In its broadest sense *synergetics* is Fuller's hypothesized
coordinate system of Universe --- both in its physical and metaphysical
aspects. Fuller's system of epistemography and mathematical-physics
attempts to disclose how Nature actually operates --- her ``operational
mathematics.'' Fuller claimed that synergetics could be understood
by children (though they probably couldn't comprehend his books on the
subject). He published this material in his essay ``Omni-directional
Halo'' (in *No More Secondhand God*), *Synergetics: Explorations
in the Geometry of Thinking*, *Synergetics 2: Further Explorations
in the Geometry of Thinking*, and *Cosmography*. *Cosmography*
is probably the easiest to read for people unfamiliar with Fuller's
prose style. An ``interleaved'' version of *Synergetics*
and *Synergetics 2* is available on the Web at
http://www.rwgrayprojects.com/synergetics/synergetics.html.

From my own study of synergetics, I'm convinced that Bucky did in fact identify the coordinate system used by Nature. But I would add the caveat that he didn't get too far along in developing it. Fuller points to what the coordinate system is in broad strokes. He gives many penetrating insights and new discoveries, but the synergetics coordinate system needs a lot more development (and integration) before it will be possible to use it as the operative model in all of Science.

Tip to students of synergetics: Build models.

The essay
Reading Synergetics: Some Tips
offers useful help for anyone struggling to read *Synergetics*.

`[From Kirby Urner]`

**Synergetics**: A metaphoric language for communicating experiences using
geometric concepts.

Thinking is the tuning in/out of systems. Systems are spherical networks of interrelated points of interest. The density of points is a measure of a system's ``frequency'' -- super high frequency systems approach sphericity.

The minimal system with the fewest possible points is a tetrahedron -- four points make a primitive volume with an inside and an outside. The canonical tetrahedron has a volume of one.

The tetrahedron may be sliced into 24 irregular tetrahedra (12 left handed, 12 right handed) called ``A modules.'' The octahedron is comprised of 48 A and 48 B modules of equal volume = 4 x the volume of the tetrahedron. A and B modules may be used to assemble the cube (3 tetravolumes), rhombic dodecahedron (6 tetravolumes), and the Coupler (1 tetravolume). The Coupler, with the same volume as the tetrahedron (1), is an irregular octahedron that packs together to fill space without gaps.

Radiation is explosive outwardly while Gravitation is an implosive squeezing at 90 degrees to Radiation, i.e. is circumferential. Metaphysically, Gravity networks points of interest into systems of interrelated thoughts while Radiation drains away the sense of our systems and turns them into meaningless noise. Radiation is compression, Gravity is tension. Radiation is Entropy. Gravity is Love. Clearly this is not Physics but a more metaphorical language for communicating experiences using geometric concepts. This is Synergetics.

`[`

Ed: I think Fuller's synergetics describes real physics. Though it is
true as Kirby points out that Bucky's presentation is more
``descriptive'' than ``hard'' physics. I contend that because Fuller is
``right on'' in his description, it is up to us to find the ``hard'' physics
interpretation behind his ``metaphors.''`]`

`[Typed in by Kurt Przybilla]`

```
[From
Synergetics [900.20-900.33]]
```

900.20 Synergetics

900.21 Synergetics is a book about models: humanly conceptual models; lucidly conceptual models; primitively simple models; rationally intertransforming models; and the primitively simple numbers uniquely and holistically identifying those models and their intertransformative, generalized and special case, number-value accountings.

900.30 Model vs. Form

900.31 Model is generalization; form is special case.

900.32 The brain in its coordination of the sensing of each special case experience apprehends forms. Forms are special case. Models are generalizations of interrelationships. Models are inherently systemic. Forms are special case systems. Mind can conceptualize models. Brains can apprehend forms.

900.33 Forms have size. Models are sizeless, representing conceptuality independent of size.

- Bucky, *Synergetics 2*

Bucky went far in describing experience in terms of the experientially derived model, discovering along the way the synergetically surprising benefits to build structures based upon the special case structures designed using the generalized principles understood using this evermore useful mentality.

```
[From
Synergetics [200.001-201.03]]
```

200.001 Definition: Synergetics

200.01 Synergetics promulgates a system of mensuration employing 60-degree vectorial coordination comprehensive to both physics and chemistry, and to both arithmetic and geometry, in rational whole numbers.

200.02 Synergetics originates in the assumption that dimension must be physical; that conceptuality is metaphysical and independent of size; and that a triangle is a triangle independent of size.

200.03 Since physical Universe is entirely energetic, all dimension must be energetic. Synergetics is energetic geometry since it identifies energy with number. Energetic geometry employs 60-degree coordination because that is nature's way to closest-pack spheres.

200.04 Synergetics provides geometrical conceptuality in respect to energy quanta. In synergetics, the energy as mass is constant, and nonlimit frequency is variable.

200.05 Vectors and tensors constitute all elementary definition.

201.00 Experientially Founded Mathematics

201.01 The mathematics involved in synergetics consists of topology combined with vectorial geometry. Synergetics derives from experientially invoked mathematics. Experientially invoked mathematics shows how we may measure and coordinate omnirationally, energetically, arithmetically, geometrically, chemically, volumetrically, crystallographically, vectorially, topologically, and energy-quantum-wise in terms of the tetrahedron.

201.02 Since the measurement of light's relative swiftness, which is far from instantaneous, the classical concepts of instant Universe and the mathematicians' instant lines have become both inadequate and invalid for inclusion in synergetics.

201.03 Synergetics makes possible rational, whole-number, low-integer quantation of all the important geometries of experience because the tetrahedron, the octahedron, the rhombic dodecahedron, the cube, and the vector equilibrium embrace and comprise all the lattices of all the atoms.

-Bucky, *Synergetics*

The isotropic vector matrix provides a model for thinking - for thought - a model more flexible than the squarebox X-Y-Z cubist mindframe, a clearly defined mathematical and physical model, an organic matrix based on the closest packing of spheres, bubbles, atoms. A model which attempts to explain everything, much more that any lingually linear metaphor can ever manage.

It provides a structure in which to think about any structure or system. Whether you want to discuss why people first built dome shaped huts, why St. Peter's cathedral was the largest space man had domed until Bucky came along, why planets and stars are spherical, the structure of the atom, the structure of complex carbon molecules scientists worldwide are building, or dome homes we wish to build.

I dream of building many. The nicest is portable, made of the highest quality, light weight alloys, easily affordable and assembles almost anywhere on the planet by the average human and friends in about a day. It will utilize the best solar technology, all technology comprehensively integrated to improve living.

These are the three omni-triangulated, omni-symmetrical, stable, space structures in Universe. The tetra has 4 vertices (crossings), 6 edges (vectors) and 4 faces (openings). The octa has 6 crossings, 12 vectors, and 8 openings. The icosa has 12 crossings, 30 vectors, and 20 openings. The Greeks called these three figures ``platonic solids.'' They are very important in synergetics.

`[From Vincent J. Matsko]`

For those interested in group theory, from a group theoretical perspective, we can view the symmetry groups of the tetrahedron and the octahedron 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.

```
[From
Synergetics [101.01-102.00]]
```

``*Synergy* means behavior of whole systems unpredicted by the behavior of
their parts taken separately.

``*Synergy* means behavior of integral, aggregate, whole systems
unpredicted by behaviors of any of their components or subassemblies of
their components taken separately from the whole.''

