by **wendy** » Fri Jan 09, 2009 11:16 am

The cartesian product is a feature of euclidean geometry. You can apply it by drawing shapes in different sets of axies, and then taking the common set.

For a hexagonal prism, you could cut a length from a hexagonal bar. This gives a prism as an off-cut (prisma means offcut!). Alternately, you could take a slab of constant thickness, and cut a pentagon from it. In either way, all space -> layer -> pentagonal prism.

In four dimensions, there are four axies, and so prisms can be cut from 2+2 axies, or 3+1. The former gives for example, polygon-polygon "duoprisms", such as the triangle-triangle (duo)prism. Circles behave as polygons, and a special name is used for the bi-circular prism (duocylinder), rather like a special name is used for the circular prism in 3d (cylinder).

"Duopyramid" is confusing in four dimensions, since you can erect a pyramid on a pyramidal base, or (as in 3d), erect a pyramid on each side of a 3d base. In any case, the term is depreciated. The figure here has no special name over bi-triangular (duo)prism.

Pyramids do not occur in the rotopes.

You can indeed use powers to generate higher polytopes. The products are all discovered in just this way.

PRISM => line ^n = line, square, cube, tesseract, ...

TEGUM => line ^n = line, rhombus, octahedron, 16ch, ...

CRIND => line ^n = line, circle, sphere, glome, ...

PYRAMID => point ^(n+1) = point, line, triangle, tetrahedron, pentachoron, ...

COMB = horogon ^n-1 = horogon, square lattice, cubic lattice, tesseract lattice, ...

The dream you dream alone is only a dream

the dream we dream together is reality.