14 1. Book Overview
points, the bug needs to spin more than 2π, say 2π + δ1, before re-
turning to his original position. At the other special point, the bug
needs to spin less than 2π, say 2π−δ2, before returning to his original
position. The numbers δ1 and −δ2 might be called the angle error at
the special points.
The numbers δ1 and δ2 depend on the hexagon in question. As
one can see by adding up the interior angles of a hexagon, we have
δ1 = δ2. That is, the total angle error is 0. This result holds for any
Euclidean cone surface with Euler characteristic 0. More generally,
on a surface with Euler characteristic χ, the total angle error is 2πχ.
This result, known as the combinatorial Gauss–Bonnet Theorem is
one of the main results of Chapter 17.
Another topic in Chapter 17 is the application of Euclidean cone
surfaces to polygonal billiards. It turns out that the contemplation of
rolling a frictionless, infinitesimally small billiard ball around inside a
polygonal shaped billard table, whose angles are all rational multiples
of π, leads naturally to a certain Euclidean cone surface. One can
profitably study this surface to get information about how billiards
would work out in the polygon.
The Euclidean cone surfaces associated to polygonal billiards have
a special structure. They are called translation surfaces. A transla-
tion surface is a Euclidean cone surface, all of whose angle errors are
integer multiples of π. The square torus is the prototypical example
of a translation surface, but it is a bit too simple of an example in
this case. The octagon surface provides a better example. The oc-
tagon surface, considered from the Euclidean geometry perspective,
is a translation surface. This surface has a single cone point, and
the angle error there is 4π. Translation surfaces are nicer than gen-
eral Euclidean cone surfaces for a variety of reasons. One reason is
that, as it turns out, it is possible to speak about directions (such as
due north) on a translation surface without any ambiguity. We will
discuss these surfaces in detail in Chapter 18.
1.8. The Modular Group and the Veech Group
We have wandered away from hyperbolic geometry and complex anal-
ysis, but actually hyperbolic geometry and complex analysis are very