`[From Blaine A. D'Amico.]`

Fuller's clearest example of ``behavior of whole systems unpredicted by the behavior of the parts'' is mass attraction. The Earth and the Moon maintain their relationship through an interattraction of their respective masses. This mass attraction (gravity being a special case of mass attraction) is a function of the mass of the two bodies AND THEIR DISTANCE FROM ONE ANOTHER. The scientific law governing this attraction states that if you halve the distance between the two bodies you quadruple the attraction and vice-versa (i.e. double the distance and the attraction is 1/4 the original). This generalized principle (the law of mass attraction) is a synergy because if either body is considered separately there is no attractive force to examine. The law of mass attraction is mathematically exact and exists only as a function of the whole system. It is therefore a Synergy.

```
[From
Synergetics [301.00-302.00]]
```

``*Universe* is the aggregate of all humanity's consciously apprehended and
communicated nonsimultaneous and only partially overlapping experiences.

`` `Aggregate' means sum-totally but nonunitarily conceptual as of any one moment. `Consciousness' means an awareness of the otherness. `Apprehension' means information furnished by those wave frequencies tunable within man's limited sensorial spectrum. `Communicated' means informing self or others. `Nonsimultaneous' means not occurring at the same time. `Overlapping' is used because every event has a duration, and their initiatings and teminatings are most often of different duration. Neither the set of all `experiences' nor the set of all the words used to describe them are instantly reviewable nor are they of the same length. Experiences are either involuntary (subjective) or voluntary (objective), and all experiences, both physical and metaphysical, are finite because each begins and ends.''

```
[From
Synergetics [410.06]]
```

``So I then went on to say if all the energy conditions were everywhere the same, then all the vectors would be the same length and all of them would interact at the same angle. I then explored experimentally to discover whether this `isotropic vector matrix' as so employed in matrix calculus, played with empty sets of symbols on flat sheets of paper, could be realized in actual modeling. ...'' He than describes his kindergarten discovery of the octet-truss (octahedrons + tetrahedrons in an all-space filling array).

```
[From
Synergetics [420.01]]
```

``When the centers of equiradius spheres in closest packing are joined by most economical lines, i.e., by geodesic vectorial lines, an isotropic vector matrix is disclosed -- `isotropic' meaning `everywhere the same,' `isotropic vector' meaning `everywhere the same energy conditions.' This matrix constitutes an array of equilateral triangles that corresponds with the comprehensive coordination of nature's most economical, most comfortable, structural interrelationships employing 60-degree association and disassociation. Remove the spheres and leave the vectors, and you have the octahedron-tetrahedron complex, the octet truss, the isotropic vector matrix.''

`[Mitch Amiano]`

An Octetruss, to use the trademarked moniker, is an OCtahedral and TETrahedral complementary grid implemented in such a way as to form a structural truss. A truss is an engineering mechanism for dispersing loads across a relatively long span, to enable coverage of large, primarily horizontal areas with a minimum of underpinning supports (posts). Most trusses appear to be arranged to act independently of one another, whereas the members of an Octetruss are all part of the whole unit. Octetruss is not the only such truss system. Other geometries have been used to create omnidirectional truss systems; the more general name for any one of these systems is a `space frame'.

It should be very well known that Alexander Graham Bell built
Octahedral/Tetrahedral trusses and used them for enormous kites and
flying machines. I saw some *very* old films of Bell and one of his
octet configuration kites; it seemed to be one of his hobbies.

`[Hal Adams]`

Most trade publications periodically have articles on space frames. You
might try ``Architecture'' a monthly publication of the American Institute
of Architects, ``Engineering News Record'' an engineering weekly. You can
check the ``Art Index'' which has an index of all articles from design
publications. A good general structural book is *Why Buildings Stand
Up* by Mario Salvadori, published by McGraw Hill paperbacks.

`[Bruce T. Lael]`

The following quote comes from Hugh Kenner's book *Bucky: A Guided
tour of Buckminster Fuller*, c.1973

``*...*What are we to make, for instance, of Alexander Graham Bell's
infatuation about the tetrahedron?

``About two years after little Bucky's adventure with the toothpicks and the peas, the veteran inventor of telephony perceived in the tetrahedron a figure of singular virtue. It is the three-dimensional equivalent of the triangle, holding its form with invincible tenacity. It is the minimum space enclosure, with four identical sides nothing simpler can be envisaged. Having of all space enclosures the maximum structure in proportion to its content, it has therefore the maximum attainable strength. Bell's mind moved to performance per pound and to aeronautics, and in the very summer before the Wrights flew he wrote in his son-in-law's National Geographic of the virtues of a tetrahedronal configuration in kites. Such a kite will not easily lose lift, and Bell's idea that the future of aeronautics lay in a design which wouldn't tend to kill the pilot in case of a stall led him to hundreds of experiments with kites composed of many tetrahedral cells, as many as 1300.

``In 1905, such a kite powered by a feeble breeze, lifted a man
some thirty feet into the air *...*.

``*...*He did erect, on his Nova Scotia island, a tetrahedronal tower,
its seventy-two foot legs meeting tripod fashion five stories above
the ground. Each leg was subdivided into four-foot tetrahedral
cells of half-inch pipe, and each cell could support two tons
without signs of distress. Bell had effected about 1907 one of the
periodical rediscoveries of the oc-tet configuration Bucky
stumbled onto in kindergarten, and moreover has used it in a
practical structure. He seems not to have applied for a patent
and the tetrahedronal tower was dismantled after a decade.
Bucky had very possibly never heard of it when he came upon the
principle yet again during his geometrical work of the 1940's
and wrote to his patent lawyer.''

`[Steve Mather]`

HCP allows infinite permutations as successive layers of spheres do not need to lie over top a specific hole in the layer 2 down. HCP is not a restrictive enough packing method to generate the IVM per se.

The IVM is equivalent to the face-centered cubic packing (FCC). I believe all diamond atoms occupy FCC positions, but leave others empty. Buckminsterfullerene will pack into an IVM (appropriately), and, with potassium wedged in some of the interstices, become a superconductor. (See section What are Fullerenes? and Buckyballs for more on the fullerenes.) --- Kirby

`[From Blaine A. D'Amico.]`

I use various sized Styrofoam (yes I know its not a green material) balls in my classes. They are very easy to work with.

`[From Chris Fearnley]`

Ping pong balls are wonderful. I use a tacky tape type stuff to bind them together (this helps me to dis-assemble and then re-assemble them into new shapes). I got about two gross to work with - very fun and educational. I prefer the sticky tape to glue, less messy, reversible and pliable to adjust for physical imperfections.

`[Charlie Hendricksen]`

Some years ago I took a delightful course in ``Patterns in Nature.'' We made models of the various crystal structures and geometric models using Duco cement and the plastic beads from bead chains from the import shop (Pier One). Cheap, the right size (about 5mm), and best of all many colors.

```
[From
Synergetics [205.01]]
```

``The geometrical model of energy configurations in synergetics is developed from a symmetrical cluster of spheres, in which each sphere is a model of a field of energy all of whose forces tend to coordinate themselves, shuntingly or pulsatively, and only momentarily in positive or negative asymmetrical patterns relative to, but never congruent with, the eternality of the vector equilibrium. The vectors connecting the centers of the adjacent spheres are identical in length and angular relationship. The forces of the field of energy represented by each sphere interoscillate through the symmetry of equilibrium to various asymmetries, never pausing at equilibrium. The vector equilibrium itself is only a referential pattern of conceptual relationships at which nature never pauses.''

