The four that are allowed to cross themselves

The previous rung dropped one clause of the definition of a regular solid — all faces the same — and found thirteen more. There is another clause, and it is one nobody writes down because it seems too obvious to state: the solid must be convex, and the faces must not pass through one another.

Dropping that clause gives four more solids. They were found by Kepler in 1619 and Poinsot in 1809, and they are as regular as the Platonic five by every test except the one nobody had thought to name.

A polygon that goes round twice #

The construction starts one dimension down. Put $p$ points evenly on a circle and join every $q$-th, continuing until the path closes. When $p$ and $q$ share no factor, the path visits every point exactly once and comes back to the start, having gone round the centre $q$ times.

The result is written $\{p/q\}$, and the number $q$ is its density. The ordinary pentagon is $\{5/1\}$; the pentagram is $\{5/2\}$.

Everything that makes a polygon regular still holds. All the sides are the same length, all the angles are the same, and the whole figure is unchanged by a rotation through $2\pi/p$. What has changed is that the sides cross each other, and the interior — whatever that means now — is covered twice near the middle and once near the points.

The angle at each corner follows from the winding. Walking round any closed polygon, the total turning is $2\pi$ times the winding number, so for $\{p/q\}$ each of the $p$ exterior angles is $2\pi q/p$ and each interior angle is $\pi - 2\pi q/p$. For the pentagram that is $180° - 144° = 36°$, which is the sharp point the shape is known for.

That last remark is the reason the list of star polygons is not as long as it might be. The path closes early whenever $p$ and $q$ have a common factor, and what results is not one polygon but several superimposed — the Star of David is two triangles, not a hexagram in this sense. Only $q$ coprime to $p$ and less than $p/2$ gives a genuine new polygon.

What density is a count of #

Density deserves one more paragraph, because it is the quantity that makes all of this coherent rather than a collection of anomalies.

For a closed curve in the plane, the number of times it winds about a point is a whole number that a continuous deformation cannot change, so long as the curve never crosses the point. The pentagram’s density is that number, taken about its centre, and the figure computes it by adding up the angle turned along the path rather than by counting crossings of a ray — two routes to the same integer, and the one used here is the one that does not depend on where a ray is drawn.

The same idea in three dimensions counts how many times the surface of a star solid encloses the centre, and it is again an integer, again unchanged by deformation, and again the thing that reconciles the angle budget with the solid’s existence. A quantity that is unchanged by deformation and that distinguishes objects a picture cannot tell apart is the standing pattern of the subject; here it arrives before anyone had a name for it.

Four solids #

With star polygons available as faces, and with the possibility of arranging ordinary faces so that they pass through each other, four regular solids exist beyond the five:

• the small stellated dodecahedron: twelve pentagrams, five meeting at each of twelve points;
• the great dodecahedron: twelve ordinary pentagons, five at each point, passing through one another;
• the great stellated dodecahedron: twelve pentagrams, three at each of twenty points;
• the great icosahedron: twenty triangles, five at each of twelve points, crossing.

They come in two dual pairs, and the same duality as before swaps faces for corners. Cauchy proved in 1813 that these four and the Platonic five are the complete list of regular polyhedra in three dimensions, star faces allowed.

The two ways of counting it, and why they disagree #

Here is the fact that makes these solids more than curiosities.

The figure above builds the solid as a surface: 32 corners, 90 edges, 60 triangular faces. Euler’s alternating sum gives $32 - 90 + 60 = 2$, which is the number every sphere gives, and the surface is indeed a sphere — a spiky one, but topologically nothing unusual.

Counted as the regular solid it is, the numbers are different. Its faces are twelve pentagrams, not sixty triangles; its corners are the twelve points, not the thirty-two; and its edges are the thirty long segments each pentagram is drawn from, not the ninety short ones. That gives

$$12 - 30 + 12 = -6.$$

Minus six. No convex solid can produce that, and no sphere can. A surface with characteristic $-6$ is a sphere with four handles.

The two counts are not in conflict; they are counts of two different things, and which one is right depends entirely on what question is being asked. For how does light fall on a physical model, the sixty triangles are the answer. For is this a regular polyhedron, the twelve pentagrams are, because regularity is a statement about faces being congruent regular polygons transitively permuted by the symmetries — and sixty triangles meeting three, three and five at various corners are not that. The spiky surface is what the drawing shows. The regular solid is what the definition describes, and its faces intersect, so the “surface” it defines is not embedded in space — it is a map drawn on a genus-four surface and then folded into three dimensions in a way that makes it cross itself.

This is where Euler’s formula was found to have limits, and the discovery is due to Poinsot and to the long argument that followed about what the formula is actually a statement about. The answer, eventually, was that it is a statement about the surface and not about the solid, and that a formula which had looked like a fact about polyhedra was a fact about topology waiting for the subject to be invented. The same lesson arrives from a different direction when the solid has a hole through it.

What density does #

Each of the four has a density — the number of times its surface wraps around the centre — just as a star polygon does. The small stellated dodecahedron and the great dodecahedron have density 3; the great stellated dodecahedron and the great icosahedron have density 7.

