Topology in Molecular Biology

220 R. Brooks

Fig. 12.7.Building on the tetrahedron

Fig. 12.8.The tetrahedron with one orientation reversed

may take the standard orientation and then reverse the orientation on zero,
one, or two vertices (Fig. 12.8).
We begin by drawing four triangles glued together, with one in the center,
as shown:
The usual orientation then tells us to glue each side to the side adjoining
it so that it does not lie on the same triangle. As before, it is easy to see that
one obtains in this way a sphere, this time with four singular points.
Now let us reverse the orientation on the center triangle, or what amounts
to the same thing, keep the orientation on the center triangle, and reverse it
in the other three.
Now we have that each side is glued to the opposite side. It is clear that
there are now two cusps rather than four, so the surface is a torus.
One way to compute which torus it is, is as follows: from symmetry con-
siderations, the central triangle must be an equilateral triangle. If we choose
the remaining triangles to be isosceles, then the condition on the midpoints
matching up is realized. This is because the conformal center of the base is