Cube · 6 squares · χ = 2

V − E + F = 2

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V = 0/ 8

E = 0/ 12

F = 0/ 6

V − E + F = 0

Euler Characteristic

V − E + F = 2 for every nice polyhedron

Count vertices, subtract edges, add faces. For any polyhedron without holes the answer is always 2 — cube, dodecahedron, soccer ball, your house. The number is a topological invariant: stretch the shape any way you like, the answer is fixed. Add a hole and it drops to 0. Topology in one equation.

Polyhedron

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Running equation

0 − 0 + 0 = 0

final · 8 − 12 + 6 = 2 (χ)

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