by Dr. Greg Gbur
I’ve been enjoying a bit of reminiscing about my childhood lately, hunting down old copies of role-playing games I enjoyed in my youth as well as exploring newer games that have come out since then. One thing that has changed dramatically since my gaming days is the proliferation of types of dice. Most human beings never go beyond ordinary 6-sided dice, which we in the gaming world call a “d6.” Classic Dungeons & Dragons players, however, are familiar with the d4, d6, d8, d10, d12, and d20.
But these days, there are even more imaginative varieties! I’ve starting collecting dice of every shape and size, and my current collection is shown below, in order¹:
This is a really amazing variety of dice! This is my “special dice” collection in its entirety, and includes some duplicates (don’t ask how I got 4 identical d60s), but in some cases, such as the d7, d12 and d24, there are varieties in shapes even with the same number of faces!
This variety got me wondering: how does one design dice with a weird number of faces? What mathematical strategies does one use to make them? What other types of dice are possible? And, perhaps most important: are these dice “fair”?
I thought it would be fun to answer these questions with a blog post, in which we discuss the geometry of dice!
We will start with the most familiar types of dice, and work ourselves gradually into strange and unfamiliar territory. Before we begin, we should note that a major consideration for any type of die is that it be “fair”: that is, every number on the die should be equally likely to be rolled. The most obvious way to do that is to make the die have a lot of symmetry in its shape, which brings us to our first category…
The Platonic Solids
The Platonic solids are the only polyhedra (multi-sided objects) which are convex (have no concavities) and regular. A “regular” polyhedron is one for which not only are all faces equivalent to one another, but so are all edges and all vertices (points). To put it another way: a Platonic solid can be rotated to make any edge, vertex or face look exactly like any other one. Obviously, this is ideal for making a fair die — there is no preferred edge, vertex, or face on the solid.
The Platonic solids are somewhat profound, in that there are a small number of them, and no (obvious) reason why they have the number of faces that they do. This profundity captured the imagination of Plato, who gave the solids a fundamental role in his dialogue Timaeus (c. 360 BCE), assigning each of them to one of the elements. From Timaeus,
To earth, then, let us assign the cubical form; for earth is the most immoveable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature. Now, of the triangles which we assumed at first, that which has two equal sides is by nature more firmly based than that which has unequal sides; and of the compound figures which are formed out of either, the plane equilateral quadrangle has necessarily, a more stable basis than the equilateral triangle, both in the whole and in the parts. Wherefore, in assigning this figure to earth, we adhere to probability; and to water we assign that one of the remaining forms which is the least moveable; and the most moveable of them to fire; and to air that which is intermediate. Also we assign the smallest body to fire, and the greatest to water, and the intermediate in size to air; and, again, the acutest body to fire, and the next in acuteness to, air, and the third to water. Of all these elements, that which has the fewest bases must necessarily be the most moveable, for it must be the acutest and most penetrating in every way, and also the lightest as being composed of the smallest number of similar particles : and the second body has similar properties in a second degree, and the third body in the third degree. Let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire; and let us assign the element which was next in the order of generation to air, and the third to water. We must imagine all these to be so small that no single particle of any of the four kinds is seen by us on account of their smallness: but when many of them are collected together their aggregates are seen. And the ratios of their numbers, motions, and other properties, everywhere God, as far as necessity allowed or gave consent, has exactly perfected, and harmonised in due proportion.
So fire is a d4, earth is a d6, air is a d8, and water is a d20. Of the 5th Platonic solid, the d12 dodecahedron, Plato vaguely says, “There was yet a fifth combination which God used in the delineation of the universe.”
Later thinkers also thought the Platonic solids played a fundamental role in the cosmos. In 1596, the German astronomer Johannes Kepler published the book Mysterium Cosmographicum, in which he speculated that the positions of the known planets corresponded to the Platonic solids inscribed within one another, as illustrated at right.
Obviously, this model did not work out, but it illustrates how the set of 5 solids captured the imagination of philosophers trying to make sense of the universe.
There is a big problem with the d4 when being used as a die, though: it doesn’t roll very well! One solution in recent years has been to truncate the tips of the tetrahedron and place the actual numbers on the tiny flat surfaces. Such dice are easier to roll and to read the result. A traditional d4 and a newer one are shown at right.
There is a curious illusion associated with the d8 that is worth noting before moving on. You may note that the d8 can be created by taking two square-base pyramids made of equilateral triangles and fixing their bases together. Looking at such a die, it seems like it is mirror-symmetric across the plane of the bases, but not so symmetric that every edge and vertex is the same!
I interpret this as an illusion due to the way the numbers are put on the die. They are all placed with their bottoms towards a single base plane, which makes it look like this is the only symmetry. However, if you look at the die from other edges, you can see that it looks the same! There are 3 different base planes.
If this isn’t obvious from looking at actual dice, it is much clearer looking at a transparent model. See if you can trace the 3 squares formed by the edges in the image at left.
