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
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
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