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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.

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

Also named here as platonic solids — the same set of essays touches all of them, so they are one junction rather than several.

tetrahedron4 trianglescube6 squaresoctahedron8 trianglesdodecahedron12 pentagonsicosahedron20 triangles

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 90°cornersgap eachtotaltetrahedron180°720°cube90°720°octahedron120°720°dodecahedron36°720°icosahedron60°720°3 squares at a corner of the cube come to 270°, leaving a gap of 90°8 corners × 90° = 720°, and every one of the five gives the same 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

Named alongside it

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

Euler characteristicPlatonic solidsPolyhedronConvexityCurvatureDihedral angleDualityGauss bonnetTiling

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