But the idea of mirror symmetry provides a way to make dice of any large even number of sides! We can make a “pyramid” of sorts with any number of faces and a base and glue two of them base to base; as noted, a d8 can be thought of as gluing two 4-faced pyramids together. In this way, we can construct the final member of the original Dungeons & Dragons dice, the d10. We can also, however, make a different form of d12, as well as a d14, d16, or higher! Such a structure is formally known as a trapezohedron. My d10 and d16 are shown below.
You may notice a slight difference between the d10 and d16: the faces of the d10 are shifted off-center across the mirror plane, while the faces of the d16 are aligned across this plane. This shift of the d10 is necessary because there are 5 faces on either side of the mirror plane. If the faces across the plane were lined up, then the die would come to rest on one face, but an edge would be facing upward! The shift means that a face is upward when the die comes to rest. This is necessary for any mirror-symmetric die with an odd number of faces on either side of the mirror plane.
One can take the idea of mirror symmetry to a ridiculous extreme. At right is pictured a d50, which uses the same principle. Note that it also has its faces shifted, because there are an odd number of faces on either side of the edge!
So far, all the dice we’ve considered have had an even number of faces. The trapezohedrons, however, suggest a very straightforward, if unimaginative, way to make dice with an odd number of faces: simply double up on the lower numbers! In this way, a d6 becomes a d3, a d10 becomes a d5, and a d14 becomes a d7, as shown below.
D&D players were doing this for years before actual dice were made: players simply said, for instance “a roll of 1-2 on a d6 is 1, 3-4 is 2, and 5-6 is 3.” We were certainly able to handle that, but it is just nice to have dice explicitly marked for this strategy.
My favorite of this type is the d2: the 2-sided die! As can be seen below, it is in essence a die for which 3 sides in a “U”-shape represent 1 and the other 3 sides in a “U” represent 2. Each trio of sides has been rounded together to make the die have, effectively, 2 rounded sides!
Lots of people will say, “Why do you need a 2-sided die? Can’t you just flip a coin?” You could, but gamers such as myself love to roll dice. Flipping a coin just doesn’t have the same feeling to it!
We will have more to say about odd-numbered dice in a few moments, but first let’s return back to dice with an even number of faces. Is there any way, other than trapezohedron, to create more varieties?
In fact, we can, by removing some of the symmetries of the Platonic solids. The Platonics, if you recall, were symmetric in faces, edges, and vertices: that is, every face, edge and vertex was equivalent to every other one. But, for a fair die, it would seem that we really only need every face of the die to be equivalent: the Platonic solids have, in a sense, more symmetry than we need. If we throw out edge and vertex symmetry, this leads us to the d24, d48, d30, d60 and d120!
These objects are members of a group of solids known as Catalan solids, after the Belgian mathematician Eugène Charles Catalan who described them in 1865. The complicated names of those pictures above are, respectively, the tetrakis hexahedron, the disdyakis dodecahedron, the rhombic triacontahedron, the deltoidal hexecontahedron, and the disdyakis triacontahedron.
That’s a lot of wordage, but we can understand most of these Catalan solids as extensions of the Platonic solids. For example: suppose we take each face of a cube, and “pull” it outward to make it a narrow pyramid, as shown below.
We have now created a tetrakis hexahedron, as there are 4 sides per pyramid and six faces to the cube! 4 times 6 = 24, obviously! It should be noted that the vertices are no longer all the same: the peak of each pyramid is not equivalent to one of the corners of the former cube.
We can similarly think of the d48, the disdyakis dodecahedron, as making an 8-sided pyramid out of each face of a cube. Alternatively, we can think of it as making a 6-sided pyramid out of each face of a d8 (octahedron).
There is another Catalan solid that is a d24, as well; it is known as the deltoidal icositetrahedron, and is shown at right. This 24-sided die can be viewed as making a triangular pyramid out of each face of an octahedron.
The d60 and d120, the deltoidal hexecontahedron and the disdyakis triacontahedron, may similarly be thought of as modifications of a d20. For the d60, we make a 3-sided pyramid out of each face of the d20, and for the d120, we make a 6-sided pyramid. I highlight these pyramids in the figure below, if it isn’t clear.
Some Catalan solids cannot be simply fashioned from the Platonic solids, however. The d30 is one of those, as is a different version of a d12, a shape known as a rhombic triacontahedron. These two are shown below.
The rhombic dodecahedron has the curious property that its shape can be used to perfectly fill a volume in three-dimensional space. That is: if you had enough of these d12s of the same size, you could stack them together to fill a region of space without any gaps, just as you could with a bunch of d6s.
There are a few other tricks we can do with these two. If we take the rhombus that forms the faces of the d30 and pyramid it into four equal sections, we end up with a d120! If we take the rhombic d12 and pyramid up each of the faces right in the middle, we end up with the d24. These divisions are also illustrated below.
