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Synergetics

Explorations in the Geometry of Thinking

Synergetics & Language/Code Models

R. Buckminster Fuller: A 20th Century Philosopher

4D: New World Game

Noel Coughlin

Struppi Pohl

⁠

Synergetics @ BFI⁠

—

http://www.rwgrayprojects.com/synergetics/⁠

Wikipedia:

⁠

Synergetics (Fuller)Last edited: Sun, Jan 28, 2024 Synergetics is the empirical study of systems in transformation, with an emphasis on whole system behaviors unpredicted by the behavior of any components in isolation. R. Buckminster Fuller (1895–1983) named and pioneered the field. His two-volume work Synergetics: Explorations in the Geometry of Thinking, in collaboration with E. J. Applewhite, distills a lifetime of research into book form. Since systems are identifiable at every scale, synergetics is necessarily interdisciplinary, embracing a broad range of scientific and philosophical topics, especially in the area of geometry, wherein the tetrahedron features as Fuller's model of the simplest system. Despite mainstream endorsements such as the prologue by Arthur Loeb, and positive dust cover blurbs by U Thant and Arthur C. Clarke, along with the posthumous naming of the carbon allotrope "buckminsterfullerene", synergetics remains an off-beat subject, ignored for decades by most traditional curricula and academic departments, a fact Fuller himself considered evidence of a dangerous level of overspecialization. His oeuvre inspired many developers to further pioneer offshoots from synergetics, especially geodesic dome and dwelling designs. Among Fuller's contemporaries were Joe Clinton (NASA), Don Richter (Temcor), Kenneth Snelson (tensegrity), J. Baldwin (New Alchemy Institute), and Medard Gabel (World Game). His chief assistants Amy Edmondson and Ed Popko have published primers that help popularize synergetics, Stafford Beer extended synergetics to applications in social dynamics, and J.F. Nystrom proposed a theory of computational cosmography. Research continues. See more en.wikipedia.org

⁠

Library: Buckyverse.org

⁠

Martian Math @ Reed College: [

Classic⁠

][

Restyled⁠

]

Martian Math⁠

within Digital Math (Silicon Forest)

⁠

Synergetics⁠

@ TrimTab Coda

⁠

Invisible > Visible⁠

⁠

AGENDA

KU: In our Knowledge Engineering meetup of 2024.3.13, DAF expressed an interest in:

distilling the original Synergetics (1970s) to axioms

systematizing and integrating (also correcting) Synergetics with what has developed since

developing quadrays further

DISTILLING SYNERGETICS

⁠

I want to take it as far as core concepts, or let’s call them the cognitive frameworks, but they’re more geometrical exhibits, perhaps short animations (e.g. hypertoons) hyperlinked in a “memory palace” (a mnemonic structure).

Fuller does write in one passage about axioms (986.044), tracing them etymologically to axes and maxims. Wearing my anthropology hat, I note multiple meanings of “axiom” from “rule on the back of a game box i.e. a rule in ‘how to play’” to “set self evident [sic] logical propositions designed to compel agreement such that all subsequent statements are seen to follow.” I would consider it an abuse of Synergetics to make it seem too compulsory in the sense of “commanding assent”. That’s a disposable piece of cultural baggage. I’d prefer the Pythonic “fits your brain” slogan, if we’re talking about level of persuasiveness.

I’d claim that treating Synergetics as a work in the humanities doesn’t sacrifice its being nevertheless precise and logical, but we should adjust our expectations in swapping out one form of precision, that of a textbook geometry, for another, that of a metaphysics or philosophy. We’re free to overlap on our PATH and STEAM in the realm of

extended precision computational geometry⁠

(natural language processing has that too).

⁠

BEAST Hypertoon: Intertransforming Voxels defined in Synergetics by RBF

⁠

Figure 1: We might call this the big bang conceptual beginning of our 4D IVM ocean blue.

1, 12, 42, 92...⁠

or cumulative

1, 13, 55, 147...⁠

Connect sphere centers for a uni-length edged scaffolding, a honeycomb. Each ball has diameter D, radius R.

