On a related topic, given how the numbers are arranged on ordinary dice, and to alternate high and low numbers, the arragements of numbers shown here seem to me to be appropriate ones for polyhedral dice; and the existence of an arrangement not devised by a manufacturer seeking to keep its competitors from copying its own arrangement might even lead to all polyhedral dice being made the same way, the way cubical dice are (although it could well be, particularly for the d8 and d12, that the arrangements shown here are already taken):
Of course, the question of why one wouldn't want to get the attractive special polyhedral dice, now that they are inexpensive and easily available, comes up too, but to have a way one could manage in a pinch is always useful.
The numbers are all shown upright, for ease of drawing, so the dots in the triangles, pentagons, or rhombi are present to show how the numbers might actually be oriented, indicating the corner towards which the top of the number would be directed.
It wasn't that long ago that dice were six-sided cubes, and that was that.
In ancient Rome, fourteen-sided dice in the shape of a cuboctahedron were sometimes used. The New York Metropolitan Museum of Art once exhibited two ancient Greek icosahedral dice, one in faience and one in steatite engraved with the letters of the Greek alphabet. Recently, a glass blue-green icosahedral die made the news when sold at auction. It had appeared to me that this die was engraved with unusual symbols, perhaps astrological in origin. However, the Louvre museum also has an icosahedral die with similar symbols on its faces. from Egypt during the Ptolemaic period, and images of a reproduction of that die are on the Dice Collector web page: this die clearly has Greek letters on its faces, implying that what seemed to be astrological symbols were stylized or script forms of Greek letters. A rhombic dodecahedron with letters on its faces is also in the Louvre.
The arrangement of letters on the one in the Louvre in particular, and on these dice in general (sometimes with rotations of the three segments relative to each other) is shown in the top diagram in the image at the right.
The Greek letters from alpha to upsilon are present on these dice.
The top cap of five triangles shows the letters alpha, beta, gamma, delta, and epsilon; the one unusual thing is that alpha is written in a squared form, reminiscent of runes, and in the script or lower-case form, whereas the others are in the form we are used to for Greek capital letters.
The middle band shows the letters from zeta to omicron. They are shown starting with theta, and wrapping around, so the order is theta, iota, kappa, lambda, mu, nu, xi, omicron, zeta, and eta. Two letters are in an unexpected form. Theta is written as a circle with a dot in the middle, like the astrological symbol for the Sun, rather than with a line across the middle. The letter zeta is written in a script or cursive style, with loops at the two corners in the letter, so that it cannot be confused with nu.
The bottom cap of five triangles shows the letters from pi to upsilon. They are shown starting with sigma, so the order is sigma, tau, upsilon, pi, and rho. Sigma is written like the Roman capital letter C, which avoids confusion with mu; Upsilon is written like a capital Y, which is common even in present-day Greek-language typestyles.
If one assumes that the same pattern is employed in some of the other dice of this type I have seen pictured on the Web, some parts of their arrangements can be inferred beyond the portions actually visible. Thus, the one in faince seems to have the same arrangement of letters as on the one in the Louvre, but, at least on the visible faces, seems to have the orientations of the letters arranged more systematically. This was an image I saw on Flickr; the text seemed to imply that the image was of an item on temporary loan, and another search result indicated that a faience die of this type belongs to the British Museum. Of course, more than one may have survived, so I can't infer with certainty that this was their specimen on loan.
The one in steatite at the same exhibition seems to have had the letters in a slightly different position, with the central band rotated relative to the cap with the letters from alpha to epsilon. None of the letters in stylized form were visible in the photograph, so I can't be sure what forms were used for them on this specimen.
Someone else placed a photograph on Flickr of a display case containing several icosahedral dice in a museum in Cairo that he had seen. He had done so for a particularly poignant reason, having seen these ancient d20-like dice on the same day that E. Gary Gygax passed away. One die was clearly enough visible in the photograph so that several of its letters could be made out; it, too, could follow the same pattern as the others, if one assumes that one of the letters is a zeta not drawn in a stylized script form rather than an eta.
