Euler's polyhedron formula

Statement

Let $P$ be a convex polyhedron in $\mathbb{R}^3$ with $V$ vertices, $E$ edges and $F$ faces. Then $V - E + F = 2.$

Remarks

The number $V - E + F$ is the Euler characteristic of the surface of $P$ . The value $2$ is the Euler characteristic of the sphere. The formula also holds for each polyhedron whose surface is topologically a sphere and whose faces are topologically discs. It can fail for other polyhedra. The surface of a polyhedron with the shape of a torus has $V - E + F = 0$ . For the cube, $8 - 12 + 6 = 2$ . For the regular tetrahedron, $4 - 6 + 4 = 2$ .

History

Euler stated the formula in 1758. Cauchy gave a proof in 1811.

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Polyhedron concept
Euler's polyhedron formula statement

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Tag	`09HT3P35YP`
URI	`m2://09HT3P35YP`
Aliases	`euler-formula-for-polyhedra`
Kind	statement, theorem
Writing	W1 stated
MSC	52B10
Status	This copy of the site has no build data. Run `m2 status 09HT3P35YP`.
Uses	Polyhedron

Euler's polyhedron formula

Statement

Remarks

History

Reading path

Where this leads

Further reading

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