A topologist might regard the donut and the coffee cup as identical but what
sort of surface is different from the donut? A candidate here is the rubber ball.
There is no way of transforming the donut into a ball since the donut has a hole
but the ball does not. This is a fundamental difference between the two surfaces.
So a way of classifying surfaces is by the number of holes they contain.
Let’s take a surface with r holes and divide it into regions bounded by edges
joining vertices planted on the surface. Once this is done, we can count the
number of vertices, edges, and faces. For any division, the Euler expression V – E
- F always has the same value, called the Euler characteristic of the surface:
Möbius srtip
V – E + F = 2 – 2r
If the surface has no holes (r = 0) as was the case with ordinary polyhedra,
the formula reduces to Euler’s V – E + F = 2. In the case of one hole (r = 1), as
was the case with the cube with a tunnel, V – E + F = 0.
One-sided surfaces
Ordinarily a surface will have two sides. The outside of a ball is different from
the inside and the only way to cross from one side to the other is to drill a hole
in the ball – a cutting operation which is not allowed in topology (you can stretch
but you cannot cut). A piece of paper is another example of a surface with two
sides. The only place where one side meets the other side is along the bounding
curve formed by the edges of the paper.