Density is what makes the solids possible at all, and the arithmetic is worth doing. The angle budget from the Platonic argument says that the face angles at a corner must add to less than a full turn, and five pentagrams at a point contribute $5 \times 36° = 180°$, which is comfortably under. But five ordinary pentagons at a point contribute $5 \times 108° = 540°$, which is well over — and the great dodecahedron has exactly that, which is why it can only exist by passing through itself. The excess angle is absorbed by the surface folding back through the middle.

So the budget argument is not wrong; it is a criterion for convexity. What it proves is that only five solids can be built without self-intersection, which is a smaller claim than the one usually attached to it and is exactly the claim the figure of stacked angles makes.

An immediate consequence is worth stating, because it repairs an argument rather than merely qualifying it. Every proof of there are exactly five regular solids that runs through Euler’s formula is proving something about surfaces of characteristic 2, and it is therefore proving a theorem about convex solids without saying so. The angle-budget proof has the same hidden hypothesis in a different place. Two independent arguments, one hidden assumption, and the assumption is the same one in both.

Stellation, and what it means #

The word in three of the four names is a description of how they are built. To stellate a solid is to extend its faces outwards as planes until they meet again, and to take the outer cells so formed as the new solid.

Extending the twelve face planes of a dodecahedron produces three successive shells, and each is one of the interesting solids: the small stellated dodecahedron, the great dodecahedron, and the great stellated dodecahedron. So three of Kepler’s and Poinsot’s four are the complete stellation sequence of a single Platonic solid, and the fourth is the icosahedron’s — the icosahedron has fifty-nine stellations, of which one is regular.

The figure here builds the small stellated dodecahedron by the equivalent and more computable route: raising a pyramid on each pentagonal face, at the height where the triangular sides continue the plane of a neighbouring face. That height is $\varphi^2$ times the distance to the face centre, with $\varphi$ the golden ratio — which has no business appearing in a construction described entirely in terms of planes meeting, and appears because a pentagram’s arms stand in exactly that ratio to its core.

Why nine, and not more #

The complete list — five convex and four star — is Cauchy’s theorem, and the shape of its proof is worth having even without the details.

A regular polyhedron is determined by two numbers: the kind of face, written $\{p/q\}$, and how many meet at each corner, say $r$. The pair is written $\{p/q, r\}$ and is called the Schläfli symbol, so the cube is $\{4, 3\}$ and the small stellated dodecahedron is $\{5/2, 5\}$.

Two constraints then cut the possibilities down. The faces must be genuine star polygons, which needs $q$ coprime to $p$; and the arrangement must close up into a finite object, which is a condition on $p$, $q$ and $r$ that fails in both directions — too much angle and it wraps forever, too little and it does not close.

Working through the survivors gives the five Platonic symbols and $\{5/2, 5\}$, $\{5, 5/2\}$, $\{5/2, 3\}$ and $\{3, 5/2\}$. Nine, with the four stars in two dual pairs whose symbols are each other reversed — which is what duality does to a Schläfli symbol, and is a pleasant thing about the notation rather than a coincidence.

What is worth noticing is that pentagrams and pentagons are the only faces in the whole list beyond triangles and squares, and every star solid involves the number five. That is not an accident of the enumeration: the golden ratio is the only ratio in which a regular polygon’s diagonal stands to its side in a way that lets the extended sides close up again, and five is the only polygon it happens for.

What the picture cannot show #

The drawing shows the spiky surface, which is honest about what a physical model looks like and dishonest about what the object is. The pentagram faces are not visible at all: each one is spread over five of the sixty triangles and the interior of a dodecahedron, and there is no viewpoint from which one of the twelve faces can be seen as a face.

That is a genuine gap and not a shortcoming of this particular drawing. A self-intersecting solid cannot be drawn with its faces distinguishable, because the parts of each face that would identify it are inside the others. The best any picture can do is show the boundary of the region actually occupied, and then say what has been lost — which is the same limitation that makes a one-sided surface hard to draw honestly, arriving from the other side: there the object embeds and the property does not survive being looked at, here the property is plain and the object does not embed — which is why the caption says the pentagrams are there and the picture does not show them.

The two other Kepler–Poinsot solids with hidden structure — the great dodecahedron and the great icosahedron — are not drawn here for the same reason, one degree worse: their outer surfaces are the same as solids already shown, so a picture of them would be a picture of something else.

The ladder from here #

Rungs above: the complete enumeration, and Cauchy’s proof that there are nine regular polyhedra and no more. Four dimensions, where the regular star polytopes number ten. Coxeter’s classification by reflection groups, which produces the star solids and the ordinary ones from one construction. The fifty-nine stellations of the icosahedron and the rule that decides which count. And the abstract polytopes, in which a solid is a combinatorial object and the question of whether it can be built in space is asked separately.

What the definition was hiding #

Two clauses were dropped in two essays, and the second is the more interesting because nobody had written it down.

All faces the same was in the definition, visible and available to be relaxed. No face passes through another was not in the definition at all; it was in everybody’s picture of what a solid is, and for two thousand years the picture and the definition were not distinguished. Kepler’s contribution was less a discovery of new shapes than a noticing that an unstated assumption had been doing work.

That is worth carrying because unstated assumptions are, by construction, the ones a proof cannot mention. The angle budget proves what it proves; what it was thought to prove was larger, and the gap was invisible until somebody drew an object in it.