1.5. Hyperbolic Geometry and the Octagon 9

consider surfaces, such as the octagon surface. However, in order to

“see” the picture in these cases, you have to draw it in the possibly

unfamiliar world of hyperbolic geometry. The idea is that hyperbolic

geometry does for the octagon surface (and most other surfaces as

well) what Euclidean geometry does for the square torus and what

spherical geometry does for the sphere.

We will discuss Euclidean, spherical, and hyperbolic geometry in

Chapter 8, Chapter 9, and Chapter 10 respectively. Our main goal is

to understand how these geometries interact with surfaces, but we will

also take time out to prove some classical geometric theorems, such as

Pick’s Theorem (a relative of the Euler formula) and the angle-sum

formula for hyperbolic and spherical triangles.

The Euclidean, spherical, and hyperbolic geometries are the 3

most symmetrical examples of 2-dimensional Riemannian geometries.

To put the 3 special geometries into a general context, we will discuss

Riemannian geometry in Chapter 11.

1.5. Hyperbolic Geometry and the Octagon

Now let us return to the question of unwrapping essential loops on

the octagon surface. The octagon surface looks a bit less natural than

the square torus, thanks to the special point. However, it turns out

that the octagon surface “wears” hyperbolic geometry very much in

the same way that the square torus “wears” Euclidean geometry.

We already mentioned that we will study hyperbolic geometry in

detail in Chapter 10. Here we just give the barest of sketches, in order

to give you a taste of the beauty that lies in this direction. One of the

many models for the hyperbolic plane is the open unit disk. There is

a way to measure distances in the open unit disk so that the shortest

paths between points are circular arcs that meet the boundary at

right angles. These shortest paths are known as geodesics. The left-

hand side of Figure 1.7 shows some of the geodesics in the hyperbolic

plane. The boundary of the unit disk is not part of the hyperbolic

plane and the lengths of these geodesics are all infinite. A bug living

in the hyperbolic plane would see it as unbounded in all directions.