`[`

Making this model will greatly ease understanding of the jitterbug
transformation described below. I use 6" dowels joined together with
surgical tubing. Cut the surgical tubing into 2" pieces. Use a washer
to form a four-valent, flexible vertex joining two pieces of the
surgical tubing or insert one piece of 2" tubing into a hole (you must
cut it yourself) in another 2" piece of tubing to create the vertex.
The diameter of the surgical tubing should be very slightly (1/16")
smaller in diameter than the dowels. You will need 24 such struts and
12 such vertices. Geometers call the shape of the VE a
``cuboctahedron.''`]`

```
[From
Synergetics [460.01-460.05]]
```

``The `jitterbug' is the finitely closed, external vector structuring of a vector-equilibrium model constructed with 24 struts, each representing the push-pull, action-and-reaction, local compression vectors, all of them cohered tensionally to one another's ends by flexible joints that carry only tension across themselves, so that the whole system of only-locally-effective compression vectors is comprehensively cohered by omniembracing continuous four sliced hexagonal cycles' tension.

``When the vector-equilibrium `jitterbug' assembly of eight triangles and six squares is opened, it may be hand-held in the omnisymmetry conformation of the vector equilibrium `idealized nothingness of absolute middleness.' If one of the vector equilibrium's triangles is held by both hands in the following manner - with that triangle horizontal and parallel to and above a tabletop; with one of its apexes pointed away from the holder and the balance of the jitterbug system dangling symmetrically; with the opposite and lowest triangle, opposite to the one held just parallel to and contacting the tabletop, with one of its apexes pointed toward the individual who is handholding the jitterbug - and then the top triangle is deliberately lowered toward the triangle resting on the table without allowing either the triangle on the table or the triangle in the operator's hands to rotate (keeping hands clear of the rest of the system), the whole vector equilibrium array will be seen to be both rotating equatorially, parallel to the table but not rotations its polar-axis triangles, the top one of which the operating individual is hand-lowering, while carefully avoiding any horizontal rotation of, the top triangle in respect to which its opposite triangle, resting frictionally on the table, is also neither rotating horizontally nor moving in any direction at all.

``While the equatorial rotating results from the top triangle's rotationless lowering, it will also be seen that the whole vector-equilibrium array is contracting symmetrically, that is, all of its 12 symmetrically radiated vertexes move synchronously and symmetrically toward the common volumetric center of the spherically chorded vector equilibrium. As it contracts comprehensively and always symmetrically, it goes through a series of geometrical-transformation stages. It becomes first an icosahedron and then an octahedron, with all of its vertexes approaching one another symmetrically and without twisting its axis.

``At the octahedron stage of omnisymmetrical contraction, all the vectors (strut edges) are doubled together in tight parallel, with the vector equilibrium's 24 struts now producing two 12-strut-edged octahedra congruent with one another. If the top triangle of the composite octahedron (which is the triangle hand-held from the start, which had never been rotated, but only lowered with each of its three vertexes approaching exactly perpendicularly toward the table) is now rotated 60 degrees and lowered further, the whole structural system will transform swiftly into a tetrahedron with it original 24 edges now quadrupled together in the six-edge pattern of the tetrahedron, with four tetrahedra now congruent with one another. Organic chemists would describe it as a quadrivalent tetrahedral structure.

``Finally, the model of the tetrahedron turns itself inside out and oscillates between inside and outside phases. It does this as three of its four triangular faces hinge open around its base triangle like a flower bud's petals opening and hinging beyond the horizontal plane closing the tetrahedron bud below the base triangle.''

`[From Blaine A. D'Amico.]`

For a full (and quite mind boggling) discussion of these Jitterbug Transformers see ``The Complete set of Jitterbug Transformers and the analysis of their motion'' by H.F. Verheyen in COMPUTERS, MATH AND APPLICATIONS Vol 17, No. 1-3 pp. 203-250, 1989.

`[`

From *Synergetics* - typed in by Kurt Przybilla`]`

224.07 Sphere: The Greeks defined the sphere as a surface outwardly
equidistant in all directions from a point. As defined, the Greeks'
sphere's surface was an absolute continuum, subdividing all the Universe
outside it from all the Universe inside it; wherefore, the Universe
outside could be dispensed with and the interior eternally conserved.
We find local spherical systems of Universe are definite rather than
infinite as presupposed by the calculus's erroneous assumption of
360-degreeness of surface plane azimuth around every point on a sphere.
All spheres consist of a high-frequency constellation of event points,
all of which are approximately equidistant from one central event point.
All the points in the surface of a sphere may be interconnected. Most
economically interconnected, they will subdivide the surface of the
sphere into an omnitriangulated spherical web matrix. As the frequency of
triangular subdivisions of spherical constellation of omnitriangulated
points approaches subvisibility, the *difference* between the sums
of the angles around all the vertex points and the numbers of vertexes,
multiplied by 360 degrees, remains constantly 720 degrees, which is
the sum of the angles of two times unity (of 360 degrees), which equals
one tetrahedron.

```
[From
Synergetics 2 [100.62-100.63]]
```

```[`

One reason for human incomprehensibility of the findings of
science`]`

is our preoccupation with the sense of static, fixed ``space'' as so much
unoccupied geometry imposed by square, cubic, perpendicular, and
parallel attempts at coordination, rather than regarding ``space'' as
being merely systemic angle-and-frequency information that is presently
non-tuned-in within the physical, sensorial range of tunability of the
electromagnetic sensing equipment with which we personally have been
organically endowed.

``The somethingness here and the nothingness there of statically interarrayed ``space'' conceptioning is vacated as we realize that the infratunable is subvisible high-frequency eventing, which we speak of as matter, while the ultratunable is radiation, which we speak of as space. The tunable is special case, sensorially apprehensible episoding.''

`[From Chris Fearnley]`

Space is ``systemic angle-and-frequency information'' because like all awareness it is patterned systemically and hence polyvertexially. It is information because the angle-and-frequency constituted system can be resolved into bits, 20-questions-wise.

Space is ``presently non-tuned-in within the physical, sensorial range'' because we are presently not receiving electromagnetic energy or information to our eyes, ears, nose, tongue or skin. But space is identifiable as a metaphysical system -- it is ``out there.''

`[Kirby Urner's contributions.]`

Space, the Final Untuned

Vis-a-vis whatever is in experience at the moment, is a vast otherness, which is by definition not tuned. That is space, the field of potential experience, I would say. Or maybe the field of ``unmeant meanings'' (no experience of that at this time). The trichotomy of ``outside system, system, inside system'' or ``ultra-system-infra'' is a generic description of that system (e.g. ``belief system''). The ``space of the untuned'' or ``final frontier'' of a specific system is whatever that system cannot tune in. We all live in the space of our ignorance.

Space as ``ultratunable radiation''

Before people knew about clusters of galaxies, or this galaxy for that matter, or ``outer space'' in general, they had yet to receive the energy through their instruments that would inform them of this ``space'' and its contents. The only way we have a concept of ``space'' is owing to our receiving energy. Relates to your dwelling on ``experience'' which Fuller equates with the ``tuned'' (vs untuned). What we tune is energetic. The far apartness of the galaxies, their infrequency, is what made them so ultratunable (unexperiential) for such a long time.

