A Fundamental Constant

Ok here's the deal.
Let v=# vertices
Let e=# edges
Let f=# faces

At least for the polyhedra you can make in Kaleidotile,


Isn't that amazing?! This is called the Euler Characteristic of the Sphere.

Q3 Once you've done this, do you think this constant is the same for non-convex (spiky) polyhedra? Conjecture and explain.


An Important Hint

This constant is a topological property, not a geometric one:
Q4 Does the constant change as the surface is stretched and distorted? What is an appropriate definition of face, vertex and edge that applies even after distortion?



In geometry, one is concerned with measurements like angles, lengths and areas. In geometry a circle is defined as all points a fixed distance from some fixed point and all lines are straight.
Topology is a closely related field, but is concerned more with properties that are preserved after arbitrary distortions, like whether two objects touch. To a topologist, all those strange little fellows we saw at the very beginning of the unit are completely equivalent. All simple closed curves are equivalent to circles, and all arcs that have no endpoints are equivalent to lines.

All of these objects are topologically disks:

The standard "joke" is that a topologist confuses her coffee cup for the donut:

Topological properties are intrinsic: A flat-lander could distinguish between the mobius band and the other surfaces (because it has only one edge, for example). But the first and third surfaces are topologically equivalent: both are made by taking a band and gluing opposite sides together top-to-top and bottom-to-bottom. A flat-lander could not tell any difference between the two. The two surfaces are equivalent if one ignores how they are actually placed in our space.

When dealing with surfaces, this is a handy point of view, since we aren't concerned with the specific arrangement of the surface in our space.

Other surfaces
to outline

  Chaim Goodman-Strauss
  Dept. Mathematics
  Univ. Arkansas
  Fayetteville, AR 72701