12 The Spectral Geometry of Riemann Surfaces 221again the geometric center, and, while it is not the case that the conformal
center of the sides is the geometric center, it is the same for all sides, so what-
ever point it is, the points match up. In order to get the angle sums of the
cusps to match up, the correct choice is 2π/3 for the top angle, andπ/6for
the bottom angles. When this is done, the picture looks as follows, and we
have a regular hexagon with opposite sides glued together:
It is a nice exercise to cut and paste this shape together to see that it
represents the equilateral torus. A faster way of seeing this is to notice the
obvious order 3 symmetry (also clear for the graph) and observe that the
equilateral torus is the only torus with an order 3 symmetry.
Now we consider the case where we reverse the orientation at two vertice
(Fig. 12.9). Here there are two LHT paths, one passing through all four vertices
and the other passing through all four vertices twice.
It is convenient to rearrange this picture with the four vertices from the
four different triangles meeting at the central point, and with opposite sides
identified. Using the argument with isosceles triangles we gave above, it is
clear that the correct choice is for each triangle to be a (π/ 4 ,π/ 4 ,π/2) isoseles
triangle. From this it is easy to identify which torus this is – it is thesquare
torus. This can also be seen directly from symmetry considerations.
As a final exercise, which we will not work out completely, the reader
is invited to take the three-regular graph, which is the skeleton of a cube
(Fig. 12.10). When one takes the usual orientation, then one again has a
sphere, this time with six punctures. In Fig. 12.10, we have written down a
number of possible ways of reversing orientations, which we denote by putting
a circle around the corresponding vertex. Which surfaces do these represent?
In the first two examples, it is clear that the resulting surface is a torus,
because there are four LHT paths – all of length 6 in the first example, and
of lengths 4, 4, 6, and 10 in the second. In the third and fourth examples, we
Fig. 12.9.The tetrahedron with two orientations reversed