Alternation is a non-symmetric relationship between some pairs of regular maps of the same genus. Any regular map whose faces have an even number of edges can be alternated. Any regular map whose faces have four edges, or with twice as many vertices as faces, can be alternated to yield another regular map.
Alternation is the removal of alternate vertices, cutting them right back to the nearest remaining vertices. Thus we can alternate a cube to obtain a tetrahedron.If a regular map is described by
The di-square is a special case, as it is covered by both p=q and p=2. When we cut off two of its vertices, we are left with two vertices and four edges running between them. We can choose how far to condense them, and end up with the 4-hosohedron, the 2-hosohedron or the monodigon.
As alternation only yields regular maps that we already know about (because we can obtain them as duals or by un-rectifying), these pages on regular maps do not list cases in which one regualr map can be obtained from another by alternation.
Other relationships between regular maps