Figure 2: the duo-tet cube of canonical volume 3, with face diagonals length D; the suggestion of an inside-outing at the center, twixt a positive and negative (left and right) space.

Figure 3: any IVM ball may be embraced within a concentric hierarchy of polyhedrons (see relative volumes table when IVM tetrahedron is unit).

Figure 4... (and so on)

Synergetics distills into two related geometries: that of spinning (the great circles) and that of not spinning (the concentric hierarchy, including BEAST modules).

These two are related in that the static polyhedrons of the concentric hierarchy are what get spun, to generate the corresponding great circle networks, which get juxtaposed.

⁠

The Jitterbug Transformation⁠

relates the polyhedrons to one another, and their corresponding great circle networks.

The philosophical language of Synergetics is one of connotations as well as denotations, hinging around core dual pairs such as:

gravity vs radiation

syntropy vs entropy

tension vs compression

tuned-out vs tuned-in

mind vs brain

angle vs frequency

A core cognitive framework in Synergetics is the Omnidirectional Halo model (also in No More Secondhand God) which may relate to models in Active Inference.

A self or system is seen as a dividing layer, a barrier, a membrane, between an inner (contained) and outer (excluded) region. The membrane itself constitutes a tuned-in system of a specific frequency, and is considered a “sphere of relevance” sandwiched between “twilight zones of tantalizing relevance” in the more inward and more outward directions.

Note to self: look for more ways to connect to the language of

Globes, Bubbles and Foams⁠

ala Peter Sloterdijk, a German language philosopher.

INTEGRATING SYNERGETICS

A mistake Fuller himself identified in Synergetics 2, looking back on the first volume, is the tentative settling on a spinning sphere as volume 5. Had he been working with version control software, Applewhite’s job would have had a different flavor, ditto the outcome.

The quest for a volume 5 gets a second look in Synergetics 2, where the T and E modules get introduced, and

BEAST⁠

gets rounded out (the five main modules or wedges).

⁠

Screen Shot 2024-05-06 at 2.24.41 PM.png

⁠

In the final analysis, a rhombic triacontahedron (RT) made of 120 T particles gets to star as the canonical volume 5. The T module’s volume is 1/24 (same as A and B mods).

The R&D I’ve been undertaking with David Koski has yielded some new core concepts, such as the S:E module volume ratio being worth isolating and talking about as a scale factor. We have also found it useful to reintroduce phi (the golden ratio) into Synergetics.

⁠

Screen Shot 2024-02-17 at 4.15.27 PM.png

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Screen Shot 2024-02-17 at 4.14.24 PM.png

⁠

As a self-coherent platform, Synergetics will likely continually prove itself to be a cornucopia of follow-up research projects and adjunct explorations.

⁠

Screen Shot 2024-05-14 at 1.00.08 PM.png

Nine-None: Octal Periodicity (Bow Tie mnemonic)

Nine-None: Octal Periodicity (Bow Tie mnemonic)

⁠

Exemplary of such a follow-up research project is, of course, the namespace of Quadray Coordinates.

QUADRAYS

⁠

ivm-xyz

⁠

(for more discussion)

Given the level of professionalism around mathematics that characterizes our M4W network, KU is keen to engage with peers regarding a curious vector algebra based on four primary vectors emanating from a common origin (0, 0, 0, 0) and pointing to the four vertices of a regular tetrahedron: (1, 0, 0, 0), (0, 1, 0, 0) (0, 0, 1, 0), (0, 0, 0, 1).

The angle between any two such “quadrays” is ~109.47 degrees.

⁠

Home Base Tetrahedron

Home Base Tetrahedron

⁠

four quadrays

four quadrays

⁠

Although such a “quadray coordinates” apparatus satisfies

the eight axioms of any Vector Space⁠

, the four so-called “basis vectors” fail to count as linearly independent, because they add to give the zero vector.

A more strict adherence to convention would hold any three of the four elementary quadrays be considered “basis vectors” with the fourth being the negated sum of those three.

⁠

Quadrays come with alternative notions of area and volume, which express a number of unit triangles and unit tetrahedra respectively.

