10 1. Book Overview

Figure 1.7. Gluing the octagon together

The hyperbolic plane shares many features with the Euclidean

plane. There is a unique geodesic joining any 2 distinct points, and

any 2 distinct geodesics meet in at most one point. Furthermore,

the hyperbolic plane is totally symmetric, in the sense that every

point and every direction looks exactly the same. A bug living in an

otherwise empty hyperbolic plane would not be able to tell where he

was.

On the other hand, the hyperbolic plane and the Euclidean plane

have some important differences. For instance, the sum of the angles

of a hyperbolic triangle, a shape bounded by 3 geodesic segments, is

always less than 180 degrees, or π radians. (When we discuss angles

in radians, we will often leave off the word “radians”.) Similarly,

the individual interior angles of a regular octagon can take on any

value less than 3π/8, which is the value in the Euclidean case. The

right hand side of Figure 1.4 shows a regular hyperbolic octagon. We

decrease the interior angles by making the octagon larger and we

increase the interior angles by making the octagon smaller.

In particular, we can adjust the size of the regular octagon so that

the interior angles are exactly π/8. We can then cut the resulting

octagon out of the hyperbolic plane and glue the sides together just

as in Figure 1.3. From the hyperbolic geometry point of view, the

resulting surface would be completely seamless: a myopic bug living

on the surface could not tell that he was not living in the hyperbolic

plane. With the chosen interior angles, the 8 corners fit together like