Philosophy: Is Nature Not Using Pi?
My attitude is “duh, that’s obvious” and I don’t see a need to lift a finger. We don’t need more contorted logic and/or heroic polemics at this point. That’s just my personal view of course. If you wanna be a crusader for our “nature is not using pi” meme, go right ahead.
That doesn’t mean mathematics (which needn’t be about anything “real world”) should stop investigating pi, and indeed its expression has been expanded to 100 trillion digits or so recently. Nature wasn’t waiting for humans to reach that goal, before it started making bubbles. That’s obvious, no?
On the other hand, I want readers to know Synergetics addresses the phenomenon of “incommensurability” — which is a word Fuller sometimes uses in place of “irrationality” — and not to sweep it under the rug.
As the quote below spells out, the concentric hierarchy is not and never was about “only whole number and/or rational volumes” (a deceptive rumor, a canard).
986.210 In our always-experimental-evidenced science of geometry we need only show ratio of proportion of parts, for parts of primitive polyhedra have no independent existence. Ergo, no experimental proof is required for 2nd roots and nth roots. Though those numbers are irrational, their irrationality could not frustrate the falling apart of the polyhedral parts, because the parts are nonexistent except as parts of wholes, and exact proportionality is not required in the structuring.
This is where he links to Figure 986.210 showing a cube with prime vector diagonals. In the caption:
“Proportionality exactly known to us is not required in nature’s structuring. Parts have no existence independent of the polyhedra they constitute.”
What are the consequences of the Fuller view, that nature is not using pi? Are we supposed to remove pi from the calculator keyboards or from math libraries? Are we supposed to “censor” or “cancel” pi?
If Bucky had meant something that preposterous, might he have said so somewhere? We probably wouldn’t think of him as a genius then.
I think he was knocking loudly on the door of the philosophers, and by now, they’ve let him in. Why do I think so?
Let’s say I’m seeing progress. It’ll be hard to turn the clock back, to a darker age when Fuller was not yet accepted as a legit author within the philosophy syllabus.
Some may be nostalgic for those days, but I say Synergetics is no longer all that counter culture. You’re no longer a rebel if you think in terms of unit volume tetrahedrons. That’s just another option by now.
And why pick on pi all the time, what about e? What about phi? Sure, phi (golden mean) may be expressed as the solution to a polynomial. It’s irrational but not transcendental. So what, though? In physical terms, what difference does it make?
What D.B. Koski has shown is that Synergetics gets a lot nicer (more economical) if we allow phi back into the picture. We’re definitely doing that.
We can keep picking on pi for rhetorical reasons, because it makes the point with bubbles. That these form so quickly and seemingly infinitely, in the wake of a ship, is important for imagery and intuition.
But if you’re into philosophy, you will likely appreciate the issues that Bucky is raising, about the nature of numbers more generally.
Numbers are not things. Or you don’t need them to be things, let’s put it that way. The “meaning” of a screwdriver is not the screw, which is not to discount the screw’s relevance. Like a screwdriver, the symbol “2” is a tool.
That our symbol “2” “really means the set of all sets of two members” (very Bertrand Russell sounding) is what we might call a dogma or a superstition.
Wittgenstein helped free us from such nonsense. In my view, we’re already past disagreeing with Fuller.
Philosophers are debating intelligently and Fuller’s Synergetics is currently in the process of integrating into the curriculum in the better schools. It’s no longer that much of an uphill battle.
986.213 The cubically-arrived-at spherical volume (A) of a sphere of diameter equal to the unit edge of the XYZ coordinate system’s cube is 4.188. To convert that spherical volume value (A) to that of sphere (B) whose diameter is equal to the diagonal of the face of the XYZ system’s cube, we multiply the volume of sphere (A) by the synergetics hierarchy’s volumetric constant, which is obtained by taking synergetics’ unit VE vector linear constant 1.0198 and raising it to its third-power — or volumetric — dimension, which is 1.0198 × 1.0198 × 1.0198, which equals 1.0606. Multiplying the XYZ system’s cube-edge-diametered (A) sphere’s volume of 4.1888 by the synergetics’ volumetric constant of 1.0606 gives us 4.4429, which is the sought-for volume of the sphere (B)….
By my lights, Fuller should have said “a sphere of radius equal to the unit edge of the XYZ coordinate system’s cube is 4.188” i.e. just over four such cube-fulls of water (~4.19), fill a sphere of radius 1. The well-known formula is (4/3)πRRR where R = 1 in this case.
Fuller’s unit of volume, on the other hand, is a tetrahedron of edges twice the XYZ cube’s, i.e. that same sphere’s diameter D = 2R = 2.
Said volume is nonetheless a tad smaller than a unit-edge cube’s, so it takes about 4.44 such tetrahedrons of water (vs just ~4.19) to fill that same sphere.
If you want more decimal places in the above calculations, you’ll get a concentric hierarchy sphere of about 4.442882938158366247015881 tetravolumes.
For more calculations with more surds and even more digits see:
(scroll down to The Sphere, right after The Cuboctahedron)
The “volume of a sphere” discussion in Synergetics ends up taking us into not only the “nature is not using pi” thread, but into the Synergetics Constant thread (i.e. XYZ volume * S3 = IVM volume), and the “what shape has tetravolume five?” thread, wherein he explains how he ended up with his T modules, 120 of which make a volume 5 rhombic triacontahedron.
Given the dearth of secondary literature on Synergetics since it was first published in the 1970s (in two volumes, from Macmillan), few scholars appreciate the role of Fuller’s Synergetics Constant to this day. Amy Edmondson’s A Fuller Explanation (1987), although a significant contribution, does not mention it.
In case you’re wondering why Synergetics was so slow to gain much traction, confusion around the Synergetics Constant (including Fuller’s own) was one important reason.
Nature is Not Using Pi (related thread):