It should be noted that the d60 and d120 end up pushing the limits of what might be considered a useful shape for a die. They are both quite spherical (the d120 is larger than a golf ball), and will roll for a long, long time before coming to rest on a number. Furthermore, the numbers are so closely packed together that it can be somewhat tricky, at a glance, to figure out what the actual rolled result is! It may be possible to make dice which have an even larger range of numbers, but they wouldn’t be terribly practical.
Let’s now return to dice with an odd number of sides. Dice makers are not, of course, limited to making dice that conform to some standard of geometric perfection! Inspired by the mirror symmetry dice described earlier, one can simply remove the edge across the mirror plane, effectively reducing the number of sides in half. If the faces are to remain flat, this means that the edges have to be rounded, resulting in something like the d3 pictured below.
If the image doesn’t make it perfectly clear, picture it as a filled tube with a triangular cross-section, and the edges are rounded. Provided the edges are rounded in the same way, it is still a fair die, because each of the 3 flat sides is the same.
This same idea has been used to design what are often referred to as “crystal dice,” which look like amethyst crystals or something similar. For instance, here’s a lovely set of crystal d4s, via Gamemaster Dice.
Here the edges aren’t rounded, but beveled to a peak, so that the die cannot rest stably on those flat edges. Some manufacturers have been particularly imaginative with the tube shape of these dice, making them look, for instance, like a rocket ship.
The Cylindrical d5
As we carry on to even more varieties of dice, we begin to enter truly strange territory. For instance, consider the d5 pictured at left.
We can view this as being similar to the d3 discussed above, except that the ends of the triangular cylinder have been made flat and turned into additional faces instead of being rounded. But now we have a problem: the faces of the sides of the cylinder are of a different size and shape than the faces of the ends of the cylinder: how can we possibly know if this die is fair or not?
Arguably one can do some calculations to prove it, or do brute force testing of dice of various lengths, but we will simply argue that there must be a size of die which is fair; presumably the dice-maker figured out what that size is! Imagine first a die which is made of a very, very long cylinder, for instance with the same length to thickness proportions as a pencil. Such a die will be almost certain to land on its length, on one of the three sides around it. If we now imagine a die which is made of a very, very short cylinder, like a coin, it is almost certain to land on one of its ends.
Somewhere between the long die which will land on its length and the short die which will land on its end, there must be a die of a length which will land 2/5ths of the time on an end, and 3/5ths of the time on a side. Because each of the ends are the same, and each of the sides are the same, this die must be fair! This idea is illustrated below.
In this case, we have again taken advantage of the symmetry of the triangular cylinder: because each of the ends is the same and each of the sides along the cylinder length is the same, there is only one free parameter to tune — the cylinder length — in order to get a fair die.
A 7-sided die designed in a similar manner is available, as well!
Here Be Dragons
Even after all of this, we still have a huge number of unusual dice to consider! They are shown below.
These dice have no particular symmetry at all — the different faces are different sizes and shapes. This raises two questions: how are they designed, and (of course) are they fair?
To answer the first question, we turn to the descriptions given at The Dice Shop at mathartfun.com:
This design is based on spacing points as equally as possible on a sphere and then cutting planar slices perpendicular to those directions.
In short: the plan is just to make the die as symmetric as possible, slicing faces at points that are roughly equally spaced on a sphere. It turns out that it is not even possible, except in the Platonic cases we’ve already considered, to space points equally on the sphere, and there are a number of different ways to define and calculate equal spacings; a bit of mathematical description is given here.
Are these dice fair? Odds are against it, but there’s one way to find out: roll it a lot of times and see what happens! On one lonely night in a hotel room during a recent work trip, I did just that with a d7. (I only tested the d7 because bigger dice would require many more rolls to get good statistics.)
The tabulated results are as follows:
- 1: 63
- 2: 57
- 3: 72
- 4: 113
- 5: 75
- 6: 81
- 7: 39
There were 500 total rolls; if the die was fair, I would expect about 71 rolls of each number. Clearly there is a bias towards rolling a 4 and a bias against rolling a 7! The other numbers are relatively balanced, though evidently not perfectly.
So the die is not fair. But we can ask: does it matter? As noted at the beginning of this post, such dice are usually employed in role-playing games, where there is usually a large amount of flexibility in the rules anyway. “Mildly biased” dice aren’t really an issue where the rules are often made up on the fly!
In this post, I’ve focused on the geometry of individual dice, but there’s even more interesting stuff that can be said about the mathematics of combinations of dice! There are the so-called Sicherman dice, which are differently numbered pairs of d6s that produce the same sums as two ordinary d6s! There are also non-transitive dice, in which each die of a set of 4 can always be beaten, on average, by another member of the set! I blogged about these a few years back, if you’re interested in learning more.
If you’ve read this far, congratulations! You’ve read 3500 words on the geometry of dice! Hopefully I’ve gotten across the point that the design of dice is not child’s play.
¹ I have a lot more dice, but this is my bag of “special” dice.