Regular polyhedra — the series
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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.
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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.
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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.
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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.
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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.
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Three mirrors make every solid
Every symmetry of a regular solid, reflections included, is produced by just three mirrors meeting at its centre, reflected in one another over and over — a kaleidoscope. Put a single point between the three mirrors and its reflections are the corners of a solid: the regular solid itself if the point sits in a corner, and every one of its truncated and expanded relatives if it sits anywhere else.
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One determinant for every kaleidoscope
Three mirrors close up into a kaleidoscope when the angles of their triangle add to more than half a turn. In more dimensions there is no triangle to add up, but there is a matrix — the inner products of the mirrors' directions — and the mirrors close up exactly when it is positive definite. Trying every arrangement of up to nine mirrors against that one test finds Coxeter's list: three families in every dimension, a fork that runs out at eight mirrors because its determinant reaches zero at nine, and five exceptions. The same search finds the hyperbolic kaleidoscopes, and finds that they stop at five mirrors.
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A cube through a hole in a cube
Prince Rupert of the Rhine bet that a hole could be cut through a cube large enough for an equal cube to pass through it, and won. The trick is to look at the cube along a direction near its long diagonal, where its shadow is a hexagon wide enough to hold a square face with room to spare: a cube 6% larger can pass. All five Platonic solids turn out to have such a passage, found by searching over the directions from which a solid is seen — and in 2025 a solid was found that has none.
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Where charges settle on a sphere
Put equal electric charges on a sphere and let them push each other apart until nothing moves. Four settle at the corners of a tetrahedron, six of an octahedron, twelve of an icosahedron — but eight do not make a cube, and twenty do not make a dodecahedron. The cube loses to two squares turned half a corner apart, because charges on a sphere prefer triangles; and however many charges there are, exactly twelve more of them have five neighbours than have seven.
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Valleys a search cannot count
Let equal charges on a sphere push each other apart from a random start, and they settle into an arrangement where nothing moves. Up to fifteen charges every start settles into the same one. Sixteen is the first number with two, a hundredth of a unit apart in energy; by fifty-two there are at least four, and for fifty-six charges only one start in ten finds the lowest. The other valleys sit a few parts in a million above it, and nothing in the arrangement a search returns says whether it is the bottom.
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Eight points that refuse the cube
Place eight points on a sphere so that the closest pair is as far apart as possible. A cube manages 70.53°. Rotating one face by 45° and pushing the faces together gives 74.86°. A botanist counting pores on pollen grains asked this in 1930, and proofs still stop at fourteen points: one search in five hundred finds the best thirteen, whose failure to reach 60° is why kissing spheres stop at twelve.