Message #2691

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Re: About me
Date: Tue, 19 Mar 2013 23:38:37 -0700

On 3/19/2013 10:59 PM, schuma wrote:
> […]
> About the regular polytopes in high dimensional space, Melinda said,
>> Cracking my bible (Regular Polytopes) to chapter and verse 7-8 gives …
> Believe it or not, the first thing after I got home today was also checking this book.

If I ever need to swear on a bible, I will insist on this one.

> Before this verse, Coxeter was calculating the dihedral angles. I think the argument can be explained using this example.
>
> The dihedral angle of a 3D cube is 90 degrees. When you use some cubes to make a 4D regular polytope, how many cubes can you fit around an edge? It’s necessary that the sum of dihedral angles is less than 360 degrees to make a bounded 4D polytope, or equal to 360 degrees to make an unbounded tessellation. So, two cubes are too few (the outcome is flat). Three cubes around an edge are OK (hypercube). Four cubes around an edge will form the tessellation. So three cubes around an edge is the only valid way to make a 4D bounded polytope using cubes as faces, in Euclidean space. The necessary condition is that the dihedral angle must be small enough. As I understand, equation (7.77) formalizes this argument in math.
>
> To make 5D regular polytopes, we need to pick some 4D regular polytopes to be the faces (that’s part of the definition of regularity). It turns out, the dihedral angles of all the special 4D polytopes are just too large to fit around an edge. So they cannot be used. So in 5D, we only have {3,3,3,3}, {4,3,3,3} and {3,3,3,4}.

Wow, what a great description, Nan. I’m starting to see it! Thanks for
spelling that out.

BTW, the criterion for regularity requires all the faces also be the
same regular polytopes, not just any regular polytopes such as prisms. I
know you know that. I just wanted to clarify it.

>
> In 6D, the Schlaefli symbol needs to be {p,q,r,s,t}, where {p,q,r,s} may be {3,3,3,3}, {4,3,3,3} or {3,3,3,4}, and {q,r,s,t} may be {3,3,3,3}, {4,3,3,3} or {3,3,3,4} (this is the definition of regularity). So q,r,s have to be 3,3,3. And you can’t make anything but {3,3,3,3,3}, {4,3,3,3,3} and {3,3,3,3,4}. Without fancy building blocks, you just can’t make fancy stuff. By the same argument (induction), these polytopes are the only regular ones in higher dimensions.
>
> You may argue that the definition of regularity is too strong. If we relax that, maybe we get more interesting things.

I feel that when you relax a constraint you *always* get interesting
things. I feel that large parts of pure math are done by branching off
from highly symmetrical roots by carefully relaxing one constraint and
seeing what you find. That’s how I got interested in IRPs.

And then there are those really rare cases where someone goes in the
opposite direction by recognizing and describing how two sometimes
seemingly unrelated results share a previously unknown deep relationship.

-Melinda