The glass one sold at auction by Christies recently for over $17,000 uses the same stylized forms as the one in the Louvre for zeta and theta, but a different form for alpha, but the letters must follow a different order; this isn't actually particularly surprising.
I suspect these dice were used for divination rather than for a game; at this site is an ancient Greek set of divinatory meanings for the 24 letters of their alphabet.
Once, I encountered a 26-sided die in the shape of a small rhombicuboctahedron for playing a baseball game on a board in a thrift shop, but failed to snap it up, and I do have a set of octahedral poker dice, so there may well have been occasional exceptions even during the 1960s.
Sometimes, inexpensive dice would be made from wood, but usually hard plastic would be used. Casino dice had sharp edges, while some Chinese-made dice would have corners so rounded as to be spheres with six shallow domes removed to leave six flat circular faces.
Sometimes, the dice would have other markings on them than the usual sequence of from one to six spots. There were poker dice, with card symbols on them, both in sets of five identical dice, and in sets where two of the five dice had one space showing a Joker, and the other 28 available faces showed the 8, 9, 10, Jack, Queen, King, and Ace of each of the four suits, so as to allow the same hands, including flushes, to be made as in Poker played with playing cards, instead of being equivalent to playing poker dice with five ordinary dice. There were dice for Crown and Anchor, and even dice for Hoo Hey How with pictures of a fish, a prawn, a crab, a rooster, a gourd, and a coin for a similar game. (The Vietnamese version, Bau Cua Ca Cop, replaces the coin with a stag, and there are a number of other variations.) There were dice with letters on them for word games.
In 1966, the book Random Number Generators by Birger Jansson was published. This book was chiefly concerned with algorithms for generating pseudo-random numbers deterministically on computers for simulation purposes. The cover showed a set of three icosahedral dice of different colors, each one with the digits from 0 through 9 on them, repeated twice. These dice were made by the Japanese bureau of standards to facilitate random sampling, and were expensive due to the limited production.
When I came across this book in the early 1970s, I felt the hope that such dice would someday become conveniently available.
Of course, this did happen. But it did not happen in isolation, because some educational toy company decided to make a probability kit available. No, it happened as an incidental result of an Earth-shaking social change in how people amused themselves in their spare time!
It is, of course, to the game Dungeons and Dragons that I refer, which first appeared in 1974.
Because of limited demand and limited capital to set up production facilities, at first the available polyhedral dice were somewhat crudely made. A set of dice would consist of one each of the five Platonic solids,
Later, the quality of reasonably inexpensive polyhedral dice would improve, but ones with pre-painted numerals would still be significantly more expensive. The dice would have sharp edges, and would be made from ordinary plastics, such as polystyrene. During this period, icosahedral dice with faces numbered from 1 through 20 became available, and then non-icosahedral dice with only ten faces supplanted the earlier form of icosahedral dice.
Except for the tetrahedron, the Platonic solids, when situated with one face down on a surface would have another face at the top. Tetrahedral dice, therfore, had three numbers on each face, either to indicate which of the four corners was at the top, or which of the four faces was at the bottom. One of the ways to make a perfectly fair ten-sided die would be to have two five-sided pyramids abutting each other. When such a die was resting on one face, however, an edge would be at the top.
While this could be dealt with by splitting the numbers across the edges, as has been done for five of the seven faces of (not perfectly fair) seven-sided dice in the shape of pentagonal prisms, this was unnecessary in this case. Instead, one of the two pyramids could be rotated 36 degrees, and the faces suitably extended into kite-like shapes to join properly. This is the shape currently used for the d10, and a set of dice now normally contains one icosahedral d20, and two d10s of this shape, one of which has the faces numbered 00, 10, 20... 90, so that there is no ambiguity when rolling them. This shape is called a trapezohedron, the kite shapes being trapeziums.
