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 faience 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,

with consecutively numbered sides, except for the icosahedron, which would have the digits from 0 through 9, repeated twice. Also, there could be two icosahedra in a set, different in color, one standing for tens, the other for units, and a roll of 00 would be counted as 100 so that this pair of dice would form a d100. More often, this was done by rolling one die twice.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. These dice were introduced in 1982 by The Armory, and were initially available primarily by mail order. 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).

The rhombic triacontahedron is the dual of the icosidodecahedron, as the diagram below illustrates:

The duals of many of the other Archimedian solids are also suitable for making fair dice with other numbers of sides, such as 60 sides or 24 sides. In some cases, though, the faces of an Archimedian dual (also known as a Catalan solid) instead of being absolutely identical, are identical except for parity reversal. A die made from such a solid could be rolled with a spin so as to favor half the faces over the other half.

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.

An alternate possibility for the d12 is the rhombic dodecahedron; this was used at least once for a set of astrology-based fortune-telling dice:

as can be seen from the diagram, it is closely related to the cube and the octahedron.
It can be thought of as a cube with pyramids on each of its faces, or an octahedron with pyramids
on each of *its* faces, high enough in both cases that the triangular sides of adjoining
pyramids unite to form a diamond shape.

A d24 could be made by reducing the height of these pyramids to about half of what is required to form the rhombic dodecahedron in either case, and as noted, this has been done for the cube, but not the octahedron.

The availability of unusual dice seems to be increasing of late. Alternate forms of the d24 are available.

Gamescience makes available a set of dice, called the Zocchi Pack, including three otherwise hard-to-find dice of the kinds mentioned above, the d14, d16, and d24, along with a d3 and a d5. The d3 is a rounded shape based on a prism; the d5 is a triangular prism which also uses its top and bottom as possible faces.

Koplow Games makes available a set of special dice called "Who Knew?" that includes a d3, d5, d7, d16, d24 and d30. The unusual shapes for the d3, d5, and d7 are avoided by instead creating a d3, d5 and d7 by repeating the numbers from 1 to 3, from 1 to 5, or from 1 to 7 respectively twice on a d6, a d10, and a d14. The d14 has the same category of trapezohedral bipyramidal shape as the d10.

From Genomic Games, a set is available that consists of a d3, d5 and d7 of the type noted above for Koplow's offering, and a d14, d16, d24, and d30. Except that some people may already have a d30, since it also includes the d14, it more closely matches what many people will be looking for.

Chessex, another major supplier of game dice, makes individual dice of these shapes, but does not offer a packaged set.

I've also seen a site on the web indicating that a set of ordinary RPG dice that adds a d24 and a d30 to the usual set is available for import from China.

On the other hand, I've heard reports that the d30 is becoming hard to find again.

The d3, d5, d7, d14, d16, and d24 have started to become available in response to these dice being used in the RPG Dungeon Crawl Classics, made by Goodman Games. As they note on their site, the game module #36, Talons of the Horned King, was the first one to make use of these additional dice. This module was first published in 2006.

The trapezohedral version of the d24 has been used as well, at least once, by Gamescience for their D-Total die; the principle behind its construction can be seen in this diagram:

This is a better shape for a d24 than the one most commonly seen, as it doesn't involve shallow triangular faces.

For that matter, I've also seen pictures of an alternate d4 made by repeating the numbers from 1 through 4 twice on a d8; as well, Chessex makes dice they call "Roman Dice", which are dodecahedrons (d12) with the numbers from 1 through 4 printed on them three times. If this means that the octahedron is so pointy that, although it rolls more easily than a tetrahedron, it still leaves something to be desired, I suppose it's only a matter of time before someone makes a bipyramidal d16 with the numbers 1 through 8 repeated twice as an alternative form of d8, although I think that a trapezohedral d24 with those numbers repeated three times would be a better idea.

