Half-cubes, sphere-folding, tetrahedrons...

[Andrius Kulikauskas]

Kirby, Joseph, Bradford,

I hope soon to send out my essay that I've been writing. I think it
might even touch on your "closing the lid" operator. I reinterpret the
"demicube" (demihypercube) polytope series Dn as "hemicubes" (halfcubes)
where the most opposite corners of the cube have been identified (the
cube/sphere has been folded in half... like n-dimensional circle
folding?) and so we have spiky Euclidean "coordinate systems" with
double edges, with additional double edges linking the tips of all of
the coordinate vertices, just as you describe. I just don't know how to
call these "trusses"? The point is that we get two different ways of
looking at this. On the one hand, we have a simplex that has grown out
of the "origin". (Just the angles aren't 60 degrees, they are 90
degrees or 45 degrees.) And because our "origin" could have been any
point of the half-n-cube, we get 2^(n-1) versions of these simplexes.
So each of these is an "anti-center". On the other hand, we get the big
picture of the half-cube and by taking a subset of dimensions we can
look at smaller half-cube within that. And from the big picture point of
view, it makes no difference which points we chose to fold by. But it
is a folded volume, so it is an "anti-volume". So the four series will be:
* An simplex (tetrahedrons) Center and Volume
* Bn cubes No-Center and Volume
* Cn cross-polytopes (orthogons) Center and No-Volume
* Dn half-cubes No-Center and No-Volume
These correspond to the four families of classical groups / Lie Algebras
/ Lie groups. That is, they express the symmetries of the above
structures in terms of actions. Some day I'll understand...

All of this to say that your mathematical taste is excellent and keep
following your mathematical sensibility! It's very helpful, inspiring
and encouraging.

Kirby, but I wanted to share with you a long history by John Baez and
Aaron Lauda that I'm looking at, "A Prehistory of n-Categorical Physics".
http://arxiv.org/pdf/0908.2469v1.pdf
On page 33, they mention the work by Ponzano-Regge in 1968 on there 3d
model of quantum gravity, where spacetime is made of tetrahedra. And
in searching on "tetrah" I also see that Kapranov-Voevodsky studied the
Zamolodchikov tetrahedron equation. Keep searching on "tetrah" and you
will find... I'm curious whatever you find interesting.

Thank you!

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665