Graphics Reference
In-Depth Information
CHAPTER 6
Combinatorial Topology
6.1
Introduction
Topology is a relatively new field in mathematics. In the last chapter we considered
some basic concepts from what is called point set topology. This chapter is the first
of three that will introduce us to a broader view of the subject and we begin with a
brief history. Before the mid-19
th
century the field as a whole consisted mainly of a
collection of isolated facts. The word “topology” itself appeared first in 1847 in the
topic
Vorstudien zur Topologie
, by J.B. Listing, a student of Gauss. Before that, one
used the term “geometria situs” for the study of certain qualitative properties of geo-
metric figures that would be considered topological today. The term was introduced
by Leibniz, although he did not contribute much to the subject himself. In the 1800s
and early 1900s topology was usually referred to as “analysis situs.”
Probably the earliest significant topological observation concerned a relationship
between the number of faces, edges, and vertices of a simple polyhedron, which was
already known to Descartes around 1620. By a
simple polyhedron
we mean a convex
three-dimensional linear polyhedron, that is, a convex solid figure without holes that
is bounded by planar faces. The five standard well-known regular simple polyhedra
are shown in Figure 6.1. A
regular polyhedron
is a polyhedron with the property that
every face has the same number of edges and every vertex has the same number of
edges emanating from it. For now we are only interested in the boundary of a simple
polyhedron. It is easy to show that a simple polyhedron is homeomorphic to
D
3
, so
that its boundary is homeomorphic to the 2-sphere
S
2
.
Given a simple polyhedron, let n
v
, n
e
, and n
f
denote the number of its vertices,
edges, and faces, respectively. Obviously, as Figure 6.1 shows, the numbers n
v
, n
e
, and
n
f
themselves vary wildly from polyhedron to polyhedron, but consider the alternat-
ing sum n
v
- n
e
+ n
f
. One can check that this sum is 2 for all the polyhedra in Figure
6.1. Is this accidental? No, we have just discovered the first combinatorial invariant.
6.1.1. Theorem
(Euler's Formula). n
v
- n
e
+ n
f
= 2 for every simple polyhedron.
