Duotegum

A duotegum is a class of polytopes formed as the tegum product of two polytopes. The simplest non-trivial duotegum is the triangular duotegum, which is the tegum product of two triangles. The dual of a duotegum is a duoprism. The cross polytopes are duotegums made from lower-dimensional cross polytopes.

If one of the polytopes is a point, then the resulting polytope is identical to the other polytope. If one of the polytopes is a line segment, then the resulting polytope is the bipyramid of the other polytope. Neither of these cases are usually considered duotegums.

The vertex coordinates of a duotegum is determined by the coordinates of two polytopes a and b in mutually distinct coordinate sets. As such, they have a number of vertices equal to the sum of the number of vertices of each polytope.

A duotegum has a vertex count equal to the sum of the vertex counts of the components. Its facets are pyramid products of the the facets of the bases, with a count equal to the product of the facet counts of the bases. In particular, the 4D duotegums generally have digonal disphenoids as cells, with tetragonal disphenoids if the bases are equivalent.

The duotegum of any two isotopic polytopes is also isotopic. In addition, the duotegum of two identical isogonal polytopes is also isogonal.

Unlike simple bipyramids, full duotegums generally do not have regular triangular faces. The only cases where they do are the orthoplexes, and a limited number of other cases such as the pentagonal-pentagrammic duotegum, which has 25 regular tetrahedra as cells.