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Concept

Gauss bonnet

The theorem that the total curvature of a closed surface equals 2π times its Euler characteristic. Curvature can be moved about by bending, and the total cannot change, so a purely local quantity adds up to a topological one.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The gap at a corner of the cube. The 3 faces meeting at one corner of the cube, unfolded onto the page. They leave a gap of 90.0 degrees, and the 8 gaps come to 720 degrees in total.

Seven hundred and twenty degrees of gap

Unfold the faces around any corner of a solid and they do not close up. The gap left over is different at every corner and on every solid, and the gaps always add to two full turns.

topology · Euler characteristic Peaks, pits and passes. Height along one direction on two bumpy surfaces: the torus has 5 peaks, 5 pits and 10 passes, total 0; the sphere has 1, 2 and 1, total 2.

Peaks, pits and passes

Take any lumpy closed surface and any direction of height. Peaks plus pits minus passes is 2 on a sphere and 0 on a torus, whatever the lumps, in every direction. Average one vertex's score over all directions and it becomes the corner's angle defect, computed from nothing but comparisons. So Gauss–Bonnet follows from a count of which neighbours lie higher.

topology · Euler characteristic

Named alongside it

The objects these essays reach for when they reach for this one.

CurvatureEuler characteristicPolyhedronAngle defectDihedral anglePlatonic solidsSimulationTorus

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