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 definition is meaningful because there is an obvious metric to use in a sphere.
There are two different "nice" metrics which can be put on the torus: the "square" one associated with Coxeter group R2, and the "hexagonal" one associated with Coxeter group V2. I do not know if there is a nice metric for the genus-2 orientable surface. I think that there is a nice metric on the genus-3 orientable surface, associated with the Klein quartic.
The geographers' definition of "antipodal" is two points are antipodal if and only if no point is further from either of them than the other is. This definition assumes the existence of a metric.
In these pages I use these words "antipodes" and "antipodal" in a way not based on a distance measure. My definition applies to all compact 2-manifolds including the sphere. It is If a symmetry operation on a regular map 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.
In the genus-0 surface, the sphere, it feels obvious that the definitions coincide. If you hold an icosahedron in your hands, and turn it so that one of the faces rotates, then the opposite face also rotates – this does not surprise us.
But things are different in other genera. If it were possible to embed S3:{7,3} in 3-space, and turn it in your hands so as to rotate one of the heptagons, you would find two of the other heptagons remaining in place and rotating, all three at different speeds. With a suitable metric, these three faces are also antipodal in the geographers' sense.
In the cases where I have been able to check, I find that the two definitions given above coincide. These are
For the sphere, this definition is consistent with the usual one. We find that the five Platonic solids have the following sets of antipodes
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.
One of the more interesting regular maps on the torus is
{6,3}(1,3) which has
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.
The genus-2 regular map S2:{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 |
The genus-3 regular map S3:{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 |