A Petrie polygon is a polygon found in a regular map by travelling along its edges, turning sharp left and sharp right at alternate vertices.

A hole has a general definition, which applies to polytopes in any number of dimensions. But for our 2-dimensional purposes, we consider it as a polygon found in a regular map by travelling along its edges, taking the second-sharpest left at each vertex. These concepts are related, and can be generalised as follows.A 1st-order hole is just a face.

A 1st-order Petrie polygon is an ordinary Petrie polygon.

A 2nd-order hole is an ordinary hole.

A

A

A

A

Etc.

This can be summarised:

| ||||||||||||||||||||||||||||||||||||||||||||||||||

Icosahedron with one face shown in red. | Icosahedron with one Petrie polygon shown in red. | |||||||||||||||||||||||||||||||||||||||||||||||||

Icosahedron with one hole shown in red. | Icosahedron with one 2nd-order Petrie polygon shown in red. |

We will use the abbeviations in the final column of the table
when specifying the sets of antipodes
of a polyhedron. Thus we would list the antipodal sets of the
icosahedron as

(2V, 2H, P) | two vertices, two holes, one Petrie polygon |

(2E) | two edges |

(2F, P_{2}) | two faces, one 2nd-order Petrie polygon |

Some regular maps

Some pages on groups

Copyright N.S.Wedd 2009