Another innovation from that period was the d30, based on the rhombic triacontahedron. Since 30 equals 2 times 3 times 5, and the number of faces on the five Platonic solids are 4 (2*2), 6 (2*3), 8 (2*2*2), 12 (2*2*3), and 20 (2*2*5), the roll of any of them could be replaced by from one to three rolls of a d30 (4: 2 rolls, 6: 1 roll, 8: 3 rolls, 12: 2 rolls, 20: 2 rolls).
Today, the mass manufacture of polyhedral dice has made them economical, with pre-painted numerals standard. The edges are more rounded, and harder plastics, similar to those used for ordinary dice, are employed in their manufacture.
At least one firm makes available a d24, made from four shallow pyramids on each face of a cube (one could put the pyramids on the faces of a tetrahedron, or rotate those on the face of the cube by 45 degrees each as alternatives), a d14, a trapezohedron based on seven-sided pyramids instead of five-sided ones, and a d16 made from two eight-sided pyramids joined without a twist. In general, though, after being available easily for a while, even the d30 is now hard to find even in specialized stores.
While it would be possible to make a conversion table for simulating the rolls of other combinations of dice through the use of three six-sided dice, it does have to be admitted that it is easier to use three ten-sided dice of different colors, used as a d1000, for this purpose.
The chart below shows the proportionate probabilities for the totals produced by various dice combinations often used for weapons damage, the effects of healing spells, and the like:
1d4 2d4 3d4 4d4 5d4
1 1 .250
2 1 .500 1 .063
3 1 .750 2 .187 1 .016
4 1 1.0 3 .375 3 .063 1 .004
5 4 .625 6 .156 4 .020 1 .001
6 3 .813 10 .313 10 .059 5 .006
7 2 .937 12 .500 20 .137 15 .021
8 1 1.0 12 .687 31 .258 35 .055
9 10 .844 40 .414 65 .118
10 6 .937 44 .586 101 .217
11 3 .984 40 .742 135 .349
12 1 1.0 31 .863 155 .500
13 20 .941 155 .651
14 10 .980 135 .783
15 4 .996 101 .882
16 1 1.0 65 .945
17 35 .979
18 15 .994
19 5 .999
20 1 1.0
1d6 2d6 3d6 4d6 1d8 2d8 3d8
1 1 .167 1 .125
2 1 .333 1 .028 1 .250 1 .016
3 1 .500 2 .083 1 .005 1 .375 2 .047 1 .002
4 1 .667 3 .167 3 .019 1 .001 1 .500 3 .094 3 .008
5 1 .833 4 .278 6 .046 4 .004 1 .625 4 .156 6 .020
6 1 1.0 5 .417 10 .093 10 .012 1 .750 5 .234 10 .039
7 6 .583 15 .162 20 .027 1 .875 6 .328 15 .068
8 5 .722 21 .259 35 .054 1 1.0 7 .437 21 .109
9 4 .833 25 .375 56 .097 8 .563 28 .164
10 3 .917 27 .500 80 .159 7 .672 36 .234
11 2 .972 27 .625 104 .239 6 .766 42 .316
12 1 1.0 25 .741 125 .336 5 .844 46 .406
13 21 .838 140 .444 4 .906 48 .500
14 15 .907 146 .556 3 .953 48 .594
15 10 .954 140 .664 2 .984 46 .684
16 6 .981 125 .761 1 1.0 42 .765
17 3 .995 104 .841 36 .836
18 1 1.0 80 .903 28 .891
19 56 .946 21 .932
20 35 .973 15 .961
21 20 .988 10 .980
22 10 .996 6 .992
23 4 .999 3 .998
24 1 1.0 1 1.0
To avoid using excessive space, only the most common combinations are listed here, but this file contains a more extensive table of similar information, for dice from two to twenty sides.