Powertopes

[quickfur]

I've seen Bowers mention powertopes, and it's also mentioned in HDDB, but I can't find a definition anywhere. Can someone please supply a definition of what a powertope is?

[Keiji]

I found powertopes very difficult to understand in the first place, but I'll try my best to explain how they're formed. To form "the A of the B" or "B to the A" where A is n-D and B is k-D, we first place shape A on an n-plane, and consider the vertices. Shape A is required to have "brick symmetry" (Bowers' term), i.e. a vertex at point (x,y) implies points at (-x,y), (x,-y) and (-x,-y) (or more for 3D+). Each set of these points will obviously be a rectangle (or hypercuboid in general). For each axis in the hyperplane of shape A we create a k-plane P including that axis such that all k-planes P are perpendicular to each other. Then, for each (n-1)-facet in the hypercuboid we create a copy of shape B which lies in one of P (there can only be one P for any facet). The powertope is then the convex hull of the vertices created.

I hope that made sense, anyway. If not, let me know and I'll post up my email conversation with Bowers himself

[wintersolstice]

The thing I understand about "powertopes" is that if you've got a polytope "P" then then the "square" of it is the "P,P duo-prism" if this is called "D" then the "cube" of P is the "P,D duoprism" and so on. The "ortherplex" (cross-polytope) of "P" is made by taking the dual, call it Q, take the "hypercube" (of the same no of dimensions) of Q and then find the dual of that.

I'm not sure about others.

[wendy]

It's rather a confusing definition, but the reference to 'brick symmetry' should make it clear to me.

One can represent spheres, cubes, and orthotopes in N dimensions, as the Nth power of a line, when different products are applied. These three figures together give rise to "brick symmetry" (ie nR = n orthogonal mirrors).

One can then represent any polytope P, as a radial function, such that the centre is zero, and the surface is always one. The layer of P=k represents then the polytope P scaled to a size of K.

For example, the line from -1 to 1, gives a radial function of "abs(x)", when 0 is taken as the centre.

The general hypercube gives a radial function of max(x,y,z,...).
The general orthoplex of edge r2, gives a radial function of sum(x,y,z,...)
The general sphere of diameter 2 is given by the radial function of rss(x,y,z,...), where rss = root-sum-square.

There was some discussion on the general brick-symmetry, where one might make figures out of nested unit functions of different types.

eg cylinder = [(x,y),z] = max(rss(x,y),z)
duocylinder = [(w,x),(y,z) = max(rss(w.x),rss(y,z))

Since instead of using simple lines as the base of these products, one can use polygons, or in general, polytopes, we have

pentagonal prism = [ P(x,y), z] = max(P(x.y),z)
pentagonal crind = (P(x,y), z) = rss(P(x,y),z)
pentagonal tegum [in 3d, bipyramid], = <P(x,y),z> = sum(P(x,y),z)

One can, then of course, replace all w, x, y, z, by radial functions representing the same polygon, eg a pentagon. A pentagon ^ cylinder is then the product

pentagon ^ cylinder = [(x,y),z], where x, y, z are taken to be the radius in pairs of a six-dimensional space, containing spheres.

You have, for a start, x = P(x1, x2), y = P(y1, y2), and z = P(z1,z2)

A ray in six dimensions (x1,x2,y1,y2,z1,z2), then strikes the surface, when

max(rss(P(x1,x2),P(y1,y2)),P(z1,z2)=1.