On Convex, Dodecahedral “Fair Dice”

Posted on 28 December 2015 by RobertLovesPi

One of my early introductions to polyhedra came through playing the game Advanced Dungeons and Dragons (AD&D), which uses a standard seven-die set which includes the five Platonic Solids, plus two “d10s” (either ten-faced dipyramids or trapezohedra) which are used as a pair to generate random numbers from 1 to 100.

Assuming they are made with uniform density, these polyhedral dice are all “fair dice” — meaning that, for example, the d12 at the top of the picture has an equal chance of rolling any of 12 results, every time it is rolled.

I had also encountered, even earlier, a polyhedron which I believed (correctly) would not work as a fair die, since there is no reason to assume that rolling a cuboctahedron would result in equal probabilities for each face, given that some of the faces are squares, while others are triangles. This shape was familiar to me long before I heard (or even read) the word “cuboctahedron,” though, because I learned about it while watching “By Any Other Name,” an episode of the original Star Trek television series. At no point in this episode is the word “cuboctahedron” used, even though the entire crew of the USS Enterprise (with four exceptions) spend most of the episode in cuboctahedral form, as Lt. Uhura appears, at her bridge station, in this screenshot:

When examining uniform-density polyhedra to look for “fair dice,” therefore, one of the first things I look for is isohedrality — all faces must have the same shape, unlike the case of the cuboctahedron, but like the cases of AD&D dice. The next thing I examine is symmetry, hoping to find that a particular polyhedron’s symmetry gives no probability-advantage to any face(s). With these two tests in mind, I decided to see how many different fair, dodecahedral d12s I could find, including polyhedra which could work, whether or not such shapes had actually been used as real dice, at any time, by anyone.

When looking for dodecahedral “fair dice,” anyone with a familiarity with AD&D, or polyhedra, is likely to identify, first, the one with this shape:

The word “dodecahedron” literally refers simply to any polyhedra with twelve faces, but, in practice, most references to dodecahedra concern this twelve-faced polyhedron — the one found immediately above, and at the top of the photograph of a standard set of AD&D dice. When it is necessary to distinguish it from other twelve-faced polyhedra, this can be called, instead, the “Platonic dodecahedron.” It is the only dodecahedron to be both regular and convex; it also possesses icosidodecahedral symmetry, also known as icosahedral symmetry.

Although the pentagons in the Platonic dodecahedron are regular, it is also possible to make a fair-die polyhedron with irregular pentagons. Such a shape appears, occasionally, in the mineral pyrite, or FeS₂, explaining the origin of the term “pyritohedral symmetry,” which is the symmetry-type of this particular dodecahedron: the pyritohedral dodecahedron.

There are actually an infinite number of slightly-different pyritohedral dodecahedra, as one can easily picture, by imagining a change in the length of the longest edges in the image directly above. However, for purposes of this survey, all possible pyritohedral dodecahedra are viewed as a single answer — the second, after the Platonic dodecahedron — to the question, “Which twelve-faced polyhedra would work as ‘fair dice?'”

The third answer to this question is provided by a completely different polyhedron, rather than a “warping” of the Platonic dodecahedron. In this third dodecahedron, the faces are rhombi, explaining why it is called the rhombic dodecahedron. Johannes Kepler studied this polyhedron extensively. Not just any rhombus will work, as a face, to make this polyhedron; it must be a specific type — one with diagonal-lengths in a ratio of one, to the square root of two.

This particular “fair die” occurs in nature, as crystals of the mineral garnet. There are also twelve-sided dice of this type now being sold, although finding them isn’t the easiest thing to do; one such retail outlet (in case you’d like to buy some rhombic dodecahedral dice) is The Dice Lab, which sells such dice, on Amazon, at this website.

The rhombic dodecahedron, unlike the other dodecahedra shown above or below, has cuboctahedral symmetry, also known as octahedral symmetry. It is also the dual of the cuboctahedron — the same polyhedron about which I first learned as a young child, watching Star Trek.

