Euler Characteristic

The Euler characteristic, \chi , is a topological invariant. It allows mathematicians to describe a shape or structure regardless of how it is bent or deformed. The Euler characteristic generalises Euler’s observations of 1751 that when you triangulate a sphere into F regions, E edges and V vertices: VE + F = 2.

Source: quantamagazine.org

Euler’s characteristic is defined by the formula:

\chi =V-E+F\,\!

where V = the number of vertices, E = the number of edges and F = the number of faces of a polyhedron.

For a convex polyhedron, \chi  = 2.

Source: storyofmathematics.com

Genus

 

Put simply, the genus is equal to the number holes in the shape: “an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.”

As topology is concerned only with what is essential about the shape, there are many surfaces with various interesting shapes that are actually the same topologically.

Topologically the Same

What separates each topological shape from one another is the number of holes it has. This is because no matter what is done to a shape, the number of holes will remain constant, as long as no cuttings are taken. Hence, the genus is another topological invariant, just like Euler’s characteristic. In fact, these two properties are linked by the formula:

\chi =2-2g.\

Sources: 1 | 2 | 3

Hope you enjoyed today’s post. Let me know if you have any ideas for future blog posts. M x

 

 

 

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2 comments

  1. I’ve always found fascinating, and delightful, proofs that the Euler characteristic is constant, by way of how adding or removing faces or edges or vertices affects the value. It strikes me as the sort of reasoning that’s real mathematics. You prove a couple simple little things, and show those simple little things cover everything you might do — turn a simple solid into as complicated a one as you want — and voila! It’s almost magic.

    Liked by 2 people

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