The figure to the right shows the regular map S2:{8,3},
with a foursome of antipodal vertices marked with the letters a, b, c, d.
These vertices are antipodal in the sense that any rotational symmetry that fixes one of them fixes
each of them.
For a geographer, the words "antipodes" and "antipodal" are 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. Or, two points in a sphere are antipodal if they lie on opposite ends of a diameter.
In these pages I use these words "antipodes" and "antipodal" in a way that appears quite different to the above: If every symmetry operation on a regular map that fixes some structure (e.g. a vertex, an edge, a face, a Petrie polygon) of it, 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 all mutually antipodal.
These definitions turn out to be similar in effect:
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.
The geographers' definition uses the word "distance" ‒ so to apply it, we need a metric on the manifold we are considering. This should be a "nice" metric, where "nice" is defined as follows: A metric A on a manifold M is said to be nice iff there exists no metric B on M such that the symmetry group of M with A applied to it is a proper subgroup of the symmetry group of M with B applied to it.
For the sphere, there is an obvious, and unique, nice metric, which can be derived from its standard embedding in ℝ3. This is of course the metric implicit in the geographers' sense of "antipodal". The symmetry group of a sphere with this metric is SO(3).
There are two different "nice" metrics which can be put on the torus. One is obtained by
cutting a square from the real plane ℝ2, and identifying its opposite
edges. The other is obtained by cutting a regular hexagon from the real plane, and
identifying its opposite edges. For the "square" torus, the symmetry group is
A metric for the double-torus, or genus-2 orientable surface, can be found by representing it as the Bolza surface, given by y2=x5-x in ℂ2. This has a finite symmetry group isomorphic to GL(2,3). It has pairs of antipodal points; but see the note below.
A metric for the triple-torus, or genus-3 orientable surface, can be found by
representing it as the Klein
quartic, given in homogenous co-ordinates by
If it were possible to embed S3:{7,3} in 3-space in a way that preserved its symmetries, and then 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. The faces of this regular map form antipodal threesomes, its vertices form antipodal pairs, and its edges form antipodal foursomes.
A metric for the projective plane is obtained by taking the metric for the sphere, and identifying all antipodal pairs of points.
We then find that fixing a point of the projective plane also fixes a line of it. For any point on this line there is another point also on the line, such that the distances between any pair of these three points are equal, and maximal. Fixing any two such points fixes the third. Thus we have sets of three points which are antipodal in the geographers' sense, but not in the group theorists' sense.
The previous section depends on the "nice metrics" that exist for compact 2-manifolds. Some of these are listed in the following table.
| surface | source of metric | symmetry group | its order | what is antipodal to a point | example regular map | antipodal sets of this example |
|---|---|---|---|---|---|---|
| sphere | ℝ3: x2+y2+z2=1 | SO(3) | ℵ1 | one point | tetrahedron | 4*(vertex,face), 3*(edge,edge) | cube | 4*(vertex,vertex), 3*(face,face), 6*(edge,edge) | octahedron | 3*(vertex,vertex), 4*(face,face), 6*(edge,edge) | dodecahedron | 10*(vertex,vertex), 6*(face,face), 15*(edge,edge) | icosahedron | 6*(vertex,vertex), 10*(face,face), 15*(edge,edge) |
| torus | ℝ2: cut a square and identify opposite edges | (ℝ/ℤ × ℝ/ℤ) ⋊ C4 | ℵ1 | one point | {4,4}(2,1) | 5*(vertex,face), 5*(edge,edge) |
| ℝ2: cut a regular hexagon and identify opposite edges | (ℝ/ℤ × ℝ/ℤ) ⋊ C6 | ℵ1 | two points | {6,3}(1,3) | 7*(vertex,vertex,face), 7*(edge,edge,edge) | |
| genus-2 orientable surface | ℂ2: y2=x5-x (Bolza surface) | GL(2,3) | 48 | one point | S2:{8,3} | 8*(vertex,vertex), 3*(face,face), 12*(edge,edge) |
| genus-3 orientable surface | ℂ3: x3y+y3z+z3x=0 (Klein quartic) | PSL(2,7) | 168 | one point, two points, or three points from different starting points | Klein map | 28*(vertex,vertex), 8*(face,face,face), 21*(edge,edge,edge,edge) |
| projective plane | ℝ3: as for the sphere but identify (x,y,z) with (-x,-y,-z) | PSO(2) | ℵ1 | a line | hemidodecahedron | in a looser sense of antipodal: 5*(edge,edge,edge) |
The figure to the right shows the regular map S2:{8,3},
with a foursome of antipodal vertices marked with the letters a, b, c, d.
These vertices are antipodal in the sense that any rotational symmetry that fixes one of them fixes
each of them.
However, the relationships among these vertices are not all equivalent. If we rotate vertex a clockwise, we find that vertex c also rotates clockwise, while vertices b and d rotate counterclockwise. We define a further term for the relationship between vertices a and c, and between b and d: if every symmetry operation on a regular map that fixes some structure of it while rotating it through some angle, also fixes some other structure while rotating it through the same angle, these two structures are said to be strongly antipodal.
Thus vertices a and c in the figure to the right are strongly antipodal, as are vertices b and d. Vertices a and b, while antipodal, are not strongly antipodal. Opposite vertices of a regular map in the sphere are never strongly antipodal: when you rotate one clockwise, the other turns anticlockwise, viewing each from the outside.