# Antipodes

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

• tetrahedron: (vertex, face), (edge, edge).
• octahedron: (vertex, vertex), (face, face), (edge, edge).
• cube: (vertex, vertex), (face, face), (edge, edge).
• icosahedron: (vertex, vertex), (face, face), (edge, edge).
• dodecahedron: (vertex, vertex), (face, face), (edge, edge).
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.

• tetrahedron: (vertex, face), (edge, edge, Petrie polygon).
• octahedron: (vertex, vertex, hole), (face, face, Petrie polygon), (edge, edge).
• cube: (vertex, vertex, Petrie polygon), (face, face), (edge, edge).
• icosahedron: (vertex, vertex, hole, hole, Petrie polygon), (face, face, 2nd-order Petrie polygon), (edge, edge).
• dodecahedron: (vertex, vertex), (face, face, Petrie polygon), (edge, edge).

### 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

 1 normal C7, so its 3 14-sided Petrie polygons form 1 antipodal threesome 7 conjugate C3s, so its 14 vertices form 7 antipodal pairs 7 conjugate C2s, so its 21 edges form 7 antipodal threesomes
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

 4 conjugate C3s, so its 16 vertices form 4 antipodal foursomes

### 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

 8 conjugate C7s, so its 24 faces form 8 antipodal threesomes 28 conjugate C3s, so its 56 vertices form 28 antipodal pairs 21 conjugate D8s, so its 84 edges form 21 antipodal foursomes

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