Up until now, I have talked a lot about probabilities and how dice are used in games, but I have not really addressed the issue about what types of dice you can have and what these look like.

Which dice are fair?

So far, we have more or less assumed that dice of any size exist, for example d7 or d13, though these are not in the common mix of gaming dice. Some companies sell dice in non-standard shapes and sizes, but it is not immediately clear if they are all fair, i.e., if all sides are equally likely to come up.

When can we be sure a die is fair? It is not enough that every face has equal area. You can construct a polyhedron where every face has the same area, but where the polyhedron can only rest on some of these.

It is not even enough that the faces all have the same shape and size, as you can make a similar construction even then. So the only way to be sure that all faces are equally likely is to require symmetry: No matter what face the die rests on, the entire polyhedron should look the same to an observer. In other words, if there is no labeling on the faces, you should not be able to determine if the die is made to lie on another face or if it is just moved a bit and rotated without changing which face is down.

The Platonic solids (the traditional d4, d6, d8, d12 and d20) have this property, but so do the Catalan (or dual Archimedean) solids. Where the Platonic solids are defined to have all faces and all vertices are identical and symmetrically arranged, on Catalan solids only the faces are identical and symmetrically arranged, but that is enough to make them fair dice.

There 13 different Catalan solids with 12, 20, 24, 30, 48, 60 and 120 faces:

More detail about Catalan solids can be found at Mathworld.

The 30-sided Catalan solid called ``the Rhombic Triacontahedron'' (shown in the middle of the bottom row above) is found as a d30 in most game stores, and I have seen a d24 based on the Tetrakis Hexahedron (number two from the left in the top row). I would have preferred the Deltoidal Icositetrahedron or Pentagonal Icositetrahedron (the two rightmost figures in the bottom row), as the faces on these have lower aspect ratio, so larger symbols can fit inside them. They also look more cool. :-)

The astute reader will have noticed that the common d10 is not among the figures shown above, and there are indeed more fair dice than the Platonic and Catalan solids. The d10 is constructed by joining two ``pyramids'' constructed from kite-shaped sides in such a way that the convex vertices of one pyramid fits the concave vertices of the other. This construction can be generalized by joining two ``pyramids'' of N > 2 kite-shaped sides in a similar way to make a polyhedron with 2N sides. If N is odd, there will be a face facing up when the polyhedron rests on a flat surface, so it can be used to make dice that have twice an odd number of sides (i.e., d6, d10, d14 etc.). Note that for N = 3, the construction yields a normal cubical d6. You can also join two ``normal'' pyramids each made of N > 2 triangles to get a polyhedron with 2N sides. This will have an upwards-facing face if N is even. This construction includes the traditional octahedral d8 but can be used for any dice with a number (greater than 4) that divides evenly by 4, i.e., d8, d12, d16 etc.

If we require all faces on dice to be flat (i.e., no curved sides), the above are the only ways to construct polyhedra where all faces are equally likely by symmetry (see my brother's dice page for a detailed analysis of why these are the only ways). If we allow curved faces, we can also make N-sided prisms that taper towards the ends (or have rounded ends) as well as a lens-shaped d2. If N is odd, we can make ``anti-prisms''with triangular sides (and rounded ends), so one face will face up when the die rests. I have seen dice like these called ``Crystal dice'' or ``Barrel dice''. Note that both prismatic and anti-prismatic dice have (or should have) tapered or rounded ends on which they can not land.

You can find other kinds of dice in some game stores, the most common being:

• A d5 made as a triangular prism with numbers on the triangular ends as well as on the rectangular sides.
• A d7 made as a pentagonal prism with numbers on the pentagonal ends as well as on the rectangular sides.
• A d100 that looks like a golf ball (the ``Zocchihedron'').

These lack the symmetry between faces, so they are not obviously fair. But could they be?

If we look at the prismatic d5, it is easy to see that the longer the rectangular sides are, the more likely it is that the die will land on one of these. Furthermore, we can make this likelihood arbitrarily small or big by making the prism sufficiently short or long. So it would seem reasonable to assume that there exists a length where the probability is the same for landing on each end as for landing on each side.

However, what this length should be depends on how the die is rolled, so a die that is fair with one rolling method is unfair if you roll it differently. Hence, I would not accept such a die as fair no matter how well-argued the calculation of the ``fair'' length is.

The Zocchihedron is somewhat different. It is sufficiently close to being a sphere that the rolling method probably doesn't matter, but I still don't believe it to be fair, as the spacing between the circular ``faces'' is not constant. This view is supported by a test made by Jason Mills for White Dwarf magazine, which concluded that results under 8 or over 93 are considerably less likely than other results (see Wikipedia's Zocchihedron page). Also, I see little point in a d100, as the markings are too small to read easily and the ball-like shape takes forever to stop rolling. Using two d10s, one for the tens and one for the units, achieves the same purpose and is quite easy to read, especially if one of of the dice is labeled 00, 10, 20, 30, .... Most dice sets these days include both a d10 labeled this way and a d10 labeled from 0 to 9.

A more in-depth coverage of shapes for fair dice (with lots of pictures) can be found at The International Bone-Rollers' Guild.

Wish list

Not all of the Catalan solids mentioned above can be bought as dice. Some of these, i.e., the 12-sided and 20-sided Catalan solids, are redundant, as the Platonic d12 and d20 can be used instead. They could be used as novelty dice, similar to the barrel dice mentioned above, though. I quite like the idea of a rhombic dodecahedron as a d12, as I find the shape appealing. A d24 is quite useful (such as for randomly rolling the time of day) but, as mentioned, I would prefer a different shape than the one you can currently get (partly for the reason stated above, but the other shapes also look more cool). The solids with more than 30 sides are probably not as useful, but they would look impressive.

