Six in four dimensions, and three forever after

Two dimensions have infinitely many regular polygons. Three have exactly five regular solids. The natural expectation is that the count keeps falling — four, three, two — and eventually reaches something dull.

It does not. Four dimensions have six, which is more than three dimensions have, and every dimension from five upwards has exactly three, forever.

What a regular polytope is #

The definition generalises without difficulty. A regular polygon has equal sides and equal angles; a regular polyhedron has regular polygons for faces, the same at every corner; a regular four-dimensional polytope has regular polyhedra for cells, the same at every corner, and so on upwards.

The bookkeeping is done by the Schläfli symbol. A regular polygon with $p$ sides is $\{p\}$. A solid with $\{p\}$ faces meeting $q$ at a corner is $\{p, q\}$ — so the cube is $\{4, 3\}$. A four-dimensional polytope with $\{p, q\}$ cells meeting $r$ around each edge is $\{p, q, r\}$, and the notation continues as far as anybody wants.

Six symbols work in four dimensions:

The pairs are duals: reversing a symbol gives the dual polytope, so the tesseract and the 16-cell are a pair, the 120-cell and the 600-cell are a pair, and the 5-cell and the 24-cell are each their own dual — the 5-cell because its symbol reads the same backwards, and the 24-cell for the same reason.

Building them from coordinates #

Each of these can be written down, and writing them down is more convincing than any picture of them.

The 5-cell is five points, all the same distance apart: the four-dimensional simplex, the analogue of a triangle and a tetrahedron. Every pair is an edge, so it has ten.

The tesseract is the sixteen points $(\pm1, \pm1, \pm1, \pm1)$. Two are joined when they differ in exactly one coordinate, which gives thirty-two edges — sixteen points with four edges each, halved.

The 16-cell is the eight points $\pm e_i$, one on each axis in each direction. Every pair except an opposite pair is at the same distance, so there are twenty-four edges and every vertex meets six of them.

The 24-cell is the twenty-four points with two coordinates $\pm 1$ and the other two zero. Every one of them is at the same distance from the origin, and the shortest distance between two of them is $\sqrt2$, achieved ninety-six times.

The figure does not take any of those edge counts on trust. For each polytope it computes every pairwise distance, finds the shortest, and calls a pair an edge when its distance equals that shortest one. Then it checks that every vertex meets the same number of edges — which is a real test, and a construction with a mistake in it would fail it.

That split is the whole of the story about counts. Three families run forever and everything else is local, and the interesting dimensions are the ones with something local in them: three, which has two extra, and four, which has three.

The 24-cell, which has no relatives #

Of the six, one is strange, and its strangeness is the reason this rung exists.

Five of the six belong to families that run through every dimension. The 5-cell is the simplex — the shape whose corners are all mutually adjacent, of which the triangle is the two-dimensional case — and there is one in every dimension. The tesseract is the cube, and there is one in every dimension. The 16-cell is the cross-polytope — the shape whose corners are the points $\pm e_i$ — and there is one in every dimension. The 120-cell and 600-cell are the four-dimensional relatives of the dodecahedron and icosahedron, which exist in three dimensions and nowhere above.

The 24-cell belongs to nothing. It has no three-dimensional shadow, no five-dimensional analogue, and no family. It exists in four dimensions and only there.

Its existence has a clean explanation that is itself surprising: in four dimensions the cross-polytope is self-dual up to a rotation, so a 16-cell and its dual tesseract can be laid over one another and their vertices combined. The twenty-four points of the 24-cell are exactly the eight corners of a 16-cell together with the sixteen corners of a tesseract, scaled to sit on one sphere — and this works because eight plus sixteen is twenty-four and the distances happen to agree, which they do in no other dimension.

The 24-cell is also the reason four dimensions have the densest lattice packing anyone would guess at, and its vertices form a lattice with astonishing properties. None of that is visible from the picture; all of it comes from the coordinates.

What the drawings are doing #

A four-dimensional object cannot be drawn, and what is drawn instead is a projection, in two stages.

First the polytope is turned by a rotation in the plane of two of its coordinates — which is what “turning in four dimensions” means, and the reason the drawn shapes look like a solid caught mid-motion. Then the fourth coordinate is used for a perspective divide: points with a larger fourth coordinate are scaled up, exactly as a perspective drawing scales up what is nearer the eye. That leaves three coordinates, which are then drawn on the page by ordinary perspective.

The choice of projection matters more here than in three dimensions, because nothing is being hidden by opacity — every edge is drawn, and which of them cross on the page is entirely a matter of the angle. Turning the object changes the picture completely while changing the object not at all, which is why these figures name their angle of turn.

The result is a Schlegel diagram when the projection point is chosen just outside one cell: the whole polytope collapses inside that cell’s image, so a tesseract appears as a cube inside a cube with the two joined corner to corner. The inner cube is not smaller than the outer one; it is further away in the fourth dimension.

The two that are not drawn #

The 120-cell and 600-cell deserve a paragraph even though no figure here shows them, because they are what makes four dimensions the interesting case rather than merely a case.

The 600-cell has 120 vertices, 720 edges, 1,200 triangular faces and 600 tetrahedral cells, with twenty tetrahedra meeting at each vertex. Its 120 vertices are, under the right identification, exactly the 120 unit quaternions of finite order — so the multiplication that remembers its order has this polytope sitting inside it as a finite group.

Its dual, the 120-cell, has 600 vertices and 120 dodecahedral cells. Between them they are the four-dimensional analogues of the icosahedron and dodecahedron, and like their three-dimensional cousins they involve the golden ratio at every turn — the coordinates of the 600-cell’s vertices are built from $\varphi$, and the arithmetic that makes them close up is the same arithmetic that makes a pentagon close up.

Both stop here. There is no five-dimensional relative of either, for the reason the next section gives.