Just as the first solution to this twelve-faced “fair dice” quest (the Platonic dodecahedron) can be “stretched” to find a second solution (the pyritohedral dodecahedron), this third solution (the rhombic dodecahedron) can be altered slightly to give a fourth solution, which I call the strombic dodecahedron, although it has other names as well (one, for example, is the “deltoidal dodecahedron”). To make this shape, one keeps the overall pattern of the rhombic dodecahedron, but allows the rhombi to be stretched into non-equilateral kites, as shown here:

As far as I know, no one has actually made such dice — but that doesn’t matter, for the point is that such dice could be made, and would be fair, given uniform density. This is actually a family of possible solutions, as was the case with the pyritohedral dodecahedron, because different versions of the strombic dodecahedron can be created by varying the length-ratio of the long and short edges of the figure. Such variants would still have the same name and symmetry-type, however, and that symmetry-type is tetrahedral.

At least one other twelve-faced “fair die” can be made which also has tetrahedral symmetry: the Catalan solid known as the triakis tetrahedron, dual of the Archimedean truncated tetrahedron:

The triakis tetrahedron can be viewed as a Platonic tetrahedron, with each of its faces augmented by short, triangular pyramids which have lateral faces which are obtuse, isosceles triangles. The height of these short pyramids can be changed, while still leaving the overall polyhedron convex, over a range of heights; such altered versions, if not true duals of the Archimedean truncated tetrahedron, could simply be called non-Catalan tetrakis tetrahedra. The Catalan (and various convex non-Catalan) tetrakis tetrahedra are here collectively offered as the fifth type of twelve-faced polyhedra which can serve as fair dice.

The fourth and fifth solutions do have a problem, due to their tetrahedral symmetry: as a physical die, when rolled on a horizontal surface, the various strombic dodecahedra and triakis tetrahedra would land without a face pointing straight up, since they do not have parallel faces. This, however, merely means that, as fair dice, they wouldn’t be as convenient as the others; one might, for example, number their faces, and then pick up the die after it is rolled, to see which number ended up pointing straight down, rather than straight up. Other “workarounds” could also be devised. The need for such extra work, however, does not negate the fact that these polyhedra can be used as fair dice, for the problem was not set up with a convenience-requirement.

Further examination of a standard seven-piece AD&D dice set can lead to still more “fair d12s,” due to the presence of the d10s, also known, when used in pairs, as percentile dice. Most AD&D d10s have kites as faces (as seen in the metal dice set above), and are duals of pentagonal antiprisms, and so are themselves known as pentagonal trapezohedra (also known as “pentagonal deltohedra,” among other names). To get a similar fair die with twelve faces, rather than ten, one can simply start with a hexagonal antiprism, and then examine its dual: the hexagonal trapezohedron, which has six-fold dihedral symmetry:

By varying the long-to-short edge length ratio of the kite-faces in this polyhedron, the overall height of this polyhedron, as a function of its width, can be changed. This sixth solution is, therefore, another “infinite family” solution — as is the seventh solution, shown below, which can be easily made from the sixth solution (immediately above). To do so, mentally hold in place the bottom half of the hexagonal trapezohedra — but let the top half rotate for another 1/12 of a rotation before also “freezing” it. There is no need, now, for the zig-zagging “equator” in the polyhedron seen above — it can now be replaced with the coplanar edges of a hexagon, hidden inside the polyhedron, with the result that the twelve kites are replaced by a dozen isosceles triangles, turning the overall shape into a hexagonal dipyramid. These isosceles triangles must have vertex angles which measure less than 60° — in order to keep the figure from collapsing into something with no height, or with many edges which do not meet. Like the sixth solution before it, this seventh solution also has six-fold dihedral symmetry.

Next, here is a table which summarizes information about these seven possible dodecahedral “fair d12s.”

It is important to point out that this collection of seven solutions may not be complete, and I make no claim that it is. In fact, I strongly suspect it is not complete. It is simply the set of all solutions which have occurred to me — so far.

Some may wonder why I did not include the “barrel”-style d12s, which are also commercially available. This omission is no oversight. This particular style of d12 is not actually dodecahedral; it really has more than 12 faces, but is designed in such a way that it only has 12 faces which such dice can land on, and stay on — and that is not the same thing, at all. Also, I’m only looking for d12s here, which is why I did not include the “barrel”-style d4, even though it is a polyhedron which does have twelve faces.

Lastly, of the pictures in this post, the seven which feature rotating polyhedra were created using Stella 4d: Polyhedron Navigator, software which may be purchased, or tried for free, at this website.

