The words "antipodes" and "antipodal" are usually used in the context of a sphere. Their meaning there is clear: two points in a sphere are antipodal if the distance between them is as great as possible.

This works because there is an obvious distance measure to use in a sphere: great circle distance. It is possible to devise at least two different sensible distance measures in a torus (or on "different toruses", if you prefer). For surfaces of genus greater than 1, it is even less clear what distance measure should be used.

In these pages I use these words "antipodes" and "antipodal" in a way not based on a distance measure. My definition applies to all finite 2-manifolds including the sphere. It is
If a symmetry operation on a regular map (which I prefer to regard as a polyhedron) fixes some structure (e.g. a vertex, edge, or face) of it, and also fixes some other structure, these two structures are said to be antipodal.
This relation is transitive, so sets of two or more structures are mutually antipodal.

Examples from genus 0

For the sphere, this definition is consistent with the usual one. We find that the five Platonic solids have the following sets of antipodes

just as with the usual definition.

The definition covers structures other than vertices, edges, and faces; e.g. holes and Petrie polygons. Here is how this applies to the Platonic solids.

An example from genus 1

One of the more interesting regular maps on the torus consists of seven hexagons, each bordering the other six. It has 14 vertices and 21 edges. Its sets of antipodes are
      (vertex, vertex), (face), (edge, edge, edge).
Its rotational symmetry group is C7⋊C6, the Frobenius group of order 42. It has as its Sylow-subgroups

1normalC7,so its314-sided Petrie polygonsform 1 antipodalthreesome
7conjugateC3s,so its14verticesform 7 antipodalpairs
7conjugateC2s,so its21edgesform 7 antipodalthreesomes
Its Petrie polygons encompass all 14 of its vertices, and are fixed by everything.

An example from genus 2

The genus-2 regular map G2:{8,3} has six octagonal faces, 16 3-valent vertices, and 24 edges. Its sets of antipodes are
      (vertex, vertex, vertex, vertex, Petrie polygon), (face, face), (edge, edge).
Its rotational symmetry group is GL(2,3). GL(2,3) has eight elements of order 3, so its Sylow-3-subgroups are

4conjugateC3s,so its16verticesform 4 antipodalfoursomes

An example from genus 3

The genus-3 regular map G3:{7,3} has 24 heptagonal faces, 56 3-valent vertices, and 84 edges. Its sets of antipodes are
      (vertex, vertex), (face, face, face), (edge, edge, edge, edge, Petrie polygon).
Its rotational symmetry group is PSL(2,7). Its Sylow-subgroups are

8conjugateC7s,so its24facesform 8 antipodalthreesomes
28conjugateC3s,so its56verticesform 28 antipodalpairs
21conjugateD8s,so its84edgesform 21 antipodalfoursomes

Some regular maps drawn on orientable 2-manifolds
Some pages on groups

Copyright N.S.Wedd 2009