Why the count collapses #

Above four dimensions there are exactly three regular polytopes in every dimension: the simplex, the cube and the cross-polytope. The dodecahedral and icosahedral shapes stop after four dimensions, the 24-cell after four, and nothing new ever appears.

The reason is an angle argument, the same one as in three dimensions and run one level up. Assembling a polytope $\{p, q, r\}$ requires several copies of the cell $\{p, q\}$ to fit round each edge, and the dihedral angle of $\{p,q\}$ — the angle between two of its faces along an edge — must multiply by $r$ to less than a full turn.

Dihedral angles grow as the cells get more complicated. A tetrahedron’s is about 70.5 degrees, so five fit round an edge and $\{3,3,5\}$ exists. A cube’s is 90, so only three fit and $\{4,3,3\}$ is the only cube-celled polytope. A dodecahedron’s is about 116.6, so three fit and $\{5,3,3\}$ exists and nothing more.

Now go up. The cells of a five-dimensional polytope are four-dimensional polytopes, whose dihedral angles are larger still — the 120-cell’s is 144 degrees, and only two fit round a face, which is not enough to close anything. Only the three families, whose dihedral angles stay small, survive; and they survive in every dimension, because the simplex, the cube and the cross-polytope have dihedral angles that stay bounded away from 180 degrees as the dimension grows.

Four dimensions is where the angle budget runs out, and the sharpness of that is the pleasing part. There is no vague sense in which higher dimensions are “too roomy”; there is one inequality, and it stops being satisfiable between four and five.

The counts, and where they come from #

It is worth putting the numbers the figures compute next to the descriptions, because they are the only part of this a reader can check.

The 5-cell has five vertices and every pair is an edge, so ten edges and four at every corner — the four-dimensional triangle, where everything is joined to everything. The tesseract has sixteen vertices and thirty-two edges, four at each corner: one for each coordinate that can be flipped. The 16-cell has eight vertices and twenty-four edges, six at each corner: everything joined except the six opposite pairs. The 24-cell has twenty-four vertices, ninety-six edges and eight at each corner.

Every one of those is computed by the figure from the coordinates alone, by measuring every pairwise distance and calling the shortest ones edges. That the resulting graph turns out to be regular — the same number of edges at every vertex — is the check, and it is the kind of check a mistyped coordinate would fail immediately.

The relation $2E = Vd$, with $d$ the number of edges at a corner, holds in all four cases and is asserted: 10 = 5·4/2, 32 = 16·4/2, 24 = 8·6/2, 96 = 24·8/2. It is the handshake count applied to a four-dimensional object, which is a reminder that most of what can be checked about these things is combinatorial rather than geometric.

The pattern of the counts #

Written out, the sequence of counts by dimension is

$$\infty,\ 5,\ 6,\ 3,\ 3,\ 3, \ldots$$

starting at dimension two. This is one of the better examples of a sequence whose small cases lie. Anybody shown the first two terms would predict a decreasing sequence; anybody shown the first three would predict something erratic, in the manner of a small case that lies; and the truth is that the interesting behaviour is confined entirely to dimensions two, three and four, after which the subject becomes uniform.

Four dimensions is the last dimension in which anything unexpected happens, and it happens twice: the count goes up, and a polytope with no family appears. Both are consequences of the same near-coincidence — that in four dimensions the angle budget is just barely large enough for the 600-cell and the numbers happen to permit the 24-cell.

What four dimensions is not #

Some care is worth taking with the word, because “the fourth dimension” carries a great deal of baggage that has nothing to do with any of this.

Nothing here is about time. The fourth coordinate is a fourth number, on the same footing as the other three, and the polytopes above are as timeless as a triangle. A four-dimensional point is a list of four numbers; a rotation is a linear map preserving lengths; a regular polytope is a combinatorial object with a geometric realisation. All of it is arithmetic that could have been done without ever drawing anything.

Nor is any of it inaccessible. The tesseract’s vertices are the sixteen strings of four plus and minus signs, and its edges join strings differing in one place — which is a description a reader can hold entirely, and is the same object as the hypercube of Boolean assignments that logic uses for a truth table. The difficulty is only in seeing it, and seeing is not the same as understanding.

What the drawings are good for, then, is not intuition about four dimensions. It is a check on the arithmetic: a figure whose edges came out with the wrong count would look wrong, and a figure whose vertices are not all at the same distance from the centre would look lopsided under rotation. That is a modest job and it is the honest one.

What the picture cannot show #

Nearly everything. A projection of a four-dimensional object into two dimensions loses two dimensions, and what it loses is precisely the information that the edges are all the same length and the cells all the same shape. Every drawing above shows a distorted object, and the assertion that the object is not distorted is made by the coordinates and checked by the arithmetic, never by the picture.

The 120-cell and 600-cell are not drawn at all. Six hundred vertices and 1,200 edges projected onto a page make a ball of ink, and a figure that cannot be read is worse than an absence. What is lost by leaving them out is the pair that makes four dimensions interesting, which is stated in the table and shown nowhere.

And the sense in which a Schlegel diagram is inside out — the inner cube of a tesseract being no smaller than the outer, the far cell of the polytope being drawn as the whole outside — cannot be conveyed by the drawing, which shows a small cube inside a large one and looks exactly like a small cube inside a large one.

The ladder from here #

Rungs above: the star polytopes of four dimensions, of which there are ten. Coxeter groups and the classification of reflection groups, which produces every regular polytope in every dimension from one diagram. The 600-cell’s relation to the quaternions, whose 120 unit elements are its vertices. Honeycombs, which are the regular tilings of space and are the case where the angle budget is exactly met. And the exceptional lattices in eight and twenty-four dimensions, which are the next place after four where a coincidence of numbers produces something with no family.