Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
Andrius: Here are my notes on... Kirby Urner Various themes
Relating to Kirby Urner and his ideas about differences in thinking comparing "human" cubes vs. "Martian" tetrahedrons. 立方体 I think your quadpod is a magnificent concept for illustrating your points. It's very vivid and fun, too. I am impressed by your geometry http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html which is intriguing and persuasive. However, if you line up the corners of the squares and also of the cubes, then you get a progression which is very helpful for teaching calculus, namely, if you consider a square x and grow it by one more bit h so you have a square of sides x+h, then: (x + h)**2 = (x + h)(x + h) = x2 + 2hx + h2 which all make geometric sense, and then you can see why you can ignore the h2 and upon subtracting x2 you are left with 2hx which, when compared with h, gives you the derivative 2x. Similarly, (x+h)**3 = (x+h)(x+h)(x+h) = x3 + 3x2h + 3xh2 + h3 and discarding the small stuff and substracting x3 you are left with 3x2h and dividing by h gives the derivative 3x2. This for me is a very powerful way to illustrate differentiation in a very real sense. And also these binomial expansions are very worthwhile to spend time with and very meaningful for problems in probability, heads and tails: (h+t)**3 or recessive and dominate genes, blue eyes b and brown eyes B (b+B)(b+B) for example. So I'm curious if your triangular thinking has a nice way to talk about this all, perhaps? 四面体 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 ndimensional 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 halfncube, we get 2^(n1) versions of these simplexes. So each of these is an "anticenter". On the other hand, we get the big picture of the halfcube and by taking a subset of dimensions we can look at smaller halfcube 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 "antivolume". 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 nCategorical Physics". http://arxiv.org/pdf/0908.2469v1.pdf On page 33, they mention the work by PonzanoRegge in 1968 on there 3d model of quantum gravity, where spacetime is made of tetrahedra. And in searching on "tetrah" I also see that KapranovVoevodsky studied the Zamolodchikov tetrahedron equation. Keep searching on "tetrah" and you will find... I'm curious whatever you find interesting.