```
[From
Synergetics [400.011-02]]
```

``A system is the first subdivision of Universe. It divides all the Universe into six parts: first, all the universal events occurring geometrically outside the system; second, all the universal events occurring geometrically inside the system; third, all the universal events occurring nonsimultaneously, remotely, and unrelatedly prior to the system events; fourth, the Universe events occurring nonsimultaneously, remotely, and unrelatedly subsequent to the system events; fifth, all the geometrically arrayed set of events constituting the system itself; and sixth, all the Universe events occurring synchronously and or coincidentally to and with the systematic set of events uniquely considered.

``A system is the first subdivision of Universe into a conceivable entity
separating all that is nonsimultaneously and geometrically outside the
system, ergo irrelevant, from all that is nonsimultaneously and
geometrically inside and irrelevant to the system; it is the remainder
of Universe that conceptually constitutes the system's set of
conceptually tunable and geometrical interrelatibility of events.
*...*

``All systems are polyhedra. Systems having insideness and outsideness
must return upon themselves in a plurality of directions and are
therefore interiorally concave and exteriorally convex. Because
concaveness reflectively concentrates radiation impinging upon it and
convexity diffuses radiation impinging upon it, concavity and
convexity are fundamentally different, and therefore every system has
an always and only coexisting inward and outward functionally
differentialed complementarity. Any one system has only one
insideness and only one outsideness. *...*''

The tetrahedron, of course.

`[From Gary Lawrence Murphy]`

The minimum system is an entity distinct from the rest of universe. The division is between the consideration set and the irrelevant; there will be leaks because no system is an island ;-), but for design purposes, the boundary defines the extent of energy interchange as represented by the concavity of the tetrahedral interior.

The four components `[of our friend the tetrahedron]`

are four sub-tunable
systems, only resolvable as a single point, but a system none the less.
Between these, we have Euler's rules for relative abundance of
topological features, so if we can identify four stellar partners, we
can postulate 6 interaction pairs and four interaction `facets;' we can
also look at the non-simultaneousness of the pair-interaction vertex
stars as potential leak points (in reality, each is probably involved in
a myriad of other tetrally-thinkable systems) or in Fuller's terms,
shunting-off points.

`[From Chris Fearnley]`

Take a tetrahedron. Hold the opposite vertices in turn (two pairs). Spin the tetra. Use a ``knife'' to cut the tetra where the ``great circle'' from the spinning would cut it. You now have the 24 A quanta modules of the tetra (12 positive, 12 negative in orientation). Take 1/8th of an octahedron (it's simple to see that the only way to do this is to extract the tetrahedron formed by the center of the octa and the three vertices that form one of its faces). Divide this into 6 equal parts (put the octa face on the table and use the edge bisectors). Note the line from the center of the octa to the center-face of the octa in the 1/8th octa. (It will be on the inside of the last division into 6 parts.) Find it's midpoint and slice the 1/48th octas along this midpoint, dividing the original octa into 96 pieces. The piece of the 1/96th octa that is 1/6th of the face of the octa is our old friend the A module. The B module is the other part. They have the same volume though the shapes differ.

`[`

This relates to Fuller's epistemography. From
Synergetics [501.10-501.12]`]`

``Any conceptual thought is a system and is structured tetrahedrally. This is because all conceptuality is polyhedral. The sums of all the angles around all the vertexes - even crocodile, or a 10,000-frequency geodesic (which is what the Earth really is) - will always be 720 degrees less than the number of vertexes time 360 degrees.

``The difference between nonconceptual, nonsimultaneous Universe and thinkability is always two tetrahedra: one as macro, to complete the convex localness outside the system, and one as micro, to complete the concave localness inside the system, to add up to finite but nonconceptual Universe. Thus the thinkable system takeout from Universe has a 'left-out' outside irrelevancy tetrahedron and a 'left-in' inside irrelevancy tetrahedron.

``You have to have the starkly nonvisible to provide the complementary tetrahedron to account for the visibility, since concave and convex are not the same. That stark invisible reality of the nonconceptual macro- and micro-tetrahedra also have to have this 720-degree elegance. But the invisible outside tetrahedron was equally stark. The finite but nonconceptual inness and outness: that is the Omnidirectional Halo.''

`[From Kirby Urner]`

Fuller used 4D to refer to the 4 rays from a central hub that omnisymmetrically define an expanding volume (e.g. the four lines from the center of a tetrahedron to its four vertices). The Cartesian system consists of 6 rays from the origin defining an expanding cube. The expanding tetrahedron uses/defines volume more economically, Bucky claimed.

`[From Clifford J. Nelson]`

The four dimensions refer to the movement of the four enclosing planes of a tetrahedron, not to rays to the vertexes.

I've been thinking lately: Does Bucky offer in SYNERGETICS an extension of the scientific method? The definition of Universe ``The aggregate of all humanity's consciously apprehended and communicated (to self or others) experiences.'' Together with Fuller's notion of thinking as the systemic process of sorting experiences into three broad sets: the macroscopic irrelevant, the microscopic irrelevant, the lucidly relevant set. This is his omnidirectional halo. I think it provides a means of organizing our thinking to make it more effective. Isn't this what the scientific method is supposed to do? Moreover, the dynamic nature of synergetics implies that we need not get stuck permanently in paradigms as Thomas Kuhn suggests. Maybe synergetics is transparadigmatic. --- C. Fearnley

Most definitely. Fuller did not choose the name Comprehensive
Anticipatory Design SCIENCE lightly. Like all of Fuller's language the
name was carefully chosen. I feel that your characterization of
Synergetics as an extension of the scientific method is absolutely true.
In fact this is one of Fuller's main criticisms of traditional geometry,
that it is not science; meaning that it is not ``*...* setting in order the
facts of experience'' but farther constructing an imaginary Universe out
of non existent points, lines, and planes. --- Blaine A. D'Amico

The connections that I see between Synergetics and the Fullerenes are manifold. First, Carbon is a tetrahedronal atom. It would seem logical that even if there were exceptions to Fuller's tetrahedronal concept of the shape of space, Carbon would surely obey these geometric principles (if the principles are true.) Fuller's discussions of tetrahedral bonding are remarkably similar to Linus Pauling's illustrations in ``The Architecture of Molecules,'' for example. --- Blaine A. D'Amico

(See section What are Fullerenes? and Buckyballs for more on the fullerenes.)

