Alternation

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A non-uniform square antiprism, created by alternation from an octagonal prism.

Alternation (also known as alternated faceting) is a procedure by which half of some elements of a polytope are removed, thus creating a new one. Most often, and unless otherwise specified, the elements taken are vertices, but edges and other elements may sometimes be alternated too. The process of vertex alternation is closely related to snubbing. It applies to any polytope whose vertex adjacency graph is bipartite.

To alternate a polytope, one first 2-colors the chosen elements, and removes all of those of a certain color, say black. The facets of the polytope then either are alternations of the former facets in turn or are so called sefas (sectioning facets underneath the being removed element of the original polytope). In case of vertex alternations those would just be the according vertex figures.

If this process creates any degenerate facets, such as digons, these usually are removed. For instance, the alternation of a cube is considered to be the tetrahedron, but could well be treated as a tetrahedron with extra digons at each edge.

Generally, any given polytope with an alternation of some of its elements has in fact two different alternations, resulting from either color choice of elements to remove. For instance, a rhombic dodecahedron can either be vertex alternated into a cube or into an octahedron. In the case where the polytope is uniform, however, both vertex alternations result in congruent polytopes (albeit sometimes in an enantiomorph pair). 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 globally 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.

Relation to snubbing[edit | edit source]

Snubbing however adds to the process of mere vertex alternation usually also the secondary process of edge resizement back to all unit edges. It is this secondary process, which might or might not be applicable. The mere alternation however always is - at least locally, cf. the theorem below. In its oldest use of the word, snubbing was applied to omnitruncates only, but later became applied more generally.

Examples[edit | edit source]

The following are examples of polytopes resulting from alternation.

Which polytopes are alternatable?[edit | edit source]

A petrial tetrahedron cannot be alternated.

A polytope is vertex 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 vertex alternatable, even though all of its faces will be squares.

Nevertheless, the following result can be established for convex polyhedra.

Theorem — Every convex polyhedron with finitely many elements whose faces all have evenly many sides can be vertex alternated.

Proof —

A convex polyhedron's vertices, edges and faces can all be put in correspondence with those of a planar graph. Thus, it suffices to prove that any finite planar graph G whose faces all have evenly many sides is bipartite.

Suppose for sake of contradiction that G contains some cycle with an odd amount of vertices. Since G is finite, there are finitely many such cycles, and we may thus take the one that encloses the least area. Call this cycle C.

If any two vertices in C were linked by an edge interior to C, then C could be subdivided into two cycles enclosing a lesser area, one of which would necessarily have an odd amount of vertices, thus contradicting the minimality of the area of C. Thus, C must enclose a connected area, and is thus a face of G with an odd amount of sides, contradicting the assumption that G contains no such face.

Reaching a contradiction in either case, we conclude our theorem.