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Thayer Watkins
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The Theorem Dual to
Euler's Twelve Pentagon Theorem

In the 18th century the Swiss mathematician Leonhard Euler discovered that if every face of a polyhedron is either a pentagon or a hexagon then it has exactly twelve pentagonal faces. The proof of this is given elsewhere. There is a dual to this theorem; i.e.,

If every face of a polyhedron is a triangle and the degree of every vertex is five or six, then the polyhedron has exactly twelve vertices of degree five.

Proof:

Let M be a closed convex polyhedron with no holes which is composed of triangular faces. Let f, e, v be the number of faces, edges and vertices of M, respectively. Let the number of degree five vertices be denoted as w and the number of degree six vertices as x. Then v is equal to w+x.

The Euler-Poincare (oiler-pwan-kar-ray) characteristic of the polyhedron, f-e+v, is equal to 2. This is one equation constraining the values of f, e and v; i.e.,

f - e + v = 2
or, equivalently
f + w + x - e = 2
or, equivalently
f - e + w + x = 2

If we traverse the polyhedron face-by-face counting the number of edges we will get 3f. We will also count every edge twice. Therefore

3f = 2e
which can be expressed as
2(f-e) = -f

If the equation derived above from the Euler Poincare characteristic equation is multiplied by two and -f substituted for 2(f-e) the result is

-f + 2w + 2x = 4

If we traverse the vertices of the polyhedron counting edges we will get 5w+6x. In this process we will have counted each edge twice since each edge terminates in exactly two vertices. Thus

5w+6x = 2e
which from an above result
is the same as
5w+6x = 3f
or, equivalently
-3f +5w + 6x = 0

If the equation -f+2w+2x=4 is multiplied by three then we have the two equations

-3f + 6w + 6x = 12
and
-3f + 5w + 6x = 0

Subtracting the second equation from the first gives

w = 12

In words, the number of degree five vertices of the polyhedron must be exactly twelve.


For more on polyhedra and thire Euler-Poincare characteristic see Euler Poincare.


Reference:
David S. Richeson, Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press, 2008. A fantastically interesting book.


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