```
[From
Synergetics [410.02]]
```

``The omnirational associating and disassociating of chemistry - always joining in whole low-order numbers, as for instance H(2)O and never H(pi)O - persuaded me that if I could discover nature's comprehensive coordination, it would prove to be omnirational despite academic geometry's fortuitous development and employment of transcendental irrational numbers and other 'pure,' nonexperimentally demonstrable, incommensurable integer relationships.''

`[From Mitch Amiano]`

Why does he so often make use of square root of 2, and to approximations of pi? It seems in fact that he actively uses them, but only as approximations, and with a synergetic conversion factor.

`[From Kiyoshi Kuromiya]`

I believe Fuller uses synergetic conversion factors simply as handy ``fudge factors'' and, if he had his way in the world, there would be no need for conversions, because everyone would use an entirely rational number system--or even more, a system consisting entirely of whole numbers. The use of the square root of two, I believe, is simply to illustrate principles of alloying, and like in the other case, could be dispensed with, if everyone were used to proofs that only relied on whole numbers.

`[From Kirby Urner]`

Relevant here is that giant Scheherazade number -- abbrev. Sz -- Fuller suggests would rationalize trigonometry. Adding lots of primes makes some sense, and a screen with that many pixels could certainly give us adequate resolution to submolecular levels, all with whole number coordinates. Second-root-of-two would be a shorthand symbol within a computational notation with a granular, integer underpinning. Given a grainy nature, with no absolute positions as represented by continuing irrational numbers to umpteen digits, I can see how Fuller felt no need to take the Number Theory idea of irrationality as a concept implemented in nature. But our symbols, our ``root of 2'' notations, continue to be useful, just as they always have.

I don't think Sz numbers make the number theory idea of irrationals go
away. The problem was never ``too few primes'' in our base. The proof
that the 2nd root of 2 is irrational has nothing to do with primes, more
with an reductio ad absurdum showing it's neither true nor false that
the root of 2 is even or odd *...* anyway, I don't think Fuller is arguing
that mathematicians have been wrong all these years in their own terms
-- just that nature doesn't need to continue pi or other fractions
according to some infinite rules. No time for that.

`[From Mitch Amiano]`

Given that Synergetics rejects irrationals, and given a modeling of nature based upon an integer representation with a specific sub-molecular resolution, could we not calculate the square root of 2 as a rational number?

This is what carpenters do every time they check the accuracy of a square layout by measuring from corner to corner. The resolution of their measurements is at a significantly higher level, in terms of fractional inches.

An example of measurement rationalization can be seen when moving from a relatively large base unit - the inch - to a relatively small one - the millimeter. More of the numbers are represented as whole integers.

Thus, the operational square root of 2 is 1.40625" when measured in thirty-seconths of an inch on a 1"x1" square, or 37mm when measured in millimeters.

`[From Vincent J. Matsko]`

Re: The irrationality of the height of an equilateral triangle: 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!

`[From Chris Fearnley]`

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 aesthetic 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.

When reading synergetics it struck me that perhaps there are two basic phases in the Universe - tetra and icosa. Your regular polyhedra hierarchy may be just another way of looking at the three fundamental geometrical forms in Universe - my so-called canonical (rational) forms.

`[From Chris Fearnley]`

I think the simplest first-order definition of precession is the side effects of a system in motion (generally occurring at 90 degrees to the direction of motion).

`[Blaine D'Amico]`

Bucky said that precessional effects are what most people label ``side effects.'' i.e., I teach a person to fish so he can feed his family (Direct effect). One of his no longer hungry children now can focus in school and goes on to become an important scientist (precessional effect).

`[From Gerry Segal]`

My college physics books defines precession as:

``a complex motion executed by a rotating body subjected to torque, by a conical locus of the axis''That's quite a mouthful. Bucky gets even more complex. In Synergetics [533.08], he defines precession as:

``the intereffect of individually operating cosmic systems upon one another. Since Universe is an aggregate of individually operative systems, all of the intersystem effects of the Universe are precessional, and the 180-degree imposed forces usually result in redirectional resultants of 90 degrees.''A beautiful example is given in Synergetics [417.00]. Here two exact sets of 60 Closest-packed spheres (wedges) are rotated 90-degrees and twisted (torque). An unexpected and marvelous result is a perfect 8 ball edged, 7-frequency tetrahedron that is formed.

I doubt that I have been successful in helping you understand precession. But I do know that if you take the time and build the models you'll have an underlying sense of the meaning that provides the basis of understanding that the written word only hints at.

`[From Leo Elliott]`

The clearest example I recall Bucky giving of the notion of ``precession'' was that from the viewpoint of a waterbug or a jellyfish on the surface of the water, directly in the path of some big ship, which will send out precessional waves slightly ahead of the bow, thereby alerting the astute bug or jellyfish that something big is indeed on the way.

`[From David Worrall]`

Imagine a pebble dropped into a pond. The pebble goes to the bottom (closer to the centre of gravity of the earth!) The wave created moves outwards, at 90 degrees, precessionally, to the pebble.

`[From Kirby Urner]`

``Precession'' in synergetics shows up as the relationship between two sides of the same generalized principle coin. Gravity begets radiation begets gravity. Tension begets compression begets tension. Pull on two ends of a rope, and its strands are squeezed even more tightly together. Where two very general aspects of nature always and only co-exist, and their relationship is generally precessional.

Synergetics is unlike traditional physics in its insistence on gravity as a circumferential pulling together (and thereby implosive), versus a radial explosiveness emanating from the center -- a 90 degree relationship. The Sun is a giant squeeze ball. Strands of thought are likewise circumferentially implosive, nonlinear hypertext countervailing against vs the information explosion.

By extension, ``precession'' refers to nature's way of getting the job done at 90 degrees to human selfishness and ignorance. We ``do the right things for the wrong reasons.'' The graduating from Class II to Class I evolution which Fuller anticipates involves our starting to do the right things for the right reasons, like you don't need the Cold War to have the space program to have higher living standard spin-off technologies (goodies yielding at 90-degrees to ignorance and fear). We don't have time for that kind of bumbling anymore.

`[Steve Mather]`

`[From Chris Fearnley]`

The issue of volume measure is dependent on the choice of the unit of volume. I recommend choosing the tetrahedron as the unit of volume. Then by subdividing the octahedron and tetrahedron into ``building blocks'' fascinating relationships will be discovered. Try building some models too!

`[From Kirby Urner]`

I agree with Chris F. -- using Fuller's regular P-lengthed tetrahedron as a unit of volume is a good beginning. A regular P-edged octahedron will have 4 times the tetra's volume, while a cube with a P-lengthed diagonal will have 3 times the tetra's volume. That means a cube of diagonal P has 3/4 the volume of an octahedron of edge P. Say P=1. The corresponding cube of diagonal 1 has a volume of about 0.354 (conventional math), and so the octahedron has a volume of 4/3 that, or 0.471 (again, conventional math). If we multiply both results by the Synergetics Constant, we get a cube of volume 0.3750000 (precise) and an octahedron of volume 1/2. Those are the nice volumes we'd like, given a simple edge of 1. Note that the tetrahedron of edge 1 has a volume of 1/8. That's because of how the Synergetics Constant is derived. The so-called ``prime vector'' between any 2 adjacent spheres in the icomatrix is of Cartesian length 2 (these are unit-radius spheres after all). So the Synergetics Constant is the ratio between the volume of a prime vector diagonaled cube in Synergetics (=3) and the corresponding cube in XYZ geometry (2nd-root of 2 to the third power).

`[From Martin Roller]`

Kirby Urner writes: >By the way, there *is* a formula that provides the volume of >*any* tetrahedron given its 6 edges as inputs. It's a monster >formula, derived by Leonhard Euler. I doubt I could write it >understandably in ASCII.Let ABCD be a tetrahedron with sides