A Chiral Solution to the Zome Cryptocube Puzzle

Posted on 5 April 2015 by RobertLovesPi

This is my second solution to the Zome Cryptocube puzzle. In this puzzle, you start with a black cube, build a white, symmetrical, aethetically-pleasing geometrical structure which incorporates it, and then, finally, remove the cube. In addition, I added a rule of my own, this time around: I wanted a solution which is chiral; that is, it exists in left- and right-handed forms.

It took a long time, but I finally found such a chiral solution, one with tetrahedral symmetry. Above, it appears with the original black cube; below, you can see what it looks like without the black cube’s edges.

My First Solution to the Zome Cryptocube Puzzle, with Special Guest Appearances by Jynx the Kitten

Posted on 11 January 2015 by RobertLovesPi

Last month, in a special Christmas promotion, the Zometool company (www.zometool.com) briefly sold a new kit (which will return later) — a fascinating game, or puzzle, called the “Cryptocube.” Zome usually comes in a variety of colors, with each color having mathematical significance, but the Cryptocube is produced in black and white, which actually (in my opinion) makes it a better puzzle. Here’s how the Crypocube challenge works: you use the black parts to make a simple cube, and then use the smaller white parts to invent a structure which incorporates the cube, is symmetrical, is attractive, and can survive having the twelve black cube-edges removed, leaving only the cube’s eight black vertices in place. I had a lot of fun making my first Cryptocube, and photographed it from several angles.

If this was built using standard Zome colors, the round white figure inside the cube, a rhombic triacontahedron, would be red, and the pieces outside the cube, as well as those joining the rhombic triacontahedron to the cube (from inside the cube), would be yellow.

It isn’t only humans who like Zome, by the way. Jynx the Kitten had to get in on this!

Jynx quickly became distracted from the Cryptocube by another puzzle, though: he wanted to figure out how to pull down the red sheet I had attached to the wall, as a photographic backdrop for the Cryptocube. Jynx takes his feline duties as an agent of entropy quite seriously.

As usually happens, Jynx won (in his never-ending struggle to interfere with whatever I’m doing, in this case by pulling the sheet down) and it took me quite a while to get the red sheet back up, in order to take kitten-free pictures of my Cryptocube solution, after removal of the black cube’s edges.

Here’s the view from another angle.

The Cryptocube will be back, available on the Zometool website, later in 2015. In the meantime, I have advice for anyone not yet familiar with Zome, but who wants to try the Cryptocube when it returns: go ahead and get some Zome now, at the link above, in the standard colors (red, blue, and yellow, plus green in advanced kits), and have fun building things with it over the next few months. The reason to do this, before attempting to solve the Crypocube, is simple: the colors help you learn how the Zome system works, which is important before trying to solve a Zome puzzle without these colors visible. After gaining some familiarity with the differing shapes of the red, blue, yellow, and green pieces, working with them in white becomes much easier.

On a related note, Zome was recommended by Time magazine, using the words “Zometool will make your kids smarter,” as one of the 14 best toys of 2013. I give Zome my own strong, personal recommendation as well, and, as a teacher who uses my own Zome collection in class, for instructional purposes, I can attest that Time‘s 2013 statement about Zome is absolutely correct. Zome is definitely a winner!

How to Get Fair Dice with Various Numbers of Possible Results, from Two to Thirty, Using Different Polyhedra

