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vertices of a face. See Figure 6.3. Dividing a face into two does not change the alter-
nating sum n
v
¢-n
e
¢+n
f
¢ because we gain both an edge and a face in the process.
Next, list the triangular faces in a sequence
T
1
,
T
2
,...,
T
k
, such that
T
i
meets
U
X
i
=
T
-
1
j
1
£<
ji
in either one or two edges. See Figure 6.4. Our argument now proceeds to show equa-
tion (6.1) holds for
X
i
by induction on i. If i = 1, then
X
1
is a triangle and clearly
nnn
¢-
¢+
¢=
3311.
-
+ =
v
e
f
Assume now that equation (6.1) holds for
X
i-1
, i ≥ 2. As Figure 6.4 shows, when we
add
T
i
we either increase n
v
¢ and n
f
¢ by 1 and n
e
¢ by 2 (Figure 6.4(a)), or we increase
n
e
¢ and n
f
¢ by 1 and leave n
v
¢ unchanged (Figure 6.4(b)). In any case, the sum n
v
¢-n
e
¢
- n
f
¢ is still equal to 1. This finishes our sketch of the proof of Euler's formula.
The alternating sum n
v
- n
e
+ n
f
in Euler's formula is called the
Euler character-
istic
of the boundary of the polyhedron and we have just proved that it is a topolog-
ical invariant. It is the first known result in
combinatorial topology
. The term
“combinatorial” is derived from the fact that one is studying invariants based on
combinations of numbers, such as n
v
, n
e
, and n
f
, that are easily computed by simple
counting.
Euler's formula is a special case of far-reaching generalizations that have many
beautiful consequences, some of which we shall see later in this chapter and in the
next two chapters. Even this simple version is enough to prove some interesting facts.
Figure 6.3.
Subdividing faces into
triangles.
T
i
T
i
T
j
T
j
1≤j≤i
1≤j≤i
(a)
(b)
Figure 6.4.
Listing the triangles in a triangular decomposition of a disk.
