Yakking on Facebook (continued)

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.

Yakking on Facebook

Esteban Trev HNY ET. I studied philosophy in university with everyone saying “whaddya gonna do with that?”. They predicted I’d end up in IT, which is fairly correct, but I also kept doing philosophy as a hobby, and that ended up adding to my net worth.

Anyway, the philo guy I zoomed in on most was Ludwig Wittgenstein, wondering if you’ve grokked him. I live until the 1950s. He came out with essentially two famous philosophies with an interim in between. From a very rich Vienna based family, fought for Austria as an artilleryman, made a prisoner of war.

But he skipped out on being rich, ironically because his family put pressure on the siblings to really succeed, make a name for themselves, and since he was born into a rich family, becoming rich was not an option, as he was rich already. No fame and glory down that road for sure.

So he lived like a hermit so he could pal around with other philo guys, like Bertrand Russell, and investigate the meaning of language to its logical (or illogical as the case may be) core.

When it comes to your example of 5 + 5 = 12, his later talk was of “language games” and he’d give examples of simple games in his posthumous Philosophical Investigations (book) which overlapped his Remarks on the Foundations of Mathematics. Some guy says “slab” and another guy brings him a slab. He sketches these actions, sometimes involving color coding and lookup tables.

It’s branching off the latter philosophy that I’d bring up the game of Quadrays and the Alternative Volumes Table (AVT) that goes with.

Games as a concept come with rules, fouls, but also innovation, and maybe episodes of ambiguity, when people just aren’t in agreement on how the rules should extend in some special case situation.

Speaking of which, this will seem tangential: I’m interested in sports wherein someone does an amazing move that’s not explicitly against the rules, but then it’s banned right away after.

I learned of two such examples recently on YouTube. (1) A figure skater does an actual flip on the ice, heals over head. She lands gracefully but the sport’s judges don’t relish all the neck-breaking and cleanup that’d stem from many imitators failing, on and off camera. Similarly: (2) a long jumper dude set a record by including a flip in mid air. Wow. Banned. Same reasons I imagine.

A lot of people aren’t familiar with what a contrarian Cantor was. I’ve read some of his original stuff and found out he took on our notion that “space is three dimensional” big time. If you ever allow your space to be both finite and discrete for some reason, i.e. let infinity drain away, then Cantor will say: hey, I can visit all your points in sequence, like we do in computer memory, so why do we say your space is “3D” even if XYZ works for ya?

At which point the mathematicians get defensive and say “yadda yadda” and innovation occurs.

A lotta layfolks will answer “space is 3D because I only need 3 coordinates, x, y, and z” (Cantor: you sure you don’t need less?). But then an athlete stunt man comes along and says “space is 4D because 4 coordinates works just as well and the minimum inside-outside made of edges faces four ways when enclosing a center”.

At this point, I could add: 5 + 5 =12 i.e. if you make your base different, a tetrahedron instead of a cube (different base shape), or 12 instead of 10 (arithmetic base), you get new moves, new rules, and therefore new truths. You get new truths for new games. Language games. And sometimes there’s confusion about which game we’re playing.