To multiply any two vectors with a shared origin, close the lid (connect the tips) and evaluate the area in

etus⁠

(unit triangles, equilateral).

⁠

7 x 10 = 70 etus

7 x 10 = 70 etus

⁠

To multiply three vectors with a shared origin, do the same (close the lid) and evaluate the volume in terms of unit tetrahedra (tetravolumes, tvs).

⁠

2 x 2 x 5 = 20 tvs

2 x 2 x 5 = 20 tvs

⁠

The 60 degree corners of an equilateral triangle, and the 60-60-60 degree corners of the regular tetrahedron, serve to define a canonical framework (grid pattern, lattice) against which vector pairs or triples at arbitrary angles to one another, but with a common tail, will have their respective “closed lid” areas and volumes.

IVM means Isotropic Vector Matrix, the scaffolding of edges connecting CCP (or FCC) balls in a

tetrahedral-octahedral honeycomb⁠

, at one time patented by RBF as

the octet truss⁠

. Alexander Graham Bell

used an IVM⁠

to make “kites”.

Quadrays (also known as IVM vectors) are isomorphic with XYZ vectors, in providing a unique coordinate address for every point in Euclidean Space. Instead of the eight octants of XYZ space, signified by sign combinations (+,+,+), (+,-,+), (-,-,+), (-,+,+), (+, +, -), (+,-,-), (-,-,-), (-,+,-), we get four quadrants of that same space signified by which vector is dormant (set to 0), i.e. which is opposite whatever quadrant a point is within: (+,+,+,0), (+,+,0,+), (+,0,+,+), (0,+,+,+).

⁠

XREF:

DAF on TrimTab Coda⁠

⁠

The above cube would be assigned a volume of 8 in XYZ, as in each octant there is a unit cube with edges R.

The tetrahedron formed by this cube’s face diagonals, expressed in units of of D = 2R, is sqrt(2) and our volume formula for the IVM tetrahedron (here premised on D as unit) returns a volume of sqrt(2) to the 3rd power i.e. 2 sqrt(2).

The entire cube (8 in XYZ world) would be thrice that volume, given any cube face-diagonals regular tetrahedron is 1/3rd the volume of that cube in which it embeds (the theorem generalizes to generic parallelepipeds).

That ratio, then, of IVM volume to XYZ volume, or 6 sqrt(2) to 8 is S3, the Synergetics Constant, about 1.06066.

>>> Tetrahedron(sqrt(2), sqrt(2), sqrt(2), sqrt(2), sqrt(2), sqrt(2)).ivm_volume()

2*sqrt(2)

>>> 3*2*sqrt(2)

6*sqrt(2)

>>> N(6*sqrt(2))

8.48528137423857

>>> N(6*sqrt(2)/8)

1.06066017177982

The four “basis vectors” do not have unit length. Rather unit length is reserved for the six edges of the tetrahedron they form.

Also, the 12 IVM ball centers distance D from (0, 0, 0, 0) have integer coordinates {2, 1, 1, 0} meaning all 12 unique permutations of those numbers: (0, 1, 1, 2), (0, 1, 2, 1)... (2, 1, 1, 0). The upshot: all IVM ball centers all have whole number coordinates.

>>> Qvector((2,1,1,0)).length()

1.00000000000000

AK: An inner product space is a vector space with an inner product

https://en.wikipedia.org/wiki/Inner_product_space⁠

⁠

KU: ... which answers the question: a vector space with no inner product is possible, or we wouldn't distinguish those without one.

Dot product is typically introduced when expressing Euclidean distance or length. How do we deal with length with a Qvector (a subclass of Vector)?

It may at first seem horrific for our (1,0,0,0) to not be unit length (what must the distance formula look like then??), but it's a comfort that all permutations of (2, 1, 1, 0) e.g. (0, 1, 1, 2), (1, 0, 1, 2) ... 12 possibilities, have length D or 1 (or D=2 if we set R=1).

⁠

That's D from an IVM ball center to the 12 neighboring balls at the corners of a cuboctahedron. D in every of 12 directions. This is our "comfort zone": the tetrahedral-octahedral honeycomb.

