Tuesday, February 14, 2012

Hypermap representations of ordinary maps

When starting from an ordinary map M, the operation of inserting a white vertex into each original edge (to obtain the map's Walsh representation as a hypermap) can be subsumed in a composite map operation that directly gives one of the hypermap representations.

If Mp() is a map operation, denote as as Mp'(), the rotation of Mp, the composite operation Mp(Du()). Then the hypermap representations of an ordinary map M are:

W = Su(M)

X = Du(Su(M)) = Pa(Du(M)) = Pa'(M)

J = Le(Su(M)) = Be(M)

B = Du(Be(M)) = Mt(M)

C = Me(Su(M)) = Ri(M)

Q = Du(Me(Su(M)) = Ra(Su(M)) = Du(Ri(M))

where the newly referenced map operations are parallel, bevel, meta, and ring.

Here are examples for the skeletal graph of the pentakis dodecahedron (the dual of the buckyball) embedded as a hypermap in the sphere.







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