Platonic solids
Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
Twenty-four ways to set a cube down
Count the rotations of a cube from its corners and the answer is eight times three. Count from its edges and it is twelve times two; from its faces, six times four. Three different pictures give one number because each count is the same theorem — the places a thing can go, times the motions that leave it where it is — and the same theorem splits Cayley's sixteen trees into twelve and four and proves that a group of eight has a centre.
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.
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.
Every count a solid can have
Euler's formula ties the corners, edges and faces of a convex solid together, but it does not say which numbers are possible. Ernst Steinitz answered in 1906: V corners and F faces occur together exactly when neither is more than twice the other less four. The proof in one direction is two lines of counting; in the other it is two moves — cut off a corner, glue on a tetrahedron — that walk from the pyramids to every allowed pair.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Euler characteristicPolyhedronDualityAngle defectArchimedean solidClassificationConjugacy classConvexityCurvatureOptimisationProjectionRegular polygon