Why Are There Only Five Perfect Dice?

A die is a solid, so it has corners, and every corner of a perfect die is the same. So start small: forget the whole die and build just one corner. Take some equal regular polygons, put one corner of each at a single point, side by side, and see whether they can fold up into the point of a solid.

You probably noticed the rule. Three squares leave a gap of 90∘°; close it and you have the corner of a cube. Four squares fill all 360∘°, so they lie flat, like floor tiles. Five need 450∘°: they can’t even fit.

The gap is what makes a corner. Cut a wedge out of a paper circle and tape the two cut edges together: you get a cone with a point, and the bigger the wedge, the sharper the point. Cut nothing out, and the paper stays flat. A corner of a die is like the point of that cone: it points outwards, so the angles of the faces meeting there must add up to less than 360∘°, and the gap, the wedge that is missing, is what lets the faces bend round into a point. (A dent can break this rule: a corner pushed inwards can have more than 360∘° round it, like a crumpled sheet. Dice have no dents.)

Now the argument, in full. It needs only one fact from Grade 8: the angles of a polygon with n sides add up to (n − 2) × 180∘°, because lines from one corner cut it into n − 2 triangles. In a regular polygon all the angles are equal, so each is

$$\frac{(n-2) \times 180^\circ}{n}.$$

Polygon	Sides	Each angle	3 at a corner	4 at a corner	5 at a corner	6 at a corner
Triangle	3	60∘°	180∘°	240∘°	300∘°	360∘°
Square	4	90∘°	270∘°	360∘°	450∘°	540∘°
Pentagon	5	108∘°	324∘°	432∘°	540∘°	648∘°
Hexagon	6	120∘°	360∘°	480∘°	600∘°	720∘°
Heptagon	7	about 128.6∘°	about 385.7∘°	514.3∘°

Every corner of a perfect die has the same number of faces, at least 3, and their angles must add up to less than 360∘°.

• Triangles (60∘°): 3, 4 or 5 at a corner. Six make exactly 360∘°: flat.
• Squares (90∘°): only 3. Four make 360∘°.
• Pentagons (108∘°): only 3. Four make 432∘°.
• Hexagons (120∘°): three make exactly 360∘°. No corner at all.
• Seven sides or more: each angle is more than 120∘°, so even three pass 360∘°.

That leaves just five possible corners. Once the corner is chosen, the die is fixed: the faces round the first corner make the next corners, and so on, face by face, so each corner gives at most one die. So there are at most five perfect dice, and each really can be built, as the next section shows: exactly five. The solids are called the Platonic solids, or the regular polyhedra: solids whose faces are all the same regular polygon, with the same number meeting at every corner. Their names come from the Greek for their numbers of faces: the tetrahedron (4 faces), the cube (6), the octahedron (8), the dodecahedron (12) and the icosahedron (20).

The same argument in algebra

If k regular n-gons meet at a corner, the rule “less than 360∘°” is

$$k \times \frac{(n-2) \times 180^\circ}{n} < 360^\circ.$$

Divide by 180∘°, then multiply by n: k(n − 2) < 2n, that is, kn − 2k − 2n < 0. Add 4 to both sides, and the left side factorizes:

$$(n - 2)(k - 2) < 4.$$

Both n and k are at least 3, so both brackets are at least 1. Two whole numbers, each at least 1, whose product is less than 4: the products can only be 1 × 1, 1 × 2, 2 × 1, 1 × 3 or 3 × 1. Those give (n, k) = (3, 3), (3, 4), (4, 3), (3, 5), (5, 3): the five again. And the product is exactly 4 for (6, 3), (4, 4) and (3, 6): hexagons, squares and triangles lying flat, the three regular floor tilings.

Here they are. Turn each one, count what meets at a corner, and look at the last readout.

Solid	Faces	Each face	Faces at a corner	Edges	Corners
Tetrahedron	4	triangle	3	6	4
Cube	6	square	3	12	8
Octahedron	8	triangle	4	12	6
Dodecahedron	12	pentagon	3	30	20
Icosahedron	20	triangle	5	30	12

Notice the pairs. The cube and the octahedron both have 12 edges, with faces and corners swapped: 6 and 8 against 8 and 6. The dodecahedron and the icosahedron do the same with 12 and 20. That is what “Partner inside” shows: put a dot in the middle of every face of a cube, join the dots across each edge, and you get an octahedron. Each solid’s partner is called its dual. Squares, 3 at a corner, become triangles, 4 at a corner: the numbers n and k swap. The tetrahedron, with 3 triangles 3 at a corner, is its own dual.

There is another way to find the five, using Euler’s formula, which says that for every solid without holes, corners − edges + faces = 2 (you can see why in Why Do Solids Always Obey One Rule?). Write V for the corners (vertices), E for the edges and F for the faces, so V − E + F = 2.

