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Platonic solids

The five convex solids whose faces are congruent regular polygons meeting alike at every corner. There are five because the angles at a corner must fall short of a full turn, which is a counting argument rather than a search.

Named by 6 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 Solids with every corner alike and more than one kind of face. truncated tetrahedron, cuboctahedron, truncated cube, icosidodecahedron, each cut from a Platonic solid and drawn in projection; every edge in each is the same length and every vertex is surrounded by the same faces.

Thirteen more when one word is dropped

The list of regular solids stops at five because the definition asks for two things at once. Ask for only the second — every corner alike — and thirteen more appear, each of them cut off a Platonic solid at a depth found rather than chosen.

geometry · regular polyhedra The star polygon {5/2}. 5 equally spaced points joined every 2th, forming a closed path that winds 2 times about the centre with an interior angle of 36.0 degrees at each point.

The four that are allowed to cross themselves

Drop convexity from the definition of a regular solid and four more appear. Their faces are pentagrams, they pass through one another, and the alternating sum that gives two for every ordinary solid gives minus six for two of them.

geometry · regular polyhedra The regular solids of four dimensions. 5-cell, tesseract, 16-cell, 24-cell, each turned in four dimensions and projected to the page; the edges are the pairs of vertices at the shortest distance apart.

Six in four dimensions, and three forever after

The count of regular solids goes five in three dimensions, six in four, and then three in every dimension above — for good. Four dimensions is the last place anything unusual happens, and it happens twice.

geometry · regular polyhedra Every turn that leaves a cube where it was. A cube in wireframe beside a table of its rotation axes: 3 of order 4, 4 of order 3, 6 of order 2, totalling 24 turns including the one that does nothing.

The five solids as three groups

There are five regular solids and only three groups of rotations between them, because a solid and its dual share their symmetries exactly. The largest of the three is the smallest group with no way of coming apart, which is why the general equation of the fifth degree has no formula.

geometry · regular polyhedra

Named alongside it

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

DualityPolyhedronEuler characteristicAngle defectClassificationRegular polygonSymmetry groupVertex configurationAlternating groupArchimedean solidConjugacy classConvexity

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