David Koski and I have different pathways into what some might call a Sacred Geometry, which I’ll capitalize as a way of showing my open mindedness and respect for such a notion. Lets get off on the right foot with all the Sacred Geometry buffs out there.
David came in by way of the candle holder, one might say, a star-like pointy rhombic triacontahedron, a home base for Phi, the Golden Ratio. I came in through the Buckminster Fuller corpus, picking up that nomenclature.
Ironically, in an effort to rid himself of inordinate cultural bias, Fuller had steeled himself against using the ritual academic symbols, such as ancient Greek letters, in much of his thinking. The Greek letter Phi is nowhere found in Synergetics, even though the rhombic triacontahedron is everywhere.
My influence on David was to stay a stickler for Synergetics nomenclature, such that he gradually adjusted to thinking in terms of “E mods” instead of “T mods”. What was the difference between these two shapes, and its significance? That’s a question that only comes up in Synergetics, in all of world literature, which is why I’m entitled to this term “esoteric”.
Even most Sacred Geometry scholars will have no idea what I’m talking about, because for the most part, they haven’t followed Synergetics in stepped away from a more Biblical unit cube. The E and T mods are tetrahedrons, both 1/120th of a rhombic triacontahedron, but distinguished by the radial depth of the latter.
David and I both realize that the many memorable ratios between polyhedron volumes, a fixation of Fuller’s, work just as well whether we value the volumes in terms of a unit cube or unit tetrahedron. Two cubes still equal the rhombic dodecahedron’s volume where the latter has long and short diagonals for each of its twelve diamond faces.
The cube, dodecahedron, and triacontahedron are all zonohedrons, in consisting of oppositely parallel rhombuses. The cube’s squares are considered rhombuses too, as well as rectangles, and some might even say squares are trapezoids of equal bases.
We could begin with the cube itself, which is where Western Civilization likes to start, by default. Yes, two regular tetrahedrons suggest themselves immediately, as sets of face diagonals. Each one, taken alone, has one third of the cube’s volume.
You might think more people would know that, but polyhedrons as a topic have been under the rug for some time. They’re associated with Sacred Geometry, magical thinking, astrology, and a kind of dated, ancient (as in dead) form of math.
What saved mere Euclideanism from the dusty past, was Extended Euclideanism of N dimensions. Fuller recognized H.S.M. Coxeter for playing a big role in rescuing this kind of geometry and keeping it relevant. Machine Learning and linear algebra are all about N-dimensional vector spaces, and that’s where the Polytopes live.
As I was saying, each of those tetrahedrons is one third the cube’s volume, but we want that cube to have edges one, and volume one. The tetrahedron has edges 2nd root of two, and yet a thankfully simple volume of one third.
Some people thank Fuller for reminding us of this trivial fact and dismiss the balance of Synergetics as two volumes of wasted words (and pictures). “So you found some nice ratios, big deal”.
However, the model in Synergetics assigns a nice whole number edge length to said tetrahedron, two sphere radii, and a second root of two to the cube’s edge. This reversal of rationality with irrationality puts the onus on the tetrahedron to anchor the volume concept.
Edge times edge times edge looks more like three sixty degree angles instead of three ninety degree ones. Again, no one goes there, following Fuller, except a few lone philosophers such as myself. The vista seems so desolate and alien, so bereft of human breath, that I call it Martian. Welcome to Martian Math.
Koski thinks the simple ratios and Phi based relationships he’s been studying deserve a wider audience and he doesn’t want to make the shift to tetravolumes for reckoning a prerequisite mental jump. That makes some sense to me. Why narrow one’s audience to one or two?
He’s done that already, for my benefit. I use the Koski Identities in my Python programs, sometimes using extended precision numbers, way beyond IEEE floating point numbers, in their capabilities.
