Illustration of Euler characteristic

1758 Euler characteristic

The book of science

Tom Sharp

Leonhard Euler mathematics Illustration of Euler characteristic

Euler characteristic

The Euler characteristic or χ (a Greek letter pronounced like “sky” but without the “s”) of any geometric object is the number of vertices (corners) minus the number of edges plus the number of faces. The χ of any convex polyhedron (any three-dimensional geometric object with sharp vertices, straight edges, and flat faces) is 2. The χ of a cube is 2 (8 vertices - 12 edges + 6 faces). The χ of a triangular pyramid is also 2 (4 vertices - 6 edges + 4 faces). But the χ of nonconvex polyhedra is more interesting. The χ of a tetrahemihexahedron is 1 (6 vertices - 12 edges + 7 faces).

Topological invariant

Let me tell you about the χ of a sphere. You might think that the χ of a sphere would be 1 (0 vertices - 0 edges + 1 face); however, the χ of a sphere is 2. To understand this, first we need to realize no matter how you cut it, a sphere always remains a sphere. So cut it both horizontally and vertically. The two cuts cross each other in two places; these are its two vertices. The cuts also divide it into 4 faces that share 4 edges. Its χ is therefore (2 - 4 + 4) = 2. Cut the sphere vertically again at a right angle to the previous vertical cut. Now the sphere has 6 vertices, 12 edges, and 8 faces, retaining its χ of 2. No matter how you cut it, a sphere will always have a χ of 2. You can cut and flatten faces into any convex polyhedron and it still keeps its χ of 2.

Eyes

Your eyes, my dear, are jewelled spheres; each has a χ of two. Your eyes of blue, between us two, reflect the sky together here.

Science, like art, searches for the hidden unity of dissimilar things and tries to explain them.

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