by wendy » Mon Aug 17, 2009 8:23 am
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.