University:
California State University, Northridge
Course:
MATH 370 | Foundations of Geometry
Academic year:

2024
Views:

123

Pages:

1
Author:

Haylie Hunt

Polyhedra and Euler's formula A polyhedron is a connected system of polygons (in R3) arranged in such a way that exactly two polygons meet at every edge. A polyhedron is called simple if it can be continuously deformed into a sphere. For instance, the regular polyhedra, and more generally, convex polyhedra, are simple. There is an important relation between the number of vertices V, the number of edges E and the number of faces F of a simple polyhedron. It is know as Euler's formula, and states that V−E+F=2. (1) Prove Euler's formula. A simple polyhedron is regular if all the faces are regular polygons, and all the vertices are equivalent. (2) Use Euler's formula to show that there are only five regular polyhedra. The number V−E+F is called the Euler number of the polyhedron. In general, it need not be equal to 2. (3) Exhibit polyhedra of Euler number a given integer of the form 2−2h. The number h is called the connectivity number (or genus) and is a non-negative integer which has a geometric (physical) interpretation, for you to explain. There are many more things that you can talk about. For example: (4) relations between groups of isometries of the sphere and regular polyhedra. References H.S.M. Coxeter, Introduction to Geometry J. Wiley & Sons, Inc., New York, 1969. B. Grunbaum and G. C. Shephard, A New Look at Euler's Theorem for Polyhedra American Mathematical Monthly, 101 (1994), 109-128. D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, New York, 1990.