Alternation

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

M:{2p,q} V F E

(meaning, it is in manifold M, each face has 2p edges, each vertex has q edges, it has V vertices, F faces and E edges), then it can be rectified. This yields a regular map described by

M:{p&q,2q} V/2 F + V/2 E.

Here p&q means, some faces with p edges and some with q.

If p=q, the result is another regular map, and we can simplify the above. The original regular map is described by

M:{2p,p} V F E

and the resulting regular map by

M:{p,2p} V F E

. Thus it is the dual of the original.

If p=2, we obtain faces with two edges, and can convert these into edges. In this case the general formulae simplify in a differetn way. The original regular map is described by

M:{4,p} V F E

and the resulting regular map by

M:{p,p} V/2 V/2 E/2

. Thus in such cases, alternation is the reverse of rectification.

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.

The name "alternation"

The term "alternation" is defined in the Wikipedia entry alternation.

Other relationships between regular maps
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