Other Surfaces

Now, we'll take as given that for any spherical polyhedron,

 v+f-e=2

In general, v+f-e is a constant for any given surface. That is, for any particular kind of surface, no matter how it is stretched or distorted, no matter how it is turned into a polyhedron, v+f-e is fixed.

This in general is called the Euler characteristic of the surface. The Euler characteristic varies from one kind of surface to another, however:

Q4Prove, using these facts, that for any polyhedral cylinder,

 v+f-e=0

Cylinder

Hint: given any cylindrical polyhedron, how can you turn it into a spherical polyhedron?

Q5Also prove that, for any polyhedral disk,

 v+f-e=1

Disk

You CERTAINLY DON'T want to count up all those edges and vertices! So try to think of a way to answer this question using the Euler Constant for a sphere, and that Euler characteristic is a topological property.

Q6 Here is a torus and a two-holed torus. What is the appropriate constant for these surfaces?

Be cautious. It is very important that none of the faces have holes.

Q6 cont'dComplete and prove this theorem, given the facts assumed above.
You might look for a trick to make the work easier...

Theorem: If a surface S has Euler characteristic n, any surface S' made from S' by adding a tube to S has Euler characteristic ........
Specifically, an n-holed torus has Euler charcteristic ....

Note it shouldn't matter whether you add the tube to the inside or outside, whether the tube is knotted, or whether the tube passes through the surface like a ghost passes through a wall.

Q7 Deduce the Euler characteristic of a Mobius band, a Klein bottle.

Bonus Weirdness:

Q8 This polyhedron passes through itself like a ghost passes through a wall. It is made from a cuboctahedron by removing the square faces and adding in four intersecting hexagonal faces. The edges in the inside of the polyhedron are illusory; the only edges that count are the edges of the polygons, on the outside of the polyhedron. Thus there are 12 faces (four hexagons, eight triangles), 12 vertices, and 24 edges. Hmmm, what surface do you think this is?
A strange surface!
For reference, a cuboctahedron.

  Chaim Goodman-Strauss
  Dept. Mathematics
  Univ. Arkansas
  Fayetteville, AR 72701
  strauss@comp.uark.edu
  501-575-6332