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InspectorCthulhu

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Registered: 2016-06-02

Posts: 7

Euler Characteristic - is it true that...

Shapes that have the same Euler Characteristic could be turned into each other by the right set of manipulations, whereas those that have a different Euler Characteristic cannot?

I'm after a short, simple but correct explanation for children ~10 years old of why the Euler Characteristic is a 'basic idea in topology' (as I've read).  It's for use as a 'by the way, you might be interested to know...' insight into more advanced maths to conclude an exercise of counting faces, vertices and edges of common 3D shapes, and discovering that they usually have an Euler Characteristic of 2 (The only exception I've been able to think of among common 3D shapes  is a cylinder, which I think has 3 faces, 0 vertices and 2 edges, so an Euler Characteristic of 1.).

Thanks for your help!

Alg Num Theory

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Registered: 2017-11-24

Posts: 693

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Re: Euler Characteristic - is it true that...

By “could be turned into each other by the right set of manipulations”, do you mean homeomorphic?

In that case, the answer is no. It is true that two homeomorphic spaces must have the same Euler characteristic, but the converse is not true: it is possible for two non-homeomorphic spaces to have the same Euler characteristic. An example (or two) is the torus and the Klein bottle: both have Euler characteristic 0, but they are not homeomorphic.

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Me, or the ugly man, whatever (3,3,6)

InspectorCthulhu

Member

Registered: 2016-06-02

Posts: 7

Re: Euler Characteristic - is it true that...

Thanks - the text I was reading probably did mean 'homeomorphic', so you've given me the answer.