Thirteen more when one word is dropped

Five regular solids and no more is one of the oldest results in mathematics and one of the sharpest, and its sharpness comes from a definition that quietly asks for two separate things:

1. every face is the same regular polygon;
2. every corner is surrounded by the same arrangement of faces.

A list that short invites the obvious experiment. Drop one clause and count again.

Which clause to drop #

Dropping the second clause and keeping the first gives solids whose faces are all congruent regular polygons but whose corners differ. That produces the deltahedra — the eight convex solids made entirely of equilateral triangles — and it is a smaller and less interesting list, because a corner that differs from its neighbour makes the solid lopsided in a way the eye reads as a mistake.

Dropping the first clause and keeping the second is the productive move. The result is a solid that looks uniform from every corner and has faces of more than one shape, and there are thirteen of them beyond the Platonic five and beyond two infinite families of prisms.

The reason this is the interesting relaxation is worth stating. What the eye recognises as regularity is not really the faces; it is the corners. A solid all of whose corners are alike can be turned so that any corner sits where any other was, and that is a statement about a group of symmetries acting transitively. Faces being identical is a weaker and less structural condition.

What a corner is described by #

The description that does the classifying is the vertex configuration: the list of face sizes met when walking round a single corner. A cube’s corners are 4.4.4 — three squares. A truncated cube’s are 3.8.8 — a triangle and two octagons.

That is a complete description of what one corner looks like, and it is nearly a complete description of the solid, because the corners have to fit together. The classification of the Archimedean solids is essentially the enumeration of which vertex configurations can close up.

The constraint is the same angle budget that limits the Platonic solids. The angles of the faces at a corner must add to less than a full turn, or the corner will not be a corner. A triangle contributes 60 degrees, a square 90, a pentagon 108, a hexagon 120, an octagon 135, a decagon 144. Three hexagons make exactly 360 and lie flat — which is a tiling of the plane rather than a solid, and is why hexagons never appear three at a corner.

There is a second constraint that is easy to overlook and does real work: at most two different odd-sided faces can meet at a corner in a mixed configuration, and configurations like 3.4.5.4 are severely restricted, because an edge is shared by exactly two faces and walking round the solid forces the sizes to alternate consistently. A configuration such as 3.5.3.5 closes up because the two alternate; 3.4.5 does not extend to a solid at all, although its angles fit comfortably inside 360 degrees.

That is the difference between a local condition and a global one, and it is the reason the classification cannot be done by arithmetic on angles alone. The angle budget says which corners are conceivable; whether they can be sewn together into a closed surface is a separate question with a separate answer.

Cutting the corners off #

The largest family of these solids is obtained by a single operation. Take a Platonic solid and slice each corner off with a flat cut. Each original face of $n$ sides becomes a $2n$-gon, since each of its corners is replaced by an edge of the cut, and each original corner becomes a new face with as many sides as edges met there.

The cut has one free parameter: how deep to go. Almost every depth produces a solid with two different edge lengths — the remnant of an original edge, and the edge of the new face — and therefore faces that are not regular.

There is exactly one depth at which the two lengths agree, and finding it is a one-dimensional problem: the remnant shrinks as the cut deepens and the new edge grows, so the difference changes sign once. The figures here find it by bisection, ninety times over, and then assert that every edge of the resulting solid is the same length to within $10^{-9}$.

Neither of those solids was designed. The football’s pattern is not a piece of industrial design that happens to be mathematical; it is the unique way of cutting an icosahedron so that the pieces stay regular, and it was described two thousand years before anybody kicked one.

The half-cut, where the parameter disappears #

Push the cut to the midpoint of every edge and something different happens. The two cuts at the ends of an edge meet, the remnant vanishes, and the original $n$-gon becomes an $n$-gon again — its own midpoint polygon — rather than a $2n$-gon.

That operation is called rectification, and it has no free parameter at all: the midpoints are the midpoints. Rectifying a cube gives the cuboctahedron, whose corners are 3.4.3.4; rectifying an icosahedron gives the icosidodecahedron, 3.5.3.5.

That last observation is the sharpest thing in the family. Rectifying a solid and rectifying its dual give the same result, because the dual has its faces where the original has its corners and its corners where the original has its faces, and the midpoints of the edges are common to both. So the thirteen are not thirteen unrelated shapes; several of them are one construction applied to a pair of solids that were already the same object seen twice.

Counting them, and what Euler says #

Every one of these solids satisfies $V - E + F = 2$, and the figures check it for each one they draw. That is not a surprise — it holds for every convex solid, and fails only for surfaces that are not spheres — but it is the constraint that makes the enumeration finite.

Given a vertex configuration, the counts follow from arithmetic. If each corner meets faces of sizes $n_1, \ldots, n_k$, then every corner has $k$ edges, so $E = kV/2$; the faces of size $n$ number $V$ times the number of $n$-gons per corner divided by $n$; and putting those into Euler’s formula pins $V$ down. If the resulting $V$ is not a whole number, or is negative, the configuration is impossible.

