by wendy » Sun Dec 30, 2018 7:52 am
A figure with a definite surface is a solid. Polytopes in the simplest form are solids with flat faces. The names I use for these is to take the fabric (hedron), and make it a solid (hedrid = 2d, chorid = 3d, terid = 4d, petid = 5d, ectid = 6d, zettid = 7d, yottid = 8d)
Something like the cone is a solid, with 1 vertex, 1 edge, and 2 faces. It's the pyramid-product of a point (1,1) and a circle (1,1,0,1).
The point has a point and nulloid, but no surface. The circle has an edge. We add to these surfaces, a body and nulloid, and multiply the two.
So point (1,1) * circle (1,1,0,1) = cone (1,2,1,1,1). Removing the terminating 1's, gives 2 faces, an edge and a vertex.
A bicurcular tegum is the product of surfaces of a circles, ie 1,0,1 * 1,0,1 = 1,0,2,0,1. Remove the trailing 1, gives a chorid surface (3d), plus two non-intersecting edges.
Multitopes
These are an assembly of polytopes without a closure. For example, the net or unfolded cube would be a multitope. These still have the same 'complete joining' of polytopes, (ie an edge is shared entirely with everything that it is part of). These are used to demonstrate the idea of 'mounting', or correctly joining polytope bits to make a full-size model.