But what is really missing from the standard RPG dice selection are dice that continue the sequence started by d4, d6, d8, d10 and d12. There is a long gap until d20, which could be filled with a d14, a d16 and a d18. I have seen a d14 (made in the same way as a d10) and a d16 (made as two base-to-base 8-sided pyramids), but I have never seen a d18 for sale. This could be made like the d10 and d14.

Labeling dice

The most obvious way to label the sides of an N-sided die is from 1 to N, but there are examples that differ from this norm:

• A d10 is often labeled 0,...,9 or 00,...,90 instead of 1,...,10, as this makes it easier to use two d10 as a d100.
• Before the modern d10 was introduced, a d10 was made as an icosahedral d20 labeled a with 0,...,9 twice.
• Fudge-dice are cubical d6s labeled with -1, 0 and 1 twice each.
• Similarly, you can get d3s that are d6s labeled with 1, 2 and 3 twice each. Other odd-numbered dice can be made in this way, i.e., by letting each number occur twice on a die with an even number of sides.
• The doubling-die used in Backgammon is a cubical d6 showing the numbers 2, 4, 8, 16, 32 and 64. When playing Backgammon, the doubling die is not used as a randomizer, but it could be used as such in an RPG if you want an exponential progression.
• In some board games, dice are labeled with non-numerical symbols. For example, the ``Lord of the Rings'' board game by Reiner Knizia uses a d6 that is labeled on one side with The Lidless Eye of Sauron and other sides showing dots, outlines of cards or nothing at all. The Danish Ludo (similar to Parcheesi) has replaced two of the numbers (2 and 5) on a normal d6 with a star and a globe that have special meaning during the game, and you can get Poker dice that use card symbols instead of numbers.

All of the above have the same number of occurrences of all the numbers used, but you can also make dice that have different numbers appearing an unequal number of times. For example, you can have a d6 with three occurrences of 1, two occurrences of 2 and one occurrence of 3 for a d3 that is skewed towards low numbers.

``Old-fashioned'' six-sided dice don't use number-symbols, but label each side with a number of ``pips'' from 1 to 6 in standardized patterns. These patterns and the rule that opposing sides sum to 7 have been used since ancient Greece, and it is only after the introduction of non-cubic dice for role-playing games that numbers are commonly used for labeling dice. A few non-cubical dice have used pips instead of numbers, but these are not common.

Placement of symbols

The placement of symbols on dice might seem unimportant: Since all sides are of equal probability, just place the numbers in any order.

There are, however, a few special cases to consider, as well as traditions that, while not important, should be kept in mind for the sake of aesthetics.

The first special case is the d4. Unlike other commonly used dice, this does not have a face opposite the face on which it rests, so you can't read the result off a top face. Early d4s, as supplied with the original Dungeons & Dragons game, used the rule that the result is determined by the face that rests on the table, but since you can't see this, the numbers were put on the near edges of all the neighbouring faces, so you would read the value of the hidden side at the bottom edge of any visible face. Most current d4s read the value from the vertex that is on top, putting the symbols near the corners of every face (again in three copies each), so you will read the value from the top corner of any showing face. I prefer the latter.

I have seen d5s and d7s that are pentagonal and heptagonal prisms with rounded ends. Like the d4, these do not have a face on top when they rest. The ones I have seen rule that you read the value on the one of the two topmost faces that are nearest to you, which in some cases may require judgement. An alternative is to place the numbers so the span the edges, but that isn't very aesthetic (though I have seen it done).

For the d7, there is another option where you use pips: Add the number of pips on the two topmost faces. If the faces have the following numbers of pips (in sequence): 0, 1, 2, 2, 3, 4, 2, you get 0+1 = 1, 1+2 = 3, 2+2 = 4, 2+3 = 5, 3+4 = 7, 4+2 = 6 and 2+0 = 2, i.e., all the results between 1 and 7 once each. I have not seen this arrangement used on a d7, but if anyone want to use the idea, please go ahead. For the similar d5, there is no arrangement of pips that gives the numbers 1 to 5 by adding the two topmost faces. If we can do with the numbers from 0 to 4 instead of 1 to 5, the arrangement 0, 0, 1, 2, 2 will work.

Using a similar system for a tetrahedral d4 will not work for several reasons: You need to look at a d4 almost directly from above to see three sides, and you can't arrange pips so you get the range 1 to 4 by adding the pips on the three visible sides. The best you can get is the range 3 to 6, i.e., corresponding to d4+2.

As mentioned above, the numbers/pips on a d6 are traditionally placed so opposing faces add up to 7. There are only two ways of doing this, which are mirror images of each other. Most manufacturers of polyhedral dice follow the generalized version of that rule: The sum of opposing faces is constant. If you use the rule for a dN labeled 1,...,N, the sum of opposite edges should be N+1, and for a dN labeled 0,...,N-1 (e.g., a d10 labeled 0,...,9), opposing sides should add to N-1. For dice larger than d6, there are several non-symmetric ways of obeying this constant-sum rule. All the d10s that I have agree on having 0, 8, 2, 6, 4 in this order clockwise around one vertex and the corresponding odd numbers around the opposite vertex, but for the other polyhedral dice there does not seem to be any consensus about which of the possible constant-sum arrangements to use.

I have seen a few dice that don't follow the constant-sum rule, such as a d20 that has 12 opposite to 2. While these are equally good randomizers as dice that follow the rule, I find that they grate on my sense of aesthetics.

The end?

This is the last planned installment of ``Roll the Bones''. I may be inspired to write more installments later, but for now, this is it.

I will look at the discussion forum below from time to time, so if you have suggestions or questions, please write.

In the meantime, I hope you have had as much fun reading the column as I have had writing it.