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.