THE ART AND SCIENCE OF MATHEMATICS
Wednesday, March 22, 2006

Topology of Surfaces

Given a polyhedron with V vertices, E edges, and F faces, then

V-E+F=2.

This is Euler's theorem. It is proved by removing one face of the polyhedron and flattening it on the plane.

Euler's formula makes it possible to find all the regular polyhedra (Platonian solids). Let a regular polyhedron be made out of r-gons and at each vertex n of these r-gons meet. Then Fr=2E, and Vn=2E, that is, 2E/n-E+2E/r=2. It is more convenient to rewrite this equation in the form:

1/n+1/r=1/2+1/E.

It is rather easy to go through all the possible cases for n and r and find that there are only five solutions. For more general polyhedra, possibly with holes, Euler's formula gives:

V-E+F=3-g,

where the number g is called the connectedness of the polyhedron. The connectedness is defined as follows. If any simple (without self-intersections) closed curve on the surface divides it into two parts, then the surface is called simply connected and its connectedness g=1. If there is a closed curve that does not divide surface (like a circle on a torus), but any two curves, one of which is closed and another one has ends on the first curve, do divide then g=2, and so on.

Problem: Compute g for a torus (a sphere with a handle), compute g for a double torus (a sphere with two handles or a bagel in the shape of figure eight).

The issue of classification of surfaces is resolved through cutting them so that after a number of cuts a surface can be flattened. It is easy to make two cuts on the torus so that it becomes a rectangle. On the other hand if you take a rectangle, glue together its two opposite sides you will get a cylinder (a ring). After that glue the two bases of the cylinder together to get a torus.

There are several other ways of gluing sides of a rectangle. For example if you twist the square before gluing the sides you will get the Moebius strip. If you glue two opposite edges straight, and then the two resulting circles with a twist, you will get Klein's bottle. Finally with twists for both pairs of opposite sides, you will get the projective plane.

Further reading

• 19 proofs of Euler's formula (including one using interdigitating trees) are given at
  http://www.ics.uci.edu/~eppstein/junkyard/euler/
• The five regular polyhedra are listed at
  http://www.cut-the-knot.org/do_you_know/polyhedra.shtml
• Want to buy a Klein bottle? See:
  http://www.kleinbottle.com/