The reliance on Euler’s formula is worth noticing, because it is what makes the problem finite at all. Without it there is no bound on how many faces a solid with a given kind of corner could have, and the enumeration would have nowhere to stop. With it, the alternating sum that every convex solid satisfies turns a question about shapes into a question about whether a small system of equations has a whole-number solution.

That reduces the classification to checking finitely many configurations, most of which fail on the angle budget and the rest of which fail on Euler’s arithmetic. Thirteen survive.

There are also two infinite families that satisfy every clause and are usually excluded by convention rather than by a theorem: the prisms, two $n$-gons joined by a band of squares, and the antiprisms, two $n$-gons joined by a band of triangles. Every corner of a prism is 4.4.$n$ and every corner of an antiprism is 3.3.3.$n$, so both are as uniform as anything in the thirteen. They are set aside because they go on forever, which is an honest reason and not a mathematical one.

Where the thirteen came from #

The solids are named for Archimedes, whose treatise on them is lost; what survives is a description by Pappus, some five centuries later, listing thirteen and attributing them. Nobody knows what Archimedes’ argument was, or whether he had one.

The list was rediscovered piecemeal over the following millennium and a half. Kepler enumerated all thirteen in 1619 and gave the first real proof, in the same work in which he found the star polyhedra and the tilings of the plane — a single sustained effort to work out what happens when the Platonic definition is loosened in each available direction. Several of the individual solids had been drawn before that, by Piero della Francesca and by Dürer, as objects of interest to painters working out perspective, with no suggestion that they belonged to a list at all.

That history is worth a paragraph because it shows what a classification adds. Drawing a truncated icosahedron is a craft problem, solved repeatedly by people who wanted to draw one. Knowing that there are exactly thirteen requires the enumeration, and the enumeration required somebody to ask the question in the form what are all the solids with this property rather than here is a nice solid.

The one that is not a truncation #

Eleven of the thirteen are obtained by cutting, rectifying or cutting-then-cutting. Two are not, and they are the reason the classification needs a proof rather than a construction.

The snub cube and the snub dodecahedron are made by pulling the faces of a solid apart, rotating each by an angle that must be found rather than chosen, and filling the gaps with triangles. The rotation angle is the root of a cubic with no nice form, and the results come in two mirror-image versions that cannot be turned into one another — the only solids in the family with that property.

They matter here because they show what the enumeration is really doing. A list produced by applying operations to a starting set is a list of things somebody thought of; a list produced by enumerating vertex configurations and checking each is a list of everything there is. The snubs sit in the second list and not in the first, and their presence is what makes “thirteen” a theorem instead of a tally.

What is preserved, and what is not #

Cutting the corners off a solid changes almost everything countable about it and changes nothing about its symmetry.

Every rotation that leaves a cube where it was also leaves the truncated cube where it was, because the truncation is defined from the cube’s own corners and edges and so is carried along by any motion that preserves those. The corner count goes from 8 to 24, the face count from 6 to 14, and the group of rotations stays at exactly the same 24 turns.

That is why thirteen Archimedean solids, five Platonic ones, and thirteen more duals share only three rotation groups between them — the tetrahedral, the octahedral and the icosahedral. The shapes are many and the symmetries are few, and which of the three a solid has is the last rung of this ladder.

It also explains the two infinite families. A prism has the symmetry of its base polygon rather than one of the three, which is the structural reason they feel like a different kind of object — and a better reason for setting them aside than the one about the list being infinite.

What the picture cannot show #

The drawings hide their back halves, because a solid drawn with its far faces visible is unreadable. So the counts printed under each figure — twelve corners, twenty-four edges — are computed from the construction and cannot be verified against the picture by anyone looking at it. That is the ordinary and unavoidable dishonesty of drawing three dimensions on a page, and the response here is that the numbers come from the vertex list rather than from the drawing.

The classification itself is not drawable at all. Thirteen pictures do not show that there is no fourteenth, and the argument that there is not is the arithmetic above, run over every configuration the angle budget permits. It is the same gap as in the Platonic case, where five drawings and one counting argument do quite different jobs.

And the snubs are not drawn, because the family here builds solids by cutting and a snub is not a cut. Naming them and not showing them is the honest version of that limit.

The ladder from here #

Rungs above: the star polyhedra, which come from dropping convexity instead. Four dimensions, where the count of regular solids rises before collapsing. The rotation groups these solids share with the Platonic ones — a truncation does not change the symmetry, which is why thirteen solids have only three groups between them. The Catalan solids, which are the duals of these and have identical faces and differing corners. And the uniform tilings of the plane, which are the same enumeration with the angle budget set to exactly 360 rather than below it.

The move worth keeping #

A definition with two clauses was taken apart, and dropping each clause separately produced two different lists.

That is a general way of finding out what a definition is doing, and it is more informative than either clause alone. The five solids were not five because of regularity; they were five because of two conditions holding at once, and the interesting structure — corner-transitivity — turns out to belong to the clause that produces thirteen rather than the one that produces eight.

The same experiment is worth running on any classification that comes out surprisingly short. Constant width is what remains of a circle when one of its defining properties is dropped and the other kept, and the result is again a family nobody expected to exist.