Suppose every face has n sides and k faces meet at every corner. Count the sides of all the faces: nF. Every edge is the side of exactly two faces, so nF = 2E. Count the edges at every corner: kV. Every edge has two ends, so kV = 2E. So

$$F = \frac{2E}{n}, \qquad V = \frac{2E}{k}.$$

Put these into Euler’s formula, $\frac{2E}{k} - E + \frac{2E}{n} = 2$, and divide by 2E:

$$\frac{1}{n} + \frac{1}{k} = \frac{1}{2} + \frac{1}{E}.$$

The right side is more than 1/2, because E is a positive number of edges. So 1/n + 1/k must be more than 1/2. With n and k at least 3, that again leaves only the five pairs, and now the formula also says how many edges each solid has.

What happens for hexagons, 3 at a corner? Then 1/6 + 1/3 = 1/2, so 1/E = 0: no number of edges is big enough. That is the honeycomb: hexagons, 3 at every corner, go on across the floor for ever and never close up into a solid. Squares 4 at a corner and triangles 6 at a corner do the same. (Floors of tiles, and the strange ones that never repeat, are in Can a Pattern of Tiles Never Repeat?.)

Look again at the widget’s last readout, All the gaps. The cube: 8 corners with gaps of 90∘°, 8 × 90∘° = 720∘°. The tetrahedron: 4 × 180∘° = 720∘°. The octahedron: 6 × 120∘°, the dodecahedron 20 × 36∘°, the icosahedron 12 × 60∘°: always 720∘°.

This is not luck. The gap at a corner is called its angle defect, and Descartes’ theorem says that for any solid without holes, the angle defects at all its corners add up to exactly 720∘°, two full turns, however many corners it has and whatever its faces are. A solid has to turn its surface right round to close up, and the corners are where all of that turning happens: the faces themselves are flat. A sharp corner uses up a lot of the 720∘°; a blunt one very little, so the solid needs many of them.

A die doesn’t need to be perfect to be fair. What makes a die fair is that every face is just like every other face as part of the whole die: for any two faces, you can turn (or reflect) the die so that one lands exactly where the other was and the die looks the same as before. Then no face can be favored by the way the die is shaped. A solid like that is called face-transitive (or an isohedron). Every Platonic solid is face-transitive, but so are many solids whose corners are not all alike.

The triangular bipyramid is two triangular pyramids glued base to base: six equal triangles, any one of which can be turned onto any other. But 3 triangles meet at the top and bottom corners and 4 at each corner round the middle: the corners differ, so it isn’t perfect. The ten-sided die in Anish’s box is a pentagonal trapezohedron: ten identical kites, 5 meeting at each of its two points and 3 at each of the other ten corners. Both still obey Descartes. The bipyramid’s gaps: 2 × 180∘° + 3 × 120∘° = 720∘°. On a ten-sided die, made with all its corners on a sphere, the gaps are about 100.9∘° at the two points and 51.8∘° at the other ten, and they add up to 720∘° too.

Fair dice come in many more sizes than perfect ones: bipyramids and trapezohedra can be made with as many faces as you like, as long as the number is even. A convex face-transitive solid always has an even number of faces, so a fair die with 7 faces can’t be made that way. Game makers who want an odd number use long dice: a long prism, rolled along its length, which stops on one of its sides. The rounded ends stop it standing up.

The proof used two rules: the solid has no dents, and it lives in our three dimensions. Change either, and the count changes.

Allow faces to pass through each other. Use five-pointed stars as faces, or let the faces meet round a corner in a star pattern, and four more regular solids appear, spiky and self-crossing: the Kepler–Poinsot polyhedra. Johannes Kepler found two of them in about 1619; Louis Poinsot found the other two in 1809, and three years later Augustin Cauchy proved there are no others.

Go up a dimension. In four dimensions, where a “solid” is built from three-dimensional cells as a die is built from faces, the same kind of corner-counting leaves exactly six regular shapes, as the Swiss mathematician Ludwig Schläfli found in the middle of the 19th century. In five dimensions and every dimension above, there are only three: the cousins of the tetrahedron, the cube and the octahedron.

• Why Do Solids Always Obey One Rule? Euler’s formula, corners − edges + faces = 2, and why it works for every solid without holes.
• Can a Pattern of Tiles Never Repeat? The flat cases: squares, triangles and hexagons that fill a floor, and a single tile that fills it without ever repeating.
• Is the Golden Ratio Really Everywhere? The golden ratio is built into the dodecahedron and the icosahedron; elsewhere it is often a myth.
• Why Do Flights to America Fly Near the North Pole? Geometry on a sphere, which curves smoothly where a solid curves only at its corners.