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  "body": "\n## Statement\n\nLet $P$ be a convex [[polyhedron|0325SGSJAR]] in $\\mathbb{R}^3$ with $V$ vertices, $E$ edges and $F$ faces. Then\n$$V - E + F = 2.$$\n\n## Remarks\n\nThe number $V - E + F$ is the Euler characteristic of the surface of $P$. The value $2$ is the Euler characteristic of the sphere.\nThe formula also holds for each polyhedron whose surface is topologically a sphere and whose faces are topologically discs.\nIt can fail for other polyhedra. The surface of a polyhedron with the shape of a torus has $V - E + F = 0$.\nFor the cube, $8 - 12 + 6 = 2$. For the regular tetrahedron, $4 - 6 + 4 = 2$.\n\n## History\n\nEuler stated the formula in 1758. Cauchy gave a proof in 1811.\n\n## Explanations\n\n### L1 human\n\nCount the corners, the edges and the faces of any convex solid with flat faces. Corners minus edges plus faces always gives $2$. Cut off one corner of a cube: you get $2$ more corners, $3$ more edges and $1$ more face, and the count stays $2$.\n",
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    "name": "Euler's polyhedron formula",
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  "name": "Euler's polyhedron formula",
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