Posted on 13 September 2014 by RobertLovesPi

For a d2, number a cube’s faces with three ones and three twos.
For a d3, number a cube’s faces 1,1,2,2,3,3.
The standard d4 is a Platonic solid, the tetrahedron. Double-numbered (two ones, two twos, etc.) octahedra are sometimes used as d4s, also.
For a d5, an icosahedron can be renumbered with four each of the numbers one through 5. Double-numbering a pentagonal dipyramid or pentagonal trapezohedron also works.
The d6 is the familiar cube.
For a d7, one option is to roll an octahedron, but re-roll 8 each time it comes up.
For a d9, one option is to roll a d10, but reroll 10s.
For d10s, pentagonal dipyramids and pentagonal trapezohedra both work. There is also the option of double-numbering an icosahedron.
For a d11, one option is to roll a d12, but reroll 12s.
For a d12, the Platonic and rhombic dodecahedra both work.
For a d13, roll a d14, but reroll 14s.
For a d14, one option is to roll a d7 and a d2, then add 7 to the d7 result iff the d2 shows 2. Another is to roll a d15, but reroll 15s.
For a d15, simply double-number the thirty faces of a rhombic triacontahedron.
For a d16, roll a d2 and a d8 together, using the d8 result, but adding 8 to the d8 result iff the d2 result is 2.
For d17, roll a d18, but reroll 18s.
For d18, roll a d2 and a d9 together, using the d9 result, but adding 9 to the d9 result iff the d2 result is 2.
For d19, roll a d20, but reroll 19s.
For d20, icosahedra are used.
For d21, one option is to roll a d24, rerolling any result from 22-24. Another is roll a d7 and a d3 together, using the d7 result, but adding 7 to this d7 result iff the d3 result is 2, but adding 14 to the d7 result iff the d3 result is 3.
For d22, one option is to roll a d24, rerolling any 22s and 24s. Another is to roll a d2 and a d11 together, using the d11 result, but adding 11 to this d11 result iff the d2 result is 2.
For d23, use a d24, and re-roll 24s.
Options for the d24 include the triakis octahedron, the tetrakis cube, the deltoidal icositetrahedron, and the pentagonal icositetrahedron, all of which are Catalan solids (duals of the Archimedeans). Another d24 can be made by rolling a d2 and a d12 together, and using the d12 result, but adding 12 to this d12 result iff the d2 result is 2.
For a d25, roll two distinguishabale d5s, called d5a and d5b. The 1-25 random number is (d5a)+ (5)(d5b-1).
For a d26, roll a d13 and a d2, then add 13 to the d13 result if the d2 shows 2. Another is to roll a d15, but reroll 15s. The first option may require two different d2s, so they will have to be distinguishable, in that case, or one d2 must be rolled twice, each for a different purpose.
For a d27, roll a d9 and d3 together. The result is (d9) + (9)(d3-1).
For a d28, roll a d14 and d2 together, using the d14 result, but adding 14 to it iff the d2 result is 2. Another option: roll a d30, but reroll results of 28 or 29. The first option may require two different d2s, so they will have to be distinguishable, in that case, or one d2 must be rolled twice, each for a different purpose.
For a d29, roll a d30, and reroll 30s.
The most common d30 is a rhombic triacontahedron. After the Platonic solids and the d10, these d30s are the most commonly available example of polyhedral dice.

How To Make Tic-Tac-Toe Interesting

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How To Make Tic-Tac-Toe Interesting

Tic-tac-toe, played by the traditional rules, is so simple a game that few people with two-digit ages ever play it — just because it’s boring. It is so simple a game, in fact, that chickens can be trained to play it, through extensive operant conditioning. Such chickens play the game at casinos, on occasion — with the rules stating that if the game ends in a tie, or the chicken wins, the human player loses the money they paid to play the game. If the human wins, however, they are promised a large reward — $10,000, for example. Don’t ever fall for such a trick, though, for casinos only use chickens that are so thoroughly trained, by weeks or months of positive reinforcement, negative reinforcement, and punishment, that they will not ever lose. You’d be better off simply saving the same money until it’s cold, and then setting it on fire, just for the heat. At least that way you’d be warm for a little while, and that certainly beats the humiliation of being beaten, at any game, by a literal bird-brain.

With a small, simple alteration, though, tic-tac-toe can actually become a worthwhile, interesting game, even for adults. I didn’t invent this variation, but have forgotten where I read about it. I call it “mutant tic-tac-toe.”

In this variation, each player can choose to play either “x” or “o” on each play — and the first person to get three “x”s or three “o”s, in a row, wins the game. That’s it — but, if you try it, you’ll find it’s a much more challenging game. I am confident chickens will never be trained to play it successfully.

Consider the board pictured above, which happens to match a game I lost, to a high school student, earlier today. Red (the student) moved first, starting with the “o” in the center. I was playing with a blue marker, and chose to play an “x” in a corner spot. This was a mistake on my part, for the student’s next move — another “x,” opposite my “x,” effectively ended the game. I had to play next — passing is not allowed — and my playing an “x” or an “o,” in any of the six open spaces, would have led to an immediate victory by the student. If you study the board, you’ll see why this is the case.

Mutant tic-tac-toe is a great activity for semester exam week, at any school. Students who finish final exams earlier than their classmates can be taught the game quickly and quietly, and then they’ll entertain themselves with this game, rather than distracting students who are still working on their tests. What’s more, students have to really think to play this version of the game well, especially when they first learn it — and isn’t getting students to think what education is supposed to be all about, anyway?