The Euler characteristic, , 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: V – E + F = 2.
Euler’s characteristic is defined by the formula:
where V = the number of vertices, E = the number of edges and F = the number of faces of a polyhedron.
For a convex polyhedron, = 2.
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.
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:
Hope you enjoyed today’s post. Let me know if you have any ideas for future blog posts. M x