Back on my paper-writing shit. I’ve had this project for 10 years or so, and pick it up every few years. Now is one of the times apparently
It’s a sick result, but how I write the paper depends on my intended audience—I don’t really want to treat it as research-grade, because that requires a lot of engagement figuring out how much of it’s been published in different capacities. It’s a result about cubes in certain dimensions, which means about 800 topics can slide into relevancy; cubes are central mathematical objects (this is part of what makes my topic so fun)
I’d much rather popularize this cool construction and write a more laid-back expository deal, since you just need undergrad-level math to follow along. But your presentation’s gotta be real fuckin tight to get that right
And I never really learned much about the publishing environment when I was still in academia, so I’ll have to figure out appropriate journals in either case, assuming they exist. And reach out to people to get advice/confirmation the result isn’t widely-known. I’ll probably reach out to my former advisor and maybe cold-email a few randos, assuming my shit’s good enough to justify bothering people that don’t know me. For sure a librarian or something to get help with a real literature search
I wanna do it (one might even say that I’ve surpassed “want” and in fact have a compulsion at this point), but it’s a whole big thing and every part except “do math and talk about it” is pretty annoying
Still improving the paper tho so that’s good
I absolutely love that you’re so passionate about cubes that you want to publish accessible cube research papers on cubes (where ‘accessible’ means ‘people who spent three years studying cubes’), purely out of cube-interest.
If you want tight presentation and just wanna talk about something cool, you could try hitting up a math YouTuber to see if they’re interested in a collab of some kind. Idk if journals accept YouTube videos as a format for submissions, but it’s worth a go.
Thank you!
You bring up a very good point—since I really just want the construction popularized, there may not be any point to engaging formally with academia. It just depends on the channels for distributing mathematics. It may turn out that academia still maintains significant control over the “best” channels. Or those channels may turn out to be YouTube videos or blogs that are somewhat independent of academia; I’m really not sure
While it’s not central to the discussion, I do want to mention that the result isn’t strictly about cubes. It’s really about a relationship between regular polytopes. You get infinitely many in dimension 2. You get 5 in dimensions 3 (the Platonic Solids). You get 6 in dimension 4, then you only get 3 in dimensions 5 on up. In every dimension, you always get a cube, a simplex (triangular pyramid in 3D), and a cross-polytope (the octahedron in 3D). It turns out that in certain dimensions, there’s a super neat relationship between the 3, beyond the more well-known relationships
Basically it’s a riff on this shape
Is there a proof about having those 3 in every dimension after 5?
I remember there's certain math stuff you encounter in graphs where some stuff breaks at certain numbers of vertices but then the pattern continues for higher vertices.
Math is cool but caused me a lot of stress and I need to get over feeling useless in the face of how much math exists.
The proofs certainly exist, although I haven't looked at them in much detail. Higher-dimensional geometry gets pretty complicated pretty quickly, as you can imagine.
It wasn't until the mid-late 19th century that people seriously did geometry in dimensions higher than 3-4; this Wikipedia page gives decent historical overview. Coxeter's Regular Polytopes is THE book on the matter, I'm certain it's in there (incidentally, Coxeter is the guy that beat me to this result in the 1930s. But from the literature review I've done, very few people noticed this particular result, so it's more-or-less entirely unknown among the mathematical community).
But you can give a rough sketch of why those 3 exist without too much effort. I'll just describe them, and only say a bit more about about complicating factors.
The entire framework of coordinate geometry is basically premised on the existence of cubes. If you look at the vertices of the 3-cube you can see how it should work in higher dimensions:
(0,0,0) (1,0,0) (0,1,0) (0,0,1) (1,1,0) (1,0,1) (0,1,1) (1,1,1)All vertices have coordinates either 0 or 1, and there's a vertex for each possible combination. The cube is the "convex hull" of these vertices (like starting with the vertices and shrink-wrapping around them). They're gonna have 2^n vertices and 2n facets in dimension n ("facets" = faces of the highest dimension. Eg, six 2-dimensional faces in 3D). You can know everything about its faces of various dimensions if you're good with combinatorics, because the n-cube has a strong relationship with the 2^n subsets of an n-element set. For any computer science nerds out there, Karnaugh maps are just exploiting this relationship between subsets and cubes.
For the cross-polytope (aka orthoplex), it's similarly based on coordinate axis. Its vertices in 3D are even easier:
(1, 0, 0) (0, 1, 0) (0, 0, 1) (-1, 0, 0) (0, -1, 0) (0, 0, -1)In general, these guys are the convex hull of plus-or-minus the standard basis vectors. They have 2n vertices in dimension n, and 2^n facets (triangles in 3D). That's because cubes and cross-polytopes are dual to one another. That wikipedia page is specifically about 3D, but the concept holds in any dimension. So cubes and cross-polytopes exist in all dimensions as a tight pair: if you believe in one, you have to believe in the other.
Then there's the simplex (triangular pyramid in 3D). It's a little less well-known, but unequivocally simpler: they have n+1 vertices and n+1 facets in dimension n.
It's easier to jump up an extra dimension to find them. There's a 2D one hiding among the vertices of the 3-cube:
(1, 0, 0) (0, 1, 0) (0, 0, 1)
So, it's relatively easy to describe the shapes and give realizations of them. The hard part really comes talking about "regularity". In 2D, it's easy: Equal edges, equal angles. We give an iterative process for 3D: Regular faces, all faces congruent, and the same number meeting at each vertex. But how to generalize into 4D? 5D? Do we need to keep adding onto the definition each time?
So that way of thinking was judged insufficient. The whole point is that we want shapes that look “the same” from multiple viewpoints, to the highest extent possible. Thus, the current state-of-the art is to define regularity by saying that "A polytope is regular if and only if its symmetry group acts transitively on its set of flags", where a flag is a complete, nested sequence of faces (for example: A vertex contained in an edge contained in a face. If you pick two of those for the 3-cube, you can find a way to spin/reflect the cube and swap those two flags). So now you've really gotta be at least somewhat comfortable with group theory, and a whole host of other math kinda slides in at this point. It's not too bad I don't think, but it takes work.
