# Nonconvex Prisms and Antiprisms

The Archimedean Solids are convex with vertices that are identical to one another, but having faces that are 2 or more different kinds of regular polygons. The prisms and antiprisms share all the characteristics of the Archimedean Solids but have historically been described seperately.

The nonconvex Uniform Polyhedra also share all the characteristics of the Archimedean Solids, with the exception that they are nonconvex. This article describes the Nonconvex Prisms and Antiprisms, which share the same relationship to the nonconvex Uniform Polyhedra as the convex Prisms and Antiprisms have to the Archimedian solids.

## Nonconvex Prisms

The Nonconvex Prisms are formed in exactly the same way as their convex counterparts; the only difference is that the top and bottom are nonconvex regular polygons joined by squares. Specifically, starting with two identical nonconvex N-gons for top and bottom, connecting them with N squares results in a nonconvex Prism.

 Pentagrammic Prism (5/2,4,4) Heptagrammic Prism 1 (7/2,4,4) Octagrammic Prism 2 (8/3,4,4)

 As an exercise, consider the nonconvex polygon pictured at the right (this is the first stellation of the regular octagon). The nonconvex prism using it as the top and bottom is the Octagrammic Prism 1 (8/2,4,4). This polyhedron is yet another example of a polyhedron that has an alternate interpretation: it is a compound of two convex solids; can you name them?

## Nonconvex Antiprisms

The Nonconvex Antiprisms are formed in exactly the same way as their convex counterparts; the only difference is that the top and bottom are nonconvex regular polygons. Specifically, starting with two identical nonconvex N-gons for top and bottom, connecting them with 2N equilateral triangles results in a nonconvex Antiprism.

 Pentagrammic Antiprism (5/2,3,3,3) Heptagrammic Antiprism 1 (7/2,3,3,3) Octagrammic Antiprism 2 (8/3,3,3,3)

## Crossed Antiprisms

A Crossed Antiprism is simply a nonconvex antiprism in which the faces pass through the center of the polyhedron. In the two examples below only one face is colored blue to highlight it relationship to the whole.

 Pentagrammic Crossed Antiprism (5/3,3,3,3) Heptagrammic Crossed Antiprism (7/4,3,3,3)

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