Alternation
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 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.
Alternation, when taken as a local process in fact alternatingly maintains respectively rejects a vertex (say, or any other element). Whenever all 2D faces have an even count of vertices, then that local process returns to the starting point again within the right parity whichever path you went. As soon as there are 2D faces encountered with an odd count of vertices, then the same local operation might return within the wrong parity to the starting point. That then would require to maintain as well as to reject that vertex. In fact, when this local application then being continued nonetheless is just what a holoalternation (holosnubbing) is. I.e. every vertex then gets maintained as such as well as replaced by its vertex figure (or the respective sectioning facet underneath). Accordingly a snub always has half the vertex count of its starting figure. But a holosnub always has the same vertex count as its starting figure.
Alternatively one could also think of holosnubbing when one replaces the starting figure with the according Grünbaumian double cover, i.e. the one which uses 2 fully incident copies of every even 2D face and uses the doubly wound doublecover for every odd 2D face. If one then would apply usual snubbing to that replacement, the outcome surely is nothing else than the process of holosnubbing being applied to the starting figure instead.
Examples[edit | edit source]
The following are examples of polytopes resulting from alternation.
- The vertex alternation of the hexagon results in the triangle.
- The alternation of one of the 2 edge types of a ditetragon results in a rectangle.
- The vertex 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 vertex alternation of the 2n-gonal prism results in a (non-uniform) n-gonal antiprism.
- For every n, the alternation of the lacing edges of a 2n-gonal prism results in a (non-uniform) n-gonal prism.
- The alternation of the triangles of a small rhombicuboctahedron results in a (non-uniform) truncated tetrahedron.
- For every n, the alternation of the n-hypercube results in the n-demihypercube.
- The rhombic dodecahedron can be vertex-alternated in two different ways. One produces the cube while the other produces the octahedron.
- Alternating a square tiling produces another square tiling, with edges sqrt(2) times longer than the original.
Which polytopes are alternatable?[edit | edit source]
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 whose faces all have evenly many sides can be vertex alternated.
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.