Alternation

Alternation is a procedure by which half of the vertices of a polytope are removed, creating a new one. It applies to any polytope whose vertex adjacency graph is bipartite.

To alternate a polytope, one first 2-colors its vertices, and removes all of the vertices of a certain color, say black. The facets of the polytope are alternated so that their vertices coincide with the vertices of the other color, say white. The gaps created by the black vertices are finally filled by the vertex figures of the original polytope at these vertices.

If this process creates any degenerate facets, such as digons, these are often though not always removed. For instance, the alternation of a cube can either be seen as the tetrahedron or as a tetrahedron with extra digons at each edge.

Generally, any given polytope with an alternation has in fact two different alternations, resulting from both choices of vertices to remove. For instance, a rhombic dodecahedron can either be alternated into a cube or into an octahedron. In the case where the polytope is uniform, however, both alternations result in congruent polytopes. Thus, in this special case, alternation can be regarded as giving a unique result.

Though having faces with an even amount of sides is a necessary condition for a polytope to be alternatable, this turns out not to be sufficient in the general case. Nevertheless, all convex polyhedra with finitely many elements whose faces have an even amount of sides can be alternated.

Examples
The following are examples of polytopes resulting from alternation.


 * The alternation of the hexagon results in the triangle.
 * The alternation of the cube results in the tetrahedron.
 * The alternation of the great rhombicuboctahedron results in a (non-uniform) snub cube.
 * For every n, the alternation of the 2n-gonal prism results in a (non-uniform) n-gonal antiprism.

Which polytopes are alternatable?
A polytope is alternatable iff its vertex adjacency graph is bipartite. Particularly, each one of its faces must have an even amount of sides. It might be tempting to declare that conversely, every polytope whose faces all have an even amount of sides is alternatable, but this turns out not to be the case. A simple counterexample is the petrial tetrahedron, whose faces are all skew quadrilaterals, but whose vertex adjacency graph is that of the tetrahedron, and therefore is not bipartite.

There are also convex counterexamples with infinitely many faces and/or vertices. For instance, if an infinite amount of triangular prisms are joined by their triangles, the resulting apeirohedron will not be alternatable, even though all of its faces will be squares.

For convex polyhedra with finitely many elements, however, the following result can be established.

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