⁠

https://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycomb⁠

(Bucky Fuller is mentioned)

The canonical form of a quadray is the one with at least one element 0 and the other three elements non-negative i.e. no negative elements are needed. That's because our four so-called "basis vectors" are all positive.

Any point in ordinary 3D XYZ space is reachable as a linear combination of six spokes radiating from (0,0,0) i.e. X, -X, Y, -Y, Z, -Z. The 4 positive quadray vectors A, B, C, D or (1,0,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1) likewise combine linearly to span the same space, but with no need of any opposite pairings.

Per any point P, addressed as a 4-tuple, one quadray stays dormant (0), and potentially involves the other three, the ones edging the quadrant the point is in.

The XYZ "jack" apparatus (six spokes) divides space into eight octants, whereas the IVM quadrays "caltrop" (four spokes) divides space into four quadrants.

⁠

We can always "normalize" (a, b, c, d) to that canonical form, by subtracting (n, n, n, n) where n = minimum(a, b, c, d). Since (n, n, n, n) is the additive identity (much as n/n is the multiplicative identity), the effect is to single out a canonically unique 4-tuple “lowest terms” expression for an equivalence class of possible addresses.

def norm(self, arg):

"""Normalize such that 4-tuple all non-negative members."""

minarg = min(arg)

return IVM(arg[0] - minarg, arg[1] - minarg, arg[2] - minarg, arg[3] - minarg)

So, for example, if you add (1,1,0,0) + (0,1,1,1) to get (1,2,1,1), that normalizes to (0,1,0,0) by subtraction of the additive identity (1,1,1,1).

On the other hand, there's always an alternative normalization based around an (n, n, n, n) such that (e0, e1, e2, e3) + (n, n, n, n) gives coordinates that add to zero i.e. ((e0 + n) + (e1 + n)

+⁠

(e3 + n)

+⁠

(e4 + n)) = 0 i.e. the sum of the elements is now zero. The algorithm: make n be the average (e0 + e1 + e2 + e3) / 4.

Let's call this function norm0 of a Qvector type object instead of norm.

From the

qrays.py source code⁠

def norm0(self):

"""Normalize such that sum of 4-tuple members = 0"""

q = self.coords

av = (q[0] + q[1] + q[2] + q[3])/4

return IVM(q[0]-av, q[1]-av, q[2]-av, q[3]-av)

And here's the distance formula for a quadray, reading its own coordinates:

For IVM Vectors:

def length(self):

"""

Uses norm0

"""

t = self.norm0()

return sp.sqrt((1/2) * (t[0]**2 + t[1]**2 + t[2]**2 + t[3]**2))

That looks a lot like Euclidean distance. It's like: normalize around 0 using norm0 and then do like the usual distance formula but with a 1/2 under the radical. We do not feel compelled to define a “dot product” for quadrays. The most obvious definition would not signify perpendicularity if zero, which seems too surprising to fit expectations of a dot product, so why define one for Quadrays? We have length regardless.

What's true is: since every quadray has a corresponding XYZ vector, we should be able to

Convert to XYZ

Use the XYZ distance formula: 2nd root of the sum of 2nd powers of a Vector-self’s (x, y, z).

For XYZ Vectors (here we might use the standard dot product):

def length(self):

"""Return this vector's length"""

return rt2(self.x ** 2 + self.y ** 2 + self.z ** 2)

Above, obtaining the length of qray (1,0,0,0) employs the native length method involving norm0, whereas the corresponding XYZ vector employs its own length method, and it gets the same answer (symbolically algebraically, not just numerically, thanks to sympy).

We can continue calling this length measure (computed either way) Euclidean Distance. It's the same for both ivm vectors and their corresponding xyz vector counterparts in our implementation.

⁠

Screen Shot 2024-03-08 at 8.58.23 AM.png

⁠

Quadrays have length sqrt(6)/4 when D equals 1. The tetrahedron they define, of edges D, then has volume 1. We’re free to set D to 1 or to 2, R to 1/2 or 1, depending on what seems most useful and/or logical in the given circumstances. D for Diameter, R for Radius.