In our conversation, I was emphasizing how the Synergetics camp wants a simple tetrahedron volume, of unit preferably, not just because it’s topologically simpler than the cube, in terms of edges and corners, but because it’s what you get with four balls of same radius, tightly packed.
Synergetics anchors its concentric hierarchy of polyhedrons in a sphere packing matrix, the once crystallographers call the FCC, also known as the CCP (when the spheres are big enough to touch each other).
Each ball is surrounded by twelve others, inter-tangent, and no ball seems any different from any other, in how it’s surrounded, forgetting about boundaries for the moment.
The CCP is an ocean of balls packed together, and the rigid skeleton left behind, if adjacent ball centers are connected, is all tetrahedrons and octahedrons. Architects use this skeleton all the time. Alexander Bell devoted himself to its study. Fuller calls it the “IVM” (isotropic vector matrix) in contrast to the more familiar “XYZ” of everywhere face-bonded cubes.
When face-bonded cubes make your XYZ ocean, the contained tetrahedrons of volume one third have no special significance. Their edges are not the main edges of the matrix. It’s when we shift emphasis to cube diagonals, and make the IVM our home grid, that we have a strong reason to want volumes of one and four respectively for the tetrahedron and octahedron.
Why fight so much rationality?
Then, around each ball, we imagine the dodecahedron encasements, each of volume six. Triacontahedrons may enclose CCP balls too, though not in the dodecahedron’s space-filling pattern. What’s their volume?
That’s where the “gap” appears, between the triacontahedron made of Ts, and the slightly bigger one made of Es. When the anchoring matrix is the IVM and the unit of volume is a tetrahedron, then the search is on for a volume five.
The rhombic dodeahedron is volume six and encases the unit radius ball exactly. What is the volume of the corresponding triacontahedron? Just a tad over five, it turns out. Shrinking the volume to exactly five involves shrinking the triacontahedron’s radius by a factor of about 0.9994. Fuller makes a big deal of the cosmic crack, this tiny difference between E and T.
David Koski and I agreed that this tiny gap is there in the math, but only reveals itself when seen in the right light. Once we have the IVM and unit volume tetrahedron, volume four octahedron, only then does it really make sense to carve them both into “orthoschemes” whereas we’ll get to the cube next.
Put poles to the body centers, from face and edge midpoints, and net the irregular tetrahedrons so carved. The octahedron’s fragments contain the tetrahedron’s fragments. We get the As and Bs, left and right, each of volume one twenty fourth.
The T mods, being each 1/120th of a volume five shape, have volume one twenty fourth as well. That’s intellectually satisfying, but not something one would come across without the quest for a volume five. Fuller says “E” is for “Einstein” (“T” for triacontahedron) but we could also say “E” is for Esoteric.
Fuller schooled himself rather severely to no think like your typical Earthling. His commitment was to serve “omni-humanity” and to that end he could not afford to become party to some one-sided nationalism in some zero-sum game. He had his reputation to think about, his internal consistency.
The whole is more than the sum of the parts. That might mean something in terms of intellectual integrity as well. His Russian and Chinese students would not see him as an American imperialist. His American students would not see him as selling out.
Indeed, his book Critical Path creates a shared futuristic tableau in which cool headed humans are not triggered to bring on some nuclear holocaust. They still have a shot at improving the world, and yet no one world culture, nor one world government is in the wings.
We do have our global networks and shared assets, our human and computer languages, our geometries, our cities and transportation infrastructure.
I bring up Critical Path and Fuller’s World Game strategies, to suggest why I think Sacred Geometry is starting to pick up more memes from the Fuller corpus. The so-called Aquarian Conspiracy (note the potentially positive spin on “conspiracy”) is fixated on Gaia and the health of Earth as a living organism.
Geometric icons and symbols connected in complexes that are not about world domination except in a whimsical sense, ala GNU and Linux, is what many practitioners would like to have more of.
If you go out on Youtube, you’ll find more “vector equilibrium” talk this year than last year. Stay tuned.