It's also possible to obtain hollow transparent dice with an additional die inside. The dice are of the same shape; someday, someone is going to put a d30 inside a d24, or vice versa, to create a die that can simulate all the other common shapes, as well as one uncommon one, easily:

d24 | d4 d6 d8 d12 (d20) (2d12) | d30 | d6 d10 (d16) ----------------------------------------------------------------------- 1 | 1 1 1 1 0 0 | 1 | 1 1 0 2 | 2 2 2 2 +10 +6 | 2 | 2 2 +8 3 | 3 3 3 3 0 0 | 3 | 3 3 0 4 | 4 4 4 4 +10 +6 | 4 | 4 4 +8 5 | 1 5 5 5 0 0 | 5 | 5 5 0 6 | 2 6 6 6 +10 +6 | 6 | 6 6 +8 7 | 3 1 7 7 0 0 | 7 | 1 7 0 8 | 4 2 8 8 +10 +6 | 8 | 2 8 +8 9 | 1 3 1 9 0 0 | 9 | 3 9 0 10 | 2 4 2 10 +10 +6 | 10 | 4 0 (10) +8 11 | 3 5 3 11 0 0 | 11 | 5 1 0 12 | 4 6 4 12 +10 +6 | 12 | 6 2 +8 13 | 1 1 5 1 0 +6 | 13 | 1 3 0 14 | 2 2 6 2 +10 0 | 14 | 2 4 +8 15 | 3 3 7 3 0 +6 | 15 | 3 5 0 16 | 4 4 8 4 +10 0 | 16 | 4 6 +8 17 | 1 5 1 5 0 +6 | 17 | 5 7 0 18 | 2 6 2 6 +10 0 | 18 | 6 8 +8 19 | 3 1 3 7 0 +6 | 19 | 1 9 0 20 | 4 2 4 8 +10 0 | 20 | 2 0 (10) +8 21 | 1 3 5 9 0 +6 | 21 | 3 1 0 22 | 2 4 6 10 +10 0 | 22 | 4 2 +8 23 | 3 5 7 11 0 +6 | 23 | 5 3 0 24 | 4 6 8 12 +10 0 | 24 | 6 4 +8 25 | 1 5 0 26 | 2 6 +8 27 | 3 7 0 28 | 4 8 +8 29 | 5 9 0 30 | 6 0 (10) +8

A d20 is achieved by adding the (d20) number to the d10 number from the d30 (counting 0 as 10), and a second d12 is achieved by adding the (2d12) number to the d6 number from the d30. A d16 can be achieved by adding the (d16) number to the d8 as well. Since both dice can simulate a d6, 2d6 is also available.

Other arrangements for the (2d12) column are possible, as long as for each number from 1 to 12 in the d12 column, one occurrence corresponds to a 0, and the other occurrence corresponds to a +6. The most obvious arrangement would have been 12 zeroes followed by 12 instances of +6.

On further reflection, I've realized that dice with fewer sides could be used successfully. A d20 inside a d6, for example, would also be able to simulate the other common die sizes, and not only that, for the most part the rolls could be derived fairly easily in one's head, without recourse to tables:

d4 d6 is ODD: 1 d6 is EVEN: 2 d20 is EVEN: +2 d8 d6 is EVEN: +4 d20 1 5 9 13 17 | 1 2 6 10 14 18 | 2 3 7 11 15 19 | 3 4 8 12 16 20 | 4 d10 d6 - ignore d20 - use the last digit d12 d6 - use the number rolled d20 is EVEN: +6

Only the d8 requires a slightly difficult task, determining the residue class modulo 4 (or, less technically, the remainder after division by 4) of the number rolled on the d20. Still, many people are well-practiced in mental division by 4 in order to determine leap years, so it's not too bad.

One way to simplify this would be to underline all the numbers that go to 1 and 2 in the table shown above for interpreting the d20 in calculating a d8 result: not just 9, but also 1, 5, 13, and 17; not just 6, but also 2, 10, 14, and 18.

Then the rule becomes:

ODD and underlined: 1 EVEN and underlined: 2 ODD and plain: 3 EVEN and plain: 4

Of course, since an icosahedral d20 has triangular faces, underlining the 6 and 9 to avoid ambiguity is not strictly necessary, as the convention that an apex of the triangular face is the 'top', and a side is the 'bottom', but this does make use of a very simple special marking that already has a reason to be present, and falls short of the complexity found on the D-Total.

Still further reflection has brought me to realize that with a bit more imagination and flexibility, there is a way to easily use a d20 as a d4 without the need for special markings.

In addition to distinguishing between even (pair) and odd (impair) at a glance, one can also distinguish between high (passé) and low (manqué) at a glance, where the numbers from 1 to 9, and also 20, are taken as "low", and the numbers from 10 to 19 are taken as "high".

Odd and Low: 1 Even and Low: 2 Odd and High: 3 Even and High: 4

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.

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