Clearly, getting these results entails using different algorithms for area and volume, given other dimensions as input, such as the six edge lengths.

We have at least three such edge-length-based algorithms for volume, deriving from formulae by Cayley-Menger, Piero della Francesca and Gerald de Jong.

Gerald’s algorithm outputs tetravolumes directly, given any six edges, entered in a specific order: any apex 0 to opposite base corners 1, 2, 3 (so 0-1, 0-2, 0-3) and then around the base starting with the first i.e.: 1-2, 2-3, 3-1. Pass in those lengths as a Python tuple, and let the magic happen.

⁠

The other two edges-based formulae incorporate a sqrt(9/8) conversion constant in their derivation, the ratio of an R-edged unit cube in XYZ to a D-edged unit tetrahedron in the IVM where R is any IVM ball radius, D the diameter (2R).

We also have a 5x5 matrix-based method to find volume, that starts with inputting the four vertices as quadrays (i.e. not the six edge lengths) in a matrix, and computing a determinant.

This volume from tetrahedron vertices method, derived by

Tom Ace⁠

, and another volume-from-edge lengths method, derived from Cayley-Menger, are shown in the figure below:

⁠

Screen Shot 2024-03-05 at 11.09.03 PM.png

from the BEAST modules slide deck

from the BEAST modules slide deck

⁠

For further reading

⁠

The Tetrahedron: Volume from Edges Formulae⁠

(Jupyter Notebook)

Tetravolumes and Quadrays⁠

(Jupyter Notebook)

Quadray Coordinates⁠

(Wikipedia)

Qrays.py source code⁠

(source code on Github)

BEAST modules on Synergeo⁠

⁠

BEAST modules videos:

⁠

Notes

Pixels/Voxels.

Generic treatment of space/time — grid for things to exist in (XYZ, IVM).

This is one use — giving reference frame for the CCP Sphere Packing coordination, whole integer values.

⁠

https://www.perplexity.ai/search/Write-in-Latex-kv3fcmJmQjurTBEW33v3Pg⁠

⁠

https://en.wikipedia.org/wiki/Brillouin_zone⁠

— relates to Bott periodicity, physics of Topological insulators. The symmetries of physics (e.g. parity/chirality, time-reversability). In that metal — if that outer shell has overlap, there is electron movement. They study the single cell (cubic, hexagonal, etc) in the infinite lattice.

Grothendieck Site.

Topos/Sheath

Arthur Loeb (preface to

Synergetics⁠

) — prefered the Sphere nucleation approach to teaching the FCC. Coxeter saw this as elementary and also an advance in research/pedagogy. Hence.... find schools that are open to this, and share it / compete.

Ontology could go into the code comment section.

Synergetics⁠

— using the terms and vocabulary that are elsewhere (e.g. honeycomb, tetrahedron, octet truss, crystal geometries).

There was the forking/disambiguation of “Synergetics” into the Hermann Haken (multiscale “enslaving” dynamics) and the Bucky Fuller (geometry of nature/thinking).

This is bread and butter math, and higher.

Fuller — World Game, Weaponry to Livingry, Trimtab, etc. — getting on with the doing, and the desire to know.

Addressed how God views one’s life.

In the cycle — Take a stand, follow through, reflect — needing the hypothesis to work with.

Chronologies

Fed the M4W Kirby repo’s into

Cursor.sh⁠

to digest and improve it.

⁠

https://github.com/4dsolutions/m4w⁠

—

To take the

Quadrays.py⁠

and

Tetravolumes.py⁠

— bring in more volume formulas and treatments (historical formalisms, figuring out ways to compute the volume of a Tet of the edges..... what order for the edges and information?).

XYZ-IVM capabilities are there. — One important aspect is that the operations remain Symbolic algebraic, not just floating point. Hence more of the Wolfram/Mathematica approach. And then can do the arbitrary precision floating point calculations as needed.

AGENDA

DISTILLING SYNERGETICS

INTEGRATING SYNERGETICS

QUADRAYS

For further reading

Notes

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