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Concept

Angle defect

The amount by which the faces meeting at a corner of a polyhedron fall short of closing up flat, measured as an angle. Summed over every corner of a convex polyhedron it comes to a full seven hundred and twenty degrees, whatever the solid.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The five Platonic solids. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, drawn at a common scale.

Why the list of perfect solids stops at five

There are infinitely many regular polygons and exactly five regular solids. The reason is not deep, but it is very sharp, and it can be checked on a single row of corners.

geometry · Regular polyhedra 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 Twelve units of shortfall, on every solid with three faces at a corner. A bar for each of 8 polyhedra with three faces at every vertex, divided into each face's shortfall from six sides; every bar has total length twelve, and the hexagons contribute nothing.

Twelve pentagons, whatever the hexagons

A football has twelve pentagons and twenty hexagons. A molecule of sixty carbon atoms has the same pattern, a molecule of seventy has twelve pentagons and twenty-five hexagons, and a geodesic dome of any size has twelve places where the pattern of six breaks. None of this is a coincidence of design: Euler's formula, rearranged, says that faces meeting three at a corner must fall short of hexagons by exactly twelve in total, and the hexagons are free.

topology · Euler characteristic

Named alongside it

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

Euler characteristicPlatonic solidsPolyhedronCurvatureConvexityDihedral angleDualityEuler formulaGauss bonnetGenusTilingTorus

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