12 Jul 1994 07:24:06 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: Bucky Fuller FAQ In-Reply-To: Message of Mon, 11 Jul 1994 06:58:52 -0700 from On Mon, 11 Jul 1994 06:58:52 -0700 Lee Wood said: >WRT Chris Fearnley's FAQ post, "It was Greek to me". > >>...SGML...nroff...gtroff... LaTeX...html...Mosaic...GIF...jpeg. > >[Though Postscript and ASCII I've heard of.] For the FAQ you want to be able to read/view or print LaTeX or Postscript (my personal favorite is LaTeX, but postscript is easier for many). LaTeX is a free formatting program. There is a port to DOS (of course, it's on UNIX). I just checked out the Tex/LaTeX FAQ and here is a quote: OzTeX is a public domain version of TeX for the Macintosh. A DVI Previewer and PostScript driver are also included. It should run on any Macintosh Plus, SE, II, or newer model, but will not work on a 128K or 512K Mac. It was written by Andrew Trevorrow, and is available via anonymous ftp from from midway.uchicago.edu (128.135.12.73) in ./pub/OzTeX, which contains other public domain TeX-related software for the Mac as well, or on a floppy disk from TUG (see question 11). Questions about OzTeX may be directed to oztex@midway.uchicago.edu. TUG is the TeX Users Group. TUGboat is their newsletter, containing useful articles about TeX and METAFONT. TUG also distributes TeX-related microcomputer software on disks. Inquiries should be directed to: TeX Users Group P. O. Box 869 Santa Barbara, CA 93102-0869 (USA) 805-963-1338 FAX: 805-963-8358 tug@tug.org html is the hypertext formatting system used by Mosaic, the premier WWW (World Wide Web) viewing system on the internet (if you don't have full internet access this isn't too exciting). SGML is the formatting program I'm using in order to be able to convert to all the other formats. It is useless unless you need this high level of flexibility. nroff, troff gtroff, etc. are the UNIX text processing system (useless for Mac users). GIF and JPEG are two popular formats for graphics files. I've decided to use JPEG (unless someone complains) because on my system X Windows JPEG looks much better than GIF (I don't know why but it could be some flaw in GIF format - it is the older format ya' know). Please consult the alt.binaries.pictures FAQ it contains information on several JPEG viewers for the Mac and all platforms. I'd quote them here, but being ignorant on Macs I don't know if system 6 is popular or not. There are several options. The biggest problem for non-Unix types is uuencode/uudecode. Since mail is an ascii-only thing, binary picture files are a problem. uuencode converts a binary file into ASCII jibberish which uudecode then turns into a binary again. There are three programs for the Mac that do this: UUlite, UUCat, and UUTool. See above FAQ for how to get them. > >Perhaps someone can tell me which, if any, format from this list >is appropriate for a Mac user. Any software I use should have a Mac/Unix/Amiga/DOS/etc. capability (well, almost any :) Certainly if I post it to the group, anyone should be able to view/use it. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= >Lee Wood | >Lee_Wood@sfu.ca | INTJ spoken here. >=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= BTW, let's end this thread here. Get the alt.binaries.pictures FAQ to learn about the graphics viewing (and etc.) info for your platform. See your local user's group if you can't get it. E-mail me if all else fails. Do Enjoy! Chris Fearnley ========================================================================= Date: Tue, 12 Jul 1994 08:16:32 -0400 Reply-To: "Shelly R. Mault" Sender: List for the discussion of Buckminster Fuller's works From: "Shelly R. Mault" Subject: Re: DATABASE LIST DATABASE LIST ========================================================================= Date: Tue, 12 Jul 1994 09:23:00 -0400 Reply-To: "Shelly R. Mault" Sender: List for the discussion of Buckminster Fuller's works From: "Shelly R. Mault" Subject: Re: DATABASE LIST DATABASE LIST ========================================================================= Date: Tue, 12 Jul 1994 12:55:00 -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 pdx4d@teleport.com Kirby Urner" Subject: Re: The Icosahedral Projection (& ancient cartography) This thread began in sci.math. Gets to the heart of what mathematics algorithms underlie the Dymaxion Projection. Kirby Urner writes: > > In article <2vs64v$av@gaia.ucs.orst.edu> sahr@thuja.FSL.ORST.EDU (Kevin Sahr) writes: > >From: sahr@thuja.FSL.ORST.EDU (Kevin Sahr) > >Subject: Re: The Icosahedral Projection (& ancient cartography) > >Date: 11 Jul 1994 19:22:07 GMT > > >In article <2vqmu9$3to@omnifest.uwm.edu> mark@omnifest.uwm.edu (Mark Hopkins) > >writes: > >> > >...history/motivation deleted... > >> > >>(2) The Icosahedral Projection > >> This is a projection I discovered (rediscovered?) about 8 years ago. It > >>consists of 20 triangular plates that can be arranged in a variety of ways. > >>To date (to the best of my knowledge) it is the only reconfigurable > >>projection. > >> > >...description deleted... > >> > > >This projection you've discovered is extremely similar to R. Buckminster > >Fuller's Dymaxion Airocean World Map in both motivation and conception, > >though subtly (to