Surfaces and Euler characteristic

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K-12: Materials at high school level.



Surfaces

FirestoneTire.jpg Donut.jpg
A tire A donut
Doubletorus.png Pretzel.jpg
A double torus A pretzel

Note that a tire and a donut have approximately the same shape. More complicated is a so called double torus, which is made by sticking two individual torii together and for instance the surface of a pretzel. In the pretzel you see 3 "holes". The donut has only one, and the double torus has two. In mathematics we use the number of holes to identify the surface.

A fun fact is that in an area of geometry - called topology - a cup and a donut are considered to have the same shape. One can be obtained form the other by morphing the shape.

Mug and Torus morph.gif

The image above shows how the mug can be morphed into a donut and back again. The main rule is that you are not allowed to cut the shape or glue pieces together.

Euler Characteristic

As the example of the mug and the donut shows, there can be some surprises when we try to find out what type of surface we are looking at. Mathematicians often compute a number called the Euler Characteristic for a surface to identify it. For a surface we would create a pattern on the surface made up of vertices, edges and faces.

For instance for a sphere we can use 4 vertices, 6 edges and 4 faces:

Sphere-tess-tetra.gif

The Euler Characteristic formula is X = V - E + F.


For the sphere we get X = 4 - 6 + 4 = 2

We can change the pattern on the sphere:

Sphere-tess-octa.gif

Here we see 6 vertices, 12 edges and 8 faces.

And we find: X = 6 - 12 + 8 = 2

So different patterns will give the same value for X! The main idea is that different surfaces have different values of X. Below are some examples.

surface X
sphere 2
torus (= donut) 0
2-holed torus -2
3-holed torus (pretzel) -4
etc etc