This page is for my thoughts on the "tetrahedral" thinking that Kirby Urner writes about in his analogies of Martian (tetrahedral) vs. Earthling (cubic) societies.
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:
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.