1.10. Dessert 17
the grid in the hyperbolic plane associated to M. It turns out that
the modular group acts as a group of symmetries of this grid. So,
when we consider the moduli space M of unit area flat tori, we get
right back to the modular group.
We can play a similar game for the octagon surface. As we dis-
cussed above, we can create the octagon surface using a suitable cho-
sen regular octagon. However, we can also glue together other hyper-
bolic octagons to produce a surface that “looks hyperbolic” at each
point and has the same Euler characteristic. When we consider the
totality of such surfaces, we arrive at a higher-dimensional generaliza-
tion of M, also called moduli space. This higher-dimensional space
is not a surface, but it does share some features in common with a
In Chapter 20 we also discuss Teichm¨ uller space, the space that
relates to the higher-dimensional version of M in the same way that
the hyperbolic plane relates to M. Teichm¨ uller space shares some
features with the hyperbolic plane, but is much more mysterious and
somewhat less symmetric. We will discuss the group of symmetries
of Teichm¨ uller space, called the mapping class group. The mapping
class groups relate to the surfaces of negative Euler characteristic in
the same way that the modular group relates to the square torus. We
will further explore Teichm¨ uller space in Chapter 21.
There are a few topics in this book that I simply threw in because
I like them. Chapter 22 has a proof of the Banach–Tarski Paradox.
One nice thing about the proof is that it involves the modular group
in an essential way. So, in a strange way, the Banach–Tarski Paradox
has some connection to hyperbolic geometry.
Chapter 23 has a proof of Dehn’s Dissection Theorem, which says
that one cannot cut a cube into finitely many pieces, using planar
cuts, and rearrange the result into a regular tetrahedron. This result
serves as a kind of foil for the decomposition methods we use to
prove the combinatorial Gauss–Bonnet Theorem and other results.
Polyhedral decomposition is quite robust in 2 dimensions, but not in