# 1758

## Euler characteristic

## The book of science

Tom Sharp

Tom Sharp

Leonhard Euler mathematics |

- 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).

- 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.

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

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

See also in

The book of science:Euler’s number—Jacob Bernoulli, Leonhard EulerGraph theory—Leonhard EulerTopology—Leonhard Euler, Antoine-Jean Lhuilier, August Ferdinand Möbius, Johann Benedict Listing, Henri PoincaréKnot theory—Alexandre-Théophile Vandermonde, Peter Guthrie TaitReadings on wikipedia: