Alternation (geometry)

Alternation of a cube creates a tetrahedron.

Alternation of a truncated cuboctahedron creates a nonuniform snub cube.

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.^[1]

Coxeter labels an alternation by a prefixed h, standing for hemi or half. Because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge.

More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be alternated. For example, the alternation of a vertex figure with 2a.2b.2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divides in half into degenerate digons. So for example, the cube 4.4.4 is alternated as 2.3.2.3.2.3 which is reduced to 3.3.3, being the tetrahedron, and all the 6 edges of the tetrahedra can also be seen as the degenerate faces of the original cube.