```
``` a = AD, b = BD, c = CD,
p = BC, q = CA, r = AB.

The volume V of the tetrahedron can be computed from the determinant of
a 5-by-5 matrix as follows (here ^2 means taking squares).
```
``` | 0 r^2 q^2 a^2 1|
| |
|r^2 0 p^2 b^2 1|
| |
288 V^2 = |q^2 p^2 0 c^2 1|
| |
|a^2 b^2 c^2 0 1|
| |
| 1 1 1 1 0|

`[From Kirby Urner]`

Euler's equation for any Tetrahedron w/ edges p,q,r,s,t,u,v:
```
```M= (2qrt)^2
-[q^2(r^2+t^2-u^2)^2]
-[r^2(q^2+t^2-v^2)^2]
-t^2(q^2+r^2-s^2)^2

substitute above M in equation below (V=Volume)
`V= 1/12 [M + (r^2+t^2-u^2)(q^2+t^2-v^2)(q^2+r^2-s^2)]^.5`

This looks a lot more complicated than the determinant expression, but then a determinant is short-hand for a long messy expression. Anyway, both give the same answers. Then you can multiply by the Synergetics Constant to give the volume relative to a Tetrahedron defined by 4 adjacent unit-radius spheres of volume one.

`[From Kevin Sahr]`

`[Synergetics]`

seems to have the potential to be used as a
mathematical basis for ``communicating experiences'' or for otherwise
describing them in a precise way.

Though I agree with this in principle, I've never seen anyone actually use it for this purpose. For instance, can anyone out there right now communicate an experience to me using Synergetics? Even a ``toy'' example would be useful for discussion, but I'm thinking more along the lines of communicating some unique thought/constellation of thoughts to me; something that could not be well communicated using ordinary english sentences, but which would be unambiguously communicated to me by a set of geometrical relationships that, say, could be stored in a computer.

`[From Chris Fearnley]`

I think these synergetics explanations are not meant to replace ordinary english, but to supplement it. I think when you start (perhaps even at a metaphorical level) to examine the complex of interrelationships in an ``english'' story, you find deeper meaning. The synergetics patterns are there but our minds CAN (are capable of) dealing with these synergetics patters at an ``unconscious'' level. By trying to enumerate the precise geometry, you slow the normal geometric patterning - break the flow of thought. I suggest that by awareness of the inherent synergetic side of thinking, we can (as sort-of a side effect) find new relationships and understand more deeply the (initially obscured) relationships inherent in your ``story.'' In other words synergetics probably does not supply a (logicians') decision procedure for unfolding a story, but rather a ``mystics'' science for appreciating relationships which before synergetics would have been left outside of cognition realization. In this synergetics science previously ``impossible'' ideas can become clear. Now you asked for an example of synergetics' application to understanding stories. Perhaps a new paragraph is in order?

Just some random thoughts. Antivirals may cure AIDS: AZT may be the answer! Well, synergetics suggests that we need to find ALL the relationships involved in our subject of concern. So we must look at the whole system. We now discover that the human body is chock full of viruses and bacteria. So it becomes clear that just by fighting the viral component of disease, we may be missing some vital components of disease. Perhaps AIDS is not a bad virus, but a ``good'' virus that through some co-factor some problem develops that is unrelated to the actual viral mechanisms. In sum, by looking at all the factors and keeping a clear sight of inside-outside phenomena, we can begin to appreciate that the AZT craze of recent years, may be too simplistic a view of the situation. So with recent reports suggesting that AZT is mostly ineffective in improving the quality of life of AIDS patients. In conclusion, because synergetics asks us to consider the WHOLE complex of factors in inside-outside relationships, a disciplined thinker can be more skeptical of false eureka's and more sober when the false theories of yesterday are debunked. English's problem is that it doesn't provide the discipline of thinking that synergetics demands.

Hope that helps.

`[From Kevin Sahr]`

I agree that synergetics is important in the role you give it in your discussion and example, but I think reading Synergetics makes clear that Fuller was very much interested in ``trying to enumerate the precise geometry.'' Synergetics consists of the types of principles you point out (i.e., look at the whole system, etc.), but it also consists of some very precise geometric statements (ie., the A-module break-down of the tetrahedron mentioned in the original post). So I guess I might re-phrase my original question: can anyone give me an example similar to Chris' AIDS example but applying, say, anything having to do with A and/or B (or, even T & E Quanta :-)) modules? Or maybe (a little bit less esoteric) the Jitterbug?

`[From Kirby Urner]`

Is Synergetics actually useful for communicating experience? Fuller's writings suggest how bare bones Synergetics supports fleshier metaphors in Critical Path and Grunch of Giants -- Fuller considered them all one magnum opus, viewed from different angles.

The non-simultaneity of only partially over lapping events, some far apart in time and space, makes the tuning in of relations among these events a discipline. Fuller felt he was revealing some of these larger patterns in Critical Path. I find many of his visions tough to swallow, but that's another conversation. Like, where's the evidence for submarine aircraft carriers?

Since reading Fuller, I've done mental gymnastics to ``feel'' myself driving on the surface of a planet, stuck on by gravity, but not oriented in an up/down Universe. Once in New Mexico, a felt I was hanging upside down (my driving was unimpaired).

More, though, it's the word building associations, lying in the dark thinking of all the metaphysical communications going around the world (networking, diplomacy, broadcasting, satellites, telephone, exchange programs, advertising) as a circumferential countering of more physical explosions of violence (bomb blasts, big and small, gun fire). Not that some communications, such as inflaming of nationalist, racist sentiments and xenophobia, aren't also conducive to violence (wrong picture to think of communication as intrinsically beneficent -- can be entropic in the extreme).

As for whether ``precise'' or ``refined'' synergetics, using A & B mods etc, is useful for communicating experience... well, to communicate precise geometric models and pictures certainly -- my ability to visual the face centered cubic lattice of crystallography, and to understand the design of geodesic spheres and the dymaxion map certainly owes a lot to synergetics. But I think Kevin's question is more about whether electronics, molecular physics, quantum mechanics or the like may be illuminated by geometric descriptions -- is a kind of ``narrative mathematics'' possible, in which your read ``hard science'' information more the way you read english syntax, than reading the usual mathematical symbolry? Fuller makes many attempts to talk about quanta, electron orbitals, energy transfer, stellar mechanics, using his language. I think we know what such narrative, or operational mathematics might look like. The question is whether these models have any experiential validity. Part of the problem, as Fuller saw it, is that geometric modeling of the physical world fell out of favor some decades ago, when the math left experiential visualization in the dust. So getting back to geometric models is tough with or without the help of a geometric narrative-style language.

`[From Chris Fearnley]`

Synergetics is subtitled ``Explorations in the geometry of thinking'' -
not explorations in the geometrical shape of an idea, argument,
discussion. Fuller's geometry is very dynamical. It shifts even more
quickly than science's theories about the origin of life on Earth (grin).
So if I understand correctly, you're asking for an example describing
how an intertransforming isotropic vector matrix pulsating through it's
full periodicities (with A and B models as an integral part of the whole
apparatus) can model the evolving process of ``casting out
irrelevancies'' to focus thinking more-and-more on the system in
question (the requested example). I think this would be impossible to
do in a general fashion. Perhaps one could examine their own thought
patterns carefully enough to see which thoughts correspond to which
A-Module pulsations, but I think this would be difficult especially
given the fact that by examining the process of our thinking we alter it
in unpredictable ways. I think Fuller's theories on the dynamics of
thinking can only be ``proven'' with high-level, fundamental reasoning
w.r.t. the nature of the geometry itself and its ``uncovering'' of the
mysteries of thinking (the process, the verbs not the nouns). By
exploring the fundamental logic of the basis of thinking in
Synergetics, I convinced myself that in general Fuller is ``right on the
money,'' but I have been unable to apply his work in the static way in
which you would like to see it. (See Fuller's essay ``Omnidirectional
Halo'' in *No More Secondhand God* which is the essay which when
``unfolded'' turned into *Synergetics*.)

`[From Kevin Sahr]`

Again, I think this kind of thing is important, but ``geometric modeling
of the physical world'' is not what I was getting at - what I'm after is
geometric modeling of the *metaphysical* world, the world of mind. A
lot of people are interested in Synergetics as throwing light on things
which fall within the realm of the hard sciences (ie., molecular
configuration (Buckyballs), civil engineering (octet trusses), etc.),
and certainly I think Synergetics will repay any time spent with it by
practitioners in those fields. But, to come out of the closet, my own
interest is in mathematical models of how we think (and how we might
think more effectively!), specifically those models which are computer
programmable. I think that Bucky saw Synergetics as very relevant to
that type of ``science,'' and I think he would have claimed that that
relevance extends even to the more mathematically esoteric elements of
his theories. I'm just trying to figure-out what that relevance might
be!

By the way, I'm not claiming that Bucky himself always saw such a
relevance in all of the ``generalized principles'' he discovered; often I
think he was just cataloging such principles in hopes they might be
useful to someone in the future. But I *am* claiming that I think Bucky
would have argued that there *must* be some such relevance for all these
principles. And I think it's clear that he saw relevance in areas that I
do not yet.

Again, to turn to the concrete. I apologize for not having the exact quote in front of me, but Bucky wrote in Synergetics something to the effect that the Jitterbug recapitulates the phenomenology of all experience. In what way is this true? If this is so, then shouldn't it be the case that we could take any experience (including any thought or line of thought) and in some sense map it onto the Jitterbug?

`[From Leo Elliott]`

Interesting discussion on the nature of synergetics as science, as linguistics, and how Bucky may have conceived their pattern-integrity. Interesting enough to make me pull out some ancient transcriptions of a 1976 ``Being With Bucky'' gathering out in SF...

Like Kevin Sahr, I must confess that, as a non-scientific type, the
appeal of Synergetics always seemed to come more from the notion,
implied or explicit, that all Bucky's perusings and perambulations
pertained at least as much to the metaphysical as to the physical, that
somehow `thought' itself was structured in the form of A and B quanta
modules, or their equivalents... As a former devotee of *The Urantia
Book* I used to get quite excited about all the triadic expressions of
`universe reality' presented therein, things like
``thing/meaning/value'' or ``fact/idea/relation'' or
``origin/nature/destiny,'' an attraction I now view as part of a basic
rhetorical appeal that somehow reinforced, in a starry-eyed-seventies
way, Fuller's own novel rhetoric.

I suppose that if the `return to modelability' that Bucky spoke of was part of his life enterprise, then perhaps a first step was a `return to speakability' -- and as any hereon who may have been fortunate enough to hear Bucky live (for me only once) may attest, it was an experience of sitting on the edge of my chair for three hours, straining to keep up with the thinking of a man three times my age, and at the end of which left wondering if perhaps Bucky weren't receiving some kind of alien transmissions through his hearing aids. I still had that feeling when listening to these tapes again, almost twenty years down the road.

Is it fair to say that Bucky's written grammar and syntax was at least
as complex and intricate as his oral presentation? I know Bucky took a
lot of flak from those whose eyes glazed over after the same sentence
went longer than two minutes or twenty -or-so lines, but somehow
whenever I tried to find any grammatical or syntactical error(s), none
showed up, and while I never actually did so, I had the feeling that one
could even diagram his sentences. There is a great photograph in E.J.
Applewhite's *Cosmic Fishing* which shows a page of galley proof,
supposedly ready to go to press, which Bucky had filled the margins of
with corrections and revisions, never being content with saying
something, on paper or in person, just the way he had said it even the
day before.

So in answer to queries for metaphysical specimens of synergetics, I can only think of Fuller's written works, and his oral presentations as such have been preserved in various archives.

I also believe Bucky spoke of his `prayers' being different every day, and how ``it also seems illogical to remind God of anything.'' If any are interested I have a version or two of at least one specimen of the Bucky-version of the Lord's Prayer, which at least on the tape -started out- as an ``our father'' type soliloquy, but which in typical fashion mutated several paragraphs later into something else. I attach below my own specimen of fullerspeak, written in the best syner-linguistics I could muster.

Is it also fair to say that Bucky's speaking and writing styles were as close to identical as any rhetoric in the collective recollection?

