Duoantifastegiaprism

A duoantifastegiaprism or duoantiwedge is a class of scaliform polytopes formed in a similar way to antiprisms, where the bases are two congurent duoprisms in opposite orientations. The simplest non-trivial duoantifastegiaprism is the digonal-triangular duoantifastegiaprism, which has a triangular prismatic (considered as a digonal-triangular duoprismatic) base. The dual of a duoantifastegiaprism is a duoantifastegiategum. A duoantifastegiaprism with zero height is equivalent to a duoantiprism. As a consequence, the duoprism-first envelope of a duoantifastegiaprism is a duoantiprism.

If one of the polytopes is a dyad, then the resulting polytope is a duoantifastegium made out of two congurent prisms in opposite orientations. If both polytopes are dyads, then the resulting polytope is the square disphenoid, which cannot be made scaliform, as the height would be 0 if all edges were equal.

In 5 dimensions, convex scaliform duoantifastegiaprisms exist for all regular polygons with m and n sides, except the above noted case where both m and n are equal to 2. They generally have duoprisms as bases connected by antifastegiums that connect a prismatic cell of one base to a face of the other. The vertex figure is generally a square-digonal disphenoidal wedge (a variant of a polychoron that, if all edges are equal, becomes the bidiminished rectified pentachoron) or a wedge pyramid if one of m or n is equal to 2. If both polygonal bases are identical, then the vertex figure becomes a square-tetragonal disphenoidal wedge and has double the symmetry.

The height of an equilateral (and hence scaliform) n-m duoantifastegiaprism of unit edge length is given by $$\sqrt{\frac{2-\frac{1}{1+\cos\frac{\pi}{m}}-\frac{1}{1+\cos\frac{\pi}{n}}}{2}}$$. If the solution has zero height, satisfied only for (n = 2, m = 2 and n = 5, m = 5/3), then its duoantiprismatic equivalent can be made uniform, which are the hexadecachoron and the great duoantiprism respectively.

Note that nonconvex duoantifastegiaprisms also exist in 5D, if one or both of the base polygons is a star, and are scaliform if the above height formula returns a positive value.

Higher dimensional generalizations of the concept exist as well, with one simple example being the 6D triangular-tetrahedral duoantifastegiaprism.