What Makes a Shape Worth Trusting. Five shapes are perfect. Rather more than five are fair. The difference is the whole story.
What Makes a Die Fair
The obvious answer is symmetry, and the obvious answer is very nearly right. It matters which symmetry, and the usual assumption asks for far more of it than a die needs.
A die is fair by symmetry when no face can be favoured. For that, every face must be equivalent to every other: same shape, same size, sitting in the solid the same way, so the die can be turned to put any face exactly where any other face was. If a rotation or reflection swaps two faces and leaves the solid unchanged, physics cannot tell them apart, so they must come up equally often.
Notice what that does not require. It says nothing about the edges, and nothing about the corners.
The Rule Faces equivalent to each other: that is the whole requirement. A die whose corners are all different can still be perfectly fair, so long as you are reading the faces.
The Five Perfect Solids
Ask instead for everything to be equivalent (every face, every edge, every corner) and you have defined a regular polyhedron. Insist it also be convex, with no dents, and there are exactly five.
Five. Not five that happen to be convenient; five that can exist. Euclid closes the last book of the Elements by proving that "no other figure, besides the said five figures, can be constructed" [1].
The proof fits on a napkin. At least three faces must meet at each corner, and their angles must add to less than 360 degrees, or the corner goes flat:
| Faces at a corner | Angles add to | Solid | Die |
|---|---|---|---|
| 3 triangles | 180° | tetrahedron | d4 |
| 4 triangles | 240° | octahedron | d8 |
| 5 triangles | 300° | icosahedron | d20 |
| 6 triangles | 360° | flat. No solid. | none |
| 3 squares | 270° | cube | d6 |
| 4 squares | 360° | flat. No solid. | none |
| 3 pentagons | 324° | dodecahedron | d12 |
| 3 hexagons | 360° | flat. No solid. | none |
Every other combination overshoots 360. Five survivors, and four of them are in your dice bag right now.
They are called the Platonic solids, which is a little unfair. An ancient note on Euclid credits the cube, tetrahedron and dodecahedron to the Pythagoreans, and the octahedron and icosahedron to Theaetetus (c. 417–369 BCE), a younger contemporary of Plato whom Plato admired enough to name a dialogue after [2]. Plato's contribution was the physics.
Plato's Likely Story
In Timaeus Plato matches four of the solids to the four elements. He is careful about how he says it: the account is offered as a "likely story", an image of the truth rather than the truth [3].
The Four Elements
| Element | Solid | Die |
|---|---|---|
| Fire | tetrahedron | d4 |
| Air | octahedron | d8 |
| Water | icosahedron | d20 |
| Earth | cube | d6 |
Anyone who has stepped on a d4 in bare feet will agree about fire.
Why Earth Cannot Burn Underneath the imagery is a geometrical argument. Plato builds the solids from two right triangles: half an equilateral triangle, and half a square [3]. The d4, d8 and d20 are faced with equilateral triangles and share the same stock, so fire, air and water can be taken apart and rebuilt into one another. The cube's square faces come from the other triangle. Earth is made of different material and cannot transform at all [3].
A pentagon can be built from neither, which is why the d12 sits out. The dodecahedron, the roundest of the five, Plato keeps for the universe as a whole [3].
Kepler's Beautiful Mistake
In 1596 Johannes Kepler published Mysterium Cosmographicum, proposing that the five Platonic solids, nested one inside another, set the spacing of the six known planets: the cube outermost, then the tetrahedron, dodecahedron, icosahedron and octahedron [4].
Five gaps between six planets, and exactly five regular solids to fill them. Kepler could not believe that was a coincidence. The sky had not yet told him about the planets it was keeping back.
It is wrong, and it is one of the loveliest wrong ideas in astronomy. Kepler stayed proud of it all his life: it is the only one of his books he reprinted, in 1621 [4].
The Tetrahedron Problem
The d4 is the one Platonic solid that makes a poor die. With four faces and sharp points it barely rolls, and it lands with a point up rather than a face, so there is no top face to read. Makers print the number at the corners (read the one pointing up) or along the bottom edges instead.
Building Dice to Order
Face equivalence is a far cheaper requirement than regularity. Drop the demand for regular solids and you can build a fair die with almost any number of sides.
Two pyramids, base to base
Take a pyramid and glue it to its mirror image, base to base. That is a bipyramid: every face equivalent, the waist corners unlike the two tips. The d8 is the special case where the result comes out regular.
The d10's little twist
Try the same trick with five faces on each half and a problem appears: the die settles on a face, and what points at the ceiling is an edge, not a face. So the d10 twists one half against the other by half a step, which turns the triangles into kites. That shape is a pentagonal trapezohedron, and the stagger you see across its middle is what puts a number on top. Twist any bipyramid the same way and you get an even-numbered die of any size: d10, d14, d16, and on up to sizes nobody has any use for.
An illusion worth noticing
A d8 looks as though it has one waist, because the numbers are all printed upright relative to a single plane. It has three. Turn it and you will find three square cross-sections through it, at right angles to one another. The numbering invents a top and a bottom the shape does not have.
Doubling up for odd numbers
Trapezohedra are always even. The simplest fix for odd numbers is to repeat yourself: number a d6 as 1, 1, 2, 2, 3, 3 and you have a d3. A d6 numbered 1, 1, 1, 2, 2, 2 is a d2, and a coin that is harder to lose down the back of the sofa.
