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Regular polyhedra — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · geometry
  2. 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.

    part 2 · geometry
  3. 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.

    part 3 · geometry
  4. 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.

    part 4 · geometry
  5. 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.

    part 5 · geometry

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