Another tie-in with Wittgenstein is that Part 2 of his posthumous Philosophical Investigations focuses on "aspect shifts" and their importance to "getting the meaning of". Like "gestalt shifts" -- with the easy example of the duckrabbit.
It's when one is able to shift one's perceptions somehow, that it all clicks into place, and one might even say "ah hah!" or "eureka!"
Going from "5+5=12 is nonsense" to "5+5=12 makes perfect sense -- in base 8" might be like a light going on.
That's more likely to happen if you're already primed to know about bases and just need a little reminder from your post-Sputnik 2nd grade New Math or whatever. Or you went into computers and learned octal and hex.
But if they've not self-consciously studied "number bases" as a topic (as the Common Core recommended dropping, making Americans dummies again) then explaining the punchline is like explaining a joke. If they don't smile and chuckle right away then you face the ordeal of explaining about position notation, carrying, powering and all that (yawn). Not worth the effort maybe. Save your jokes for those who might get them (hard to know in advance).
Cubes may fill space, but if we wanna start by closest packing same-size marbles, and filling in space that way (with gaps), then the cubes lattice might not be where we want to end up.
In going for maximum ball-to-space density (~0.74) and omni-symmetry, we end up with one ball to start, 12 around it (6 squares, 8 triangles), then bump up the frequency (between ball intervals) to 42, then again (92), then again (162) and so on: 10 * F * F + 2 where F = 1, 2, 3, 4... See HSM Coxeter's remarks to that New Yorker fact checker, verifying Bucky got it right.
Anyway, that's where we get our skeletal frame, our iconic ghost ship (cuboctahedron) adrift in the IVM ocean. Always 12 balls around 1. 42 around that...
Said lattice is all tetrahedrons and octahedrons, of relative volume 1 to 4 (no matter how long each edge, just keep them the same) and relatively twice as many 4eyes as Richard Katrinho Rasteirinho Haileisela shows in his video, with the omni-triangulated space-filling rhombohedron of volume 1+4+1 = 6, same as the space-filling RD.
I suppose there's a chance Bucky invented "allspace" as a term, however the practice of filling space with uniform and/or complementary space-fillers, the volumetric analog of tiling a surface (without gaps), is a game played since ancient times (since Archimedes at least). Within that scope, there's the game of finding shapes that do so all by themselves, like cubes and RDs do, without need of left and right handed versions.
Some tetrahedra do that, if not the regular ones, and Synergetics does a lot to map this territory, overlapping work by mathematicians such as Sommerville and Goldberg.
Fuller's A & B make an AAB (left and right A plus a left or right B ) the so-called MITE (MInimum TEtrahedron), an important space-filler that ends up with no outward handedness.
Sure we can throw the door open to other iconic representations of surface and volume, beyond the square and cube. There's the hexagon. There's the sphere.
What's true about the triangle and tetrahedron though, is they're each more minimal than their respective counterparts, nor is it clear that either might be undercut.
In that sense, the tetrahedron makes a strong case for being primary: the simplest cage, the fewest edges to carve inside from outside (the sphere being a complex membrane relatively speaking). The tetrahedron beats the cube at its own game so to speak, with only six edges instead of twelve. Now that it's finally had a chance to strut its stuff, as unit volume, the cube feels a bit on the ropes these days. Some qyoobists are circling the wagons already.
The cube-minded orthogonalists are very much not accustomed to having their authority challenged and I enjoy seeing them get so annoyed, as they don't have a leg to stand on, if their goal is to make us go away.
Even if I have more tolerance for imaginary "fictional" structures (prefrequency) that do no load-bearing, I'm still able to appreciate the many advantages this new brand of 4D talk brings to the table.
It's a privilege to question the authority of the hypercross dogmatists, even from the perspective of another ghost ship captain.
