Klitzing wrote:quickfur wrote:So basically your segmentotopic product (A||B) # (C||D) is basically just (A x C)||(B x D) where x denotes the Cartesian product, and # is a placeholder symbol for your "segmentotopic product"? In other words, you take the Cartesian product of the "top" layer of the segmentotope A||B with the "top" layer of the segmentotope C||D, and likewise take the Cartesian product of the respective bottom layers, and then construct a new segmentotope out of the results. Right? And what you're claiming then is, if A||B and C||D exist, then (A x C)||(B x D) must also exist?

Seems reasonable to me, though it doesn't really add anything new. It's just a way of analysing a segmentotope by decomposition into Cartesian products (or conversely, of constructing a higher-dimensional segmentotope from two lower-dimensional ones). Or am I missing something?

Well, you just miss here the restriction about the height to be real and truely positive.

But if we already know that A||B and C||D exist, then (A x C)||(B x D) should also exist, right?

A different variation of this idea, is if we start with an even polytope E (where "even" means it can be alternated) -- say we call its two alternated forms E^{+}and E^{-}-- and a segmentotope F||G, then we take the convex hull of (E^{+}x F) U (E^{-}x G), where x denotes the Cartesian product and U denotes set union. Under what conditions would the result contain F||G as facets? Is it possible to make the result CRF? For example, one could start with a cube and, say, a pentagonal cupola (pentagon||decagon). Since the cube alternates into two dual tetrahedra, we form the 5D Cartesian products tetrahedron x pentagon and dual_tetrahedron x decagon, and take their convex hull. The result should contain pentagonal cupolae as surtopes. Is it possible to make the result CRF?

Union would not work, as then those would all remain within the same Hyperspace and you'd get rather a compound than a pair of stacked bases.

I think you misunderstood that (1) I was not looking for creating more segmentotopes, I was looking to create new polytopes that contain segmentotope surtopes (surface elements); (2) I defined the operation take the convex hull of the compound.

For example, I just figured out last night that the product of a cube (as convex_hull(tet, dual_tet)) with a triangle (as point||line) can produce the topological equivalent of two 16-cells sharing a cell, or a tetra-augmented tetrahedral prism (probably cannot be made CRF, though).