me, at least!) different in execution. Bucky's projection > >(which also individually projects each triangle of the spherical icosahedron) > >has the advantage that all great circle arcs parallel to any of the edges > >of a given icosahedron triangle are straight lines on the planar triangle, > >and distances along these arcs are preserved on the planar triangle. It > >has the disadvantage (big, big :( here!) that it does not seem to be > >mathematically well-defined. > >Kevin > > I believe the Fuller projection is mathematically well defined. The faces > of an icosa are subdivided into similar equilateral triangles, which are > pushed outward along radii from the sphere center to the surface (orthagonal > projection). The mathematics for doing this, same as for the domes, is > mathematically expressed and computer-implemented. And yes, more > work needs to be done to popularize this map and its methods. I don't believe your description of the projection method is correct; I don't think, for instance, that what you're saying (assuming I get your drift) would preserve distances along the great-circle arcs. I think what Fuller did was a bit more subtle than that; again I refer you to his "steel- straps and straws" illustration which appears in many of his books. If you have any references or code for doing the Fuller projection I would be very interested in seeing it. The information I have is from an unpublished paper by Robert W. Gray of IBM, "Fuller's Dymaxion Map." In it he recounts how Fuller developed what he called a "three-way great circle grid" to use as a reference system for manually transcribing points off of a globe onto a Dymaxion Map, and this is the system which appears in Fuller's 1946 patent of the Dymaxion Map. However, before his death Fuller realized that when this grid was projected to the plane the intersections of the arcs did not form points, but little triangles (_Cosmography_, pg. 236). Gray's version of the projection suggests taking the average of the location of the vertexes of these little triangles to use as the projected point location. One of the things we are exploring is how this averaging affects the properties of the projection at various scales. I do believe that the projection could be implemented "precisely" _to an arbitrary degree of precision_ by recursively sub-dividing the spherical triangle until a point of interest lies within the specified precision of one of the sub-triangle vertices and then using the corresponding vertex on the sub-divided planar triangle as the position of the planar location of the point. But I need to spend more time looking for an analytic method of accelerating this procedure before it would be sufficiently efficient for our use. But, I am more than open to being proved wrong about the mathematical nature of Fuller's projection! If you have more information I would appreciate hearing about it. Kevin -- //////////////////////////////////////////////////////////////////////// // // // Kevin Sahr Forestry Sciences Laboratory // // Research Associate/Programmer 3200 SW Jefferson Way // // Department of GeoSciences Corvallis, OR 97331 // // Oregon State University // // voice: (503) 750-7492 // // kevin@geochelone.fsl.orst.edu fax: (503) 750-7329 // // // //////////////////////////////////////////////////////////////////////// ========================================================================= Date: Tue, 12 Jul 1994 20:48:12 -0500 Reply-To: List for the discussion of Buckminster Fuller's works Sender: List for the discussion of Buckminster Fuller's works From: Richard Hendricks Organization: Kansas State University Subject: Re: Deriving PI without trig Well, as far as deriveing PI without trig, how is deriving it without a circle? :) My old calulus book (Pub. by Heath/Written by Gillett) has a quote by Lord Kelvin: A mathematician is one to whom that [Referring to the Integral of e^(-x^2)dx taken from 0 to infinity equals SQR(pi)/2] is as obvious as that twice two makes four is to you. -- Richard Hendricks, Kansas State University, Manhattan, KS Computer Engineering, KSU Consultant hendric@ksu.ksu.edu ========================================================================= Date: Wed, 13 Jul 1994 04:13:23 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: Way-Cool JPEG of a 6-strut Tensegrity! I designed this with POV-Ray, the copyrighted freeware raytracing program available as binaries for Mac, Amiga, DOS/Windows and Linux. It rendered in 58 minutes on my 486-33 Linux Workstation. I converted the 900K targa file to JPEG (JFIF) with the PBMPLUS toolkit (also freeware). Let me know what you think of it. The mathematical formulas were originally derived by John Kirk. I have the POV-Ray source and an awk script to generate a whole class of 6-strut Tensegrities that I can post. I originally wrote the awk script to animate it, but that may not happen soon :( I hope everyone gets this alright. tenseg-6.jpg (37717 bytes) Checksums: (Using GNU sum: sum -r tenseg-6.jpg; sum --sysv tenseg-6.jpg) 03426 37 21371 74 tenseg-6.jpg See the alt.binaries.pictures FAQ for info on where to obtain a uudecoder and a JPEG viewer if you don't already have them. 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