A d10 numbered in pairs is a d5. A d14 gives a d7. Players did this in their heads for years (one or two is a one, three or four is a two); printed dice just save the arithmetic.
Long dice
The other route to an odd number is a prism that rolls along its length, with pointed or rounded ends so it cannot stand up. A triangular one has three identical long faces, and it is fair for the same reason every die here is: the faces are interchangeable.
Catalan Solids
Here is where dropping the extra symmetry pays. Keep every face equivalent, abandon edge and corner equivalence, and you arrive at thirteen convex solids that the Belgian mathematician Eugène Catalan described in 1865 [5].
Five of them are Platonic solids with their faces raised into shallow pyramids. Raise each square of a cube into a four-sided pyramid: six faces times four is a fair d24. Raise each triangle of an icosahedron into a three-sided pyramid: a d60.
Every fair die size the Catalan solids give you
| Faces | Solids | Die |
|---|---|---|
| 12 | triakis tetrahedron, rhombic dodecahedron | a second d12 |
| 24 | triakis octahedron, tetrakis hexahedron, deltoidal icositetrahedron, pentagonal icositetrahedron | d24 |
| 30 | rhombic triacontahedron | d30 |
| 48 | disdyakis dodecahedron | d48 |
| 60 | triakis icosahedron, pentakis dodecahedron, deltoidal hexecontahedron, pentagonal hexecontahedron | d60 |
| 120 | disdyakis triacontahedron | d120 |
Each Catalan solid is the "dual" of an Archimedean solid: put a face where the Archimedean solid has a corner. So the face counts are just those solids' corner counts.
The corners are plainly not all alike: the peak of a raised pyramid is nothing like a corner of the original cube. It does not matter. You read faces.
Where It Stops Working
Two limits, one practical and one real.
Too round to settle
The d120 is very nearly spherical. It rolls for a long time, and when it stops you must hunt for the number among a hundred and nineteen others. Lou Zocchi's 100-sided "spherically shaped game die", patented in 1989 [6], has the same problem. Most tables read two d10s instead, one for tens and one for units.
Too irregular to trust
Then there are dice with no face symmetry: faces of different shapes and sizes, made by spacing points around a sphere as evenly as possible and slicing flat where each point lands. It is a reasonable way to approximate fairness. It is not fairness by symmetry, and nothing but a long tally tells you how close it gets.
Does It Matter? For a d7 you are using to pick a random encounter, no. A mild bias in a novelty die is not the thing that will decide your session.
The fair dice, by size
| You want | Fair by symmetry? | How |
|---|---|---|
d2, d3 |
yes | a d6 numbered twice or three times over |
d4, d6, d8, d12, d20 |
yes | the Platonic solids |
d5, d7 |
yes | a d10 or d14 numbered in pairs, or a long die |
any even number from d6 up |
yes | a trapezohedron |
d24, d30, d48, d60, d120 |
yes | a Catalan solid |
d100 in one die |
only as a 100-face trapezohedron, a spindle nobody wants | two d10s, or a Zocchi-style ball |
Roll Any of Them
The roller on the dnddiceroller.com home page has the d2, d3, d4, d6, d8, d10, d12, d20 and d100, and the 3D table throws the seven standard shapes with real physics. The home-page dice are perfectly fair, because they are not shapes at all.
This post is also a free booklet: download the PDF.
Table Talk
Rohan: Five perfect solids. Fine. I only care whether the die is fair.
Arthur: Then you care about faces. The corners can all differ; if the faces are interchangeable, it's fair.
Rohan: So the d10 isn't Platonic and it's still fine.
Arthur: Pentagonal trapezohedron. One half twisted by half a step so a face lands on top. Fair by symmetry.
Rohan: And the d4 is Platonic and it's the worst die in the bag.
Arthur: It lands point up. Some makers print the number at the corners, some along the bottom edges. Check yours before the session, not during.
Rohan: The d120. Yes or no.
Arthur: Fair, and too round to settle. You'll hunt for the number among a hundred and nineteen others.
Rohan: So two d10s for percentile. Done. Kepler would've argued.
Arthur: Kepler argued with the solar system. He lost, and reprinted the book anyway.
References
- Euclid. Elements, Book XIII, Proposition 18 and the remark following it. D. E. Joyce (ed.), Clark University. https://mathcs.clarku.edu/~djoyce/java/elements/bookXIII/propXIII18.html
- O'Connor, J. J., & Robertson, E. F. (n.d.). Theaetetus of Athens. MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Theaetetus/
- Zeyl, D., & Sattler, B. (2022). Plato's Timaeus. Stanford Encyclopedia of Philosophy, sections 4 and 8. https://plato.stanford.edu/entries/plato-timaeus/
- Linda Hall Library. Scientist of the Day: Johannes Kepler. https://www.lindahall.org/about/news/scientist-of-the-day/johannes-kepler-2/
- Catalan, E. (1865). Mémoire sur la théorie des polyèdres. Journal de l'École Polytechnique, 24, 1–71. https://hdl.handle.net/2268/194785
- Zocchi, L. (1989). Spherically shaped game die. US Design Patent D303,553, filed 17 May 1985, granted 19 September 1989. https://patents.google.com/patent/USD303553S/en