```
```Our tetrahedron
who art in geometry
hollowed by thy concavity
thy convexity come
thy system be dome
on Urth
as it is in Universe.
Give us this eternity
our daily integrity
and fore-give us our dis-integrations
as we fore-give those
who dis-integrate around us.
And lead us only in -- to syntegration
and de-livery us from entropic monofocus
on material self-interest
universe within Universe
amen.
lhe
13 May 1977

`[From Clif White]`

Try this the next time you have some time, newspaper, a dowel and some masking tape.

All that newspaper laying around your house can be made into large structures that are surprising strong as well.

Simply roll a sheet of newspaper around a 1/2" dowel secure the end of the roll with a bit of tape and slip out the dowel and then repeat procedure to make another strut.

Using your stock of newly created struts, secure the ends to form joints using more masking tape. (Don't use a lot of tape at the ends.) Form triangles and then tetrahedrons along with octahedrons and you will begin to make a large scale octet truss system that will quickly fill up your room.

You will be amazed at how strong this system is!

This is a great activity for a bunch of kids. You can make all sorts of polyhedra quickly and cheaply. A production line of strut makers, and joiners can really pump out the structures. My kids love this activity.

`[From Christopher Rywalt]`

The other day I was wandering through Star Magic -- another one of those science toy-type stores -- when I was about to complain that I never could find anything very interesting in such a store. Just as I began to speak, however, my friend said, ``Sure, you play with the useless stuff and walk right past that thing you've been looking for for months.'' And he pointed me at a little kit called a Vector Flexor. I don't know how many of you have run into this, but its rather neat. It's basically colored sticks and rubber tubes, and the rubber tubes can be assembled into an X shape and the sticks stuck into them to make a vector equilibrium. It's pretty cool, because it can be made to jitterbug and it comes with a pretty detailed insert explaining what it is and even refers the buyer to several of Bucky's books.

`[From Mitch C. Amiano]`

I note that, in Fuller's Octetruss patent, there is an implementation disclosed in which the struts are formed by the overlapping edges of aluminum triangular plates with 3 flanges. I tinkered with an alternative (overlapping faces & flanges) form of the same thing in paper. I decided that with a few extensions and mating pieces it could be a real modeling kit; the major drawback being that the paper models were not self-aligning like plastic or metal formed plates would be, so large models tended to show signs of twisting.

I then took some empty 1-gallon polyethylene water containers, stripped the labels off, cut off and layered the flat sides, and melted them together carefully in a 375-450 degree oven, to get a ~1/8th inch thick laminated sheet. I cut and shaped a plastic prototype of an octet plate of my revised design. It's about 3cm high, and looks pretty neat - but I think my wife would get X-( mad if I do it again soon: melted polyethylene smell even with the fans on and windows open.

Here I want to include references to work that shows how useful Fuller's synergetics ideas are and have been - To persuade the skeptical :)

`[From Ed Applewhite]`

Satellite sensing data displayed for first time on geodesic triangular-tetrahedral grid

``Scientific American.'' (January 1991) reported that researchers at the Los Alamos National Laboratory turned a technique for modeling explosions into one that simulates climatic change.

``It relies on meshes made of half a million tetrahedrons. . . .Every tetrahedron covers an area no wider than 30 kilometers. . . . In the event of a disturbance such as a hurricane, these meshes would twist. Conventional models which use rigid meshes of rectangular bricks, typically lack the resolution to portray such comparatively local phenomena.'' (The graphics accompanying this article demonstrate the kind of applied geodesics that Buckminster Fuller had in mind.)

`[From Kirby Urner]`

What'd be nice to have is a large inventory of artfully produced
synergetics clips *in the public domain* which personal workspace
enthusiasts (e.g. me) could inload, edit/recombine, and outload to the
network. Over time, we'd build up quite a library. In the short term,
I don't think Internet is the place to communicate these high bandwidth
scenarios so much as a place to verbally fantasize them or give info
about how to get them through other channels (e.g. the mail). Most
realistically, I think a CD-ROM of Synergetics Clip Art, stills, short
animations, pictures of artifacts, inventions, Bucky's prose and poetry,
who's who contact lists etc would be the ideal evolutionary tool to
galvanize the incipient Design Scientists among us to get to work. As
the dial-up and downloading of visual video clips becomes more
available, then we can move our collection to a more public archives.

Again, I think these metaphysical assets should be public domain (even though the CD-ROMs themselves will cost) to encourage users to incorporate them freely into works of their own, and to upload these for downloading by others in turn, and so on. That'll be the metaphysical/fantasy part: out in the real world, we'll be sharing our storyboards with TV producers to get Hollywood-style storyboards enacted big time, on a bigger scale. Any mass infusion of domed domiciles would be televised for sure.

Best to work with the entertainment industry from the inside out, rather than expecting Design Science to take off on the side some place, and have TV news people come running to ``the scene.'' No. The Design Science revolution will start right in the studio, when the map behind Dan Rather's head stops looking so stupidly distorted.

`[From Mitch C. Amiano]`

I am trying to develop a procedure for giving the ratio of the circumference to the average radius of certain circle-like polyhedrals, as a function of the number of outer chords on the polynomial edge. The constraints I have (arbitrarily) placed require that the polyhedron be formed by a whole number of equivalent triangles placed about a center point. The triangles have (at least) two identical edge lengths (of unit length) which are the radii of a circle circumscribing the polyhedron, and one (outer) edge in common with the formed polyhedral.

As an example: for a hexavertexion (hexagon) with an outer radius of 1, there are six equivalent triangles which happen to be equilateral; the outer edge also has a length of 1. The frequency of subdivision is 6 (the number of outer edges). The average diameter is 1+sqrt(3)/2. The approximation of pi for this case is 6/[1+sqrt(3)/2] = 3.2154... which itself is irrational, but at least it seems to have some relationship to the polyhedron.

`[From Robert L. Read]`

From the formula for regular polygons in _CRC Standard Mathematical Tables_, Edition #27, page 122, the inner radius of a regular polygon (the radius of the inscribed circle) is (r = 1/2 * s * cot(180/n)), where cot is cotangent, n is the number of sides, and s is the length of the chord on the outside edge. The radius of the circumscribed circle is (R = 1/2 * s * csc(180/n)), where csc is cosecant. Since the circumference of the n-gon is (n * s), we can write the ratio of circumference to ``average radius'' (if you mean by that, as you apparently do, the average of the maximum and minimum, which is not obviously the same as what you would get by calculating the average over an infinite number of rays via calculus, but it might be) as: n * s / ((r + R) / 2 , which by algebra is equal to 2 * n * s / (r + R) , which by substituting the above formulae is, 2 * n * s / (1/2 * s * cot(180/n) + 1/2 * s * csc(180/n), and so the s's can be crossed out of the top and bottom and we get: 2 * n / ( 0.5 * cot(180/n) + 0.5 * csc(180/n)), which is a function only of n, which is what we desire, and can be cleaned up to: 4*n / (cot(180/n) + csc(180/n)) which, since cot and csc are kind of a pain we can replace with sin and cos via identities that we should all remember but happily can be found on page 135 (cot x = cos(x) / sin(x) and csc(x) = 1/sin(x)). Then with a little more algebra we get the easy-to-use-if-you-have -a-calculator formula: circumference / av. radius (n) = (4 * n * sin (180 / n) / (1 + cos(180/n))And, BTW, it works, I checked it at a few values.

