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

hyperbolic surface.

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.

1.10. Dessert

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

higher dimensions.