Surfaces and Euler characteristic

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K-12: Materials at high school level.


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 torua, which is made by sticking two individual torus 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.