`[From Kirby Urner]`

Nature is not using PI, nor are humans (part of nature). All
computer-based and calculator-based representations of PI are truncated
to the number of digits internal storage permits. Even those gazillion
digit Cray monsters terminate (and besides, are not used in practice in
any calculations). Bucky's argument that nature does not use irrational
numbers is pretty straightforward: you have never used an irrational
number in your life: all computations with root-of-two, pi, e and so
forth are definite, terminated. We call them ``approximations'' just as
we say all lines are ``approximations'' of perfectly straight ones. Bucky
simply starts with what's right in front of us, in our everyday
experience, and says ``not approximations of anything, this is what
simply *is* -- no perfectly straight lines and no ultimate value of PI
actually exist or gets used for anything in nature.''

`...`

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.

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, sqrt(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 nonterminating irrationals?'' is moot in any case -- we get along fine right now.

`[From Mitch C. Amiano]`

Given that there is no infinite precision in practice and in Universe, we must decide what precision to use. The most common approach is to just use however many digits our calculator gives us, which is usually too many, and get a bigger calculator if its not enough. Generally, the precision we need is determined by the size of our bolt-holes and the elasticity of the materials we're working with.

For any working environment we could define a necessary precision (higher for steel than wood, higher for wood than plastic) and develop tables of fundamentals values, such as the ratio of circumference to diameter for an n-gon, expressed as fractions in simplest form that would be accurate enough for that application. (The decimal expansion could be used, but often we would find much simpler fractions that are accurate enough.) This would have a certain pleasing simplicity, but, on the other hand, it has no really obvious advantage over an over-precise description. However, it seems worth investigating, because if some pattern could be detected, then we would gain not only an engineering tool, but a tool for understanding, which is one of the wonderful things about many of Fuller's inventions.

`[From Vincent J. Matsko]`

It is possible, beginning with a hexagon, to perform the ``irrational''
approximations `[to pi]`

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: 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 formula for a solid angle 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.

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.

`[From Kirby Urner]`

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
<stuff deleted>
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.

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.

This method may have already been published many times, but I derived it from scratch I'm proud to say.

I've further simplified, or at least re-expressed, an algorithm for generating pi without using trig functions.

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...)
[Expansion]
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])

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.
`[From Kiyoshi Kuromiya]`

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 measurable 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.''

`[From Kirby Urner]`

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).

`[From Richard Hawkins]`

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 continuously 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.

One picture of this model is available by anonymous ftp switchboard.ftp.com:/bucky/curVE.jpg.

`[From Chris Fearnley]`

In the Humanities Citation Index I found an article by Prudence Allen,
R.S.M (Concordia University) in International Philosophical Quarterly
32(1):3-16 entitled ``Fuller's *Synergetics* and Sex Complementarity.''

The article had several very interesting features. First, she provides a test-case example of Fuller's principle that the minimum conceptual system is structured tetrahedrally.

Concept of Male Concept of Female Description ======================================================================= first vertex: male female Primarily Biological second vertex: masculinity femininity Primarily Psychic (cultural) third vertex: femininity masculinity " fourth vertex: man woman As individuality

Allen argues that this tetrahedron of Male and Female is both historically and philosophically tantalizing (if not valid).

Second, Allen looks at Fuller's concepts of complementarity and parity
(and implicitly the concepts of system, integrity, events,
inter-relationships of events in a system - well, basically the whole of
Fuller's epistemography) in application to sex identity. She also
evaluates several of Fuller's references to male-female complementarity
(*Synergetics*:
1210,
511.12,
1076.11-12,
1024.15,
and others). She quotes Fuller's article ``Goddesses of the Twenty-First
Century'' in Saturday Review 14:(2 March 1986). (Has anyone seen this
article?)

``Women are tensional and continuous. Each new female as well as male life comes from the womb of the women. We have, then, the new female life as a series of expanding waves, the new ever emerging from within the older wave. Women are continuous. ...

``Males are discontinuous. The new life is noncontiguous to the previous male life. Men are, then, islanded, individual discontinuities.''

Finally, I noticed that Allen's article is a very interesting piece of
scholarly applied synergetics and synergetics' philosophy. She really
understands Fuller's philosophy of systems and its inherent
complementarity, parity and synergy. Her only real criticism was that
Fuller didn't take the next step to viewing man as a person in community
and woman as a person in community. Although I don't recall any
references to a philosophy of humans in community in *Synergetics*
(besides the electronic voting), I think we need to review Fuller's
essays in Ekistics before we can safely claim that Fuller didn't develop
any specific thinking regarding communities. Overall, this is a good
read for the Fuller scholar looking for philosophical applications to
synergetics.

```
[BTW, there is another philosophical essay by Derek A. Kelly ``The
Philosophy of R. Buckminster Fuller'' in International Philosphical
Quarterly 22(1982): 295-314. This long essay disappointed me as the
author does not seem to have integrated all the pieces of Fuller's
philosophy. Well, in my opinion (based on a very cursory examination)
Kelly didn't comprehend the full meaning of Fuller's concepts. I'll
have to read this one more carefully before passing final judgment.]
```

http://www.cjfearnley.com/synergetics.essay.html provides some
tips for reading *Synergetics*.

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