1.6. Complex Analysis and Riemann Surfaces 11

8 slices in a pizza to make a perfect hyperbolic disk. We will consider

this construction in detail in Chapter 12.

A similar construction can be made for the surfaces obtained

by gluing together the opposite sides of a regular 2n-gon, for each

n = 5, 6, 7 . . . . All these surfaces “wear” hyperbolic geometry in a

seamless way, just like the square torus “wears” Euclidean geometry.

Now, we can tile the Euclidean plane by copies of the unit square.

The vertices of this tiling are precisely the integer grid points. In the

same way, we can move our hyperbolic octagon around the hyperbolic

plane and tile the hyperbolic plane with copies of it. When drawn in

the disk model, the picture looks like the drawings in M. C. Escher’s

Circle Woodcut series. To our Euclidean eyes, the octagons appear

to get smaller as they move out toward the boundary of the disk.

However, in the hyperbolic world, the various octagons all have the

same size.

The vertices of this tiling are a kind of hyperbolic geometry ver-

sion of the integer grid. These points are in one-to-one correspondence

with the equivalence classes of essential loops on the octagon surface.

The same kind of thing works for the surfaces corresponding to the

(2n)-gons for n = 4, 5, 6 . . . . In fact, such a construction works for

all surfaces that have negative Euler characteristic: one always gets

a grid of points in the hyperbolic plane that names the different es-

sential loops on the surface. We will explore this in detail in Chapter

12.

1.6. Complex Analysis and Riemann Surfaces

It turns out that there is a single kind of geometry which unifies

Euclidean, spherical, and hyperbolic geometry. This geometry, called

M¨ obius or conformal geometry, takes place in the Riemann sphere.

The Riemann sphere is the set C ∪ ∞, where C is the complex plane

and ∞ is an extra point that is added. For starters,

• The Euclidean plane is identified with C.

• The hyperbolic plane is identified with the open disk {z ∈

C| z 1}.