San José State University
Department of Economics
& Tornado Alley
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.,
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.,
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
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
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
If the equation -f+2w+2x=4 is multiplied by three then we have the two equations
Subtracting the second equation from the first gives
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.
David S. Richeson, Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press, 2008. A fantastically interesting book.