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Followup notes about dice and polyhedra
I got a lot of commentary about these geometric articles, and started writing up some followup notes. But halfway through I got stuck in the middle of making certain illustrations, and then I got sick, and then I went to a conference in Vienna. So I decided I'd better publish what I have, and maybe I'll get to the other fascinating points later.

• Regarding a die whose sides appear with probabilities 1/21 ... 6/21
  • Several people wrote in to cast doubt on my assertion that the probability of an irregular die showing a certain face is proportional to the solid angle subtended by that face from the die's center of gravity. But nobody made the point more clearly than Robert Young, who pointed out that if I were right, a coin would have a 7% chance of landing on its edge. I hereby recant this claim.

  • John Berthels suggested that my analysis might be correct if the die was dropped into an inelastic medium like mud that would prevent it from bouncing.

  • Jack Vickeridge referred me to this web site, which has a fairly extensive discussion of seven-sided dice. The conclusion: if you want a fair die, you have no choice but to use something barrel-shaped.

  • Michael Lugo wrote a detailed followup in which he discusses this and related problems. He says "What makes Mark's problem difficult is the lack of symmetry; each face has to be different." Quite so.

• Regarding alternate labelings for standard dice
  • Aaron Crane says that these dice (with faces {1,2,2,3,3,4} and {1,3,4,5,6,8}) are sometimes known as "Sicherman dice", after the person who first brought them to the attention of Martin Gardner. Can anyone confirm that this was Col. G.L. Sicherman? I have no reason to believe that it was, except that it would be so very unsurprising if it were true.

  • Addendum 20070905: I now see that the Wikipedia article attributes the dice to "Colonel George Sicherman," which is sufficiently clear that I would feel embarrassed to write to the Colonel to ask if it is indeed he. I also discovered that the Colonel has a Perl program on his web site that will calculate "all pairs of n-sided dice that give the same sums as standard n-sided dice".

  • M. Crane also says that it is an interesting question which set of dice is better for backgammon. Both sets have advantages: the standard set rolls doubles 1/6 of the time, whereas the Sicherman dice only roll doubles 1/9 of the time. (In backgammon, doubles count double, so that whereas a player who rolls a–b can move the pieces a total of a+b points, a player who rolls a–a can move pieces a total of 4a points.) The standard dice permit movement of 296/36 points per roll, and the Sicherman dice only 274/36 points per roll.

    Ofsetting this disadvantage is the advantage that the Sicherman dice can roll an 8. In backgammon, one's own pieces may not land on a point occupied by more than one opposing piece. If your opponent occupies six conscutive points with two pieces each, they form an impassable barrier. Such a barrier is passable to a player using the Sicherman dice, because of the 8.

  • Doug Orleans points out that in some contexts one might prefer to use a Sicherman variant dice {2,3,3,4,4,5} and {0,2,3,4,5,7}, which retain the property that opposite faces sum to 7, and so that each die shows 3.5 pips on average. Such dice roll doubles as frequently as do standard dice.

  • The Wikipedia article on dice asserts that the {2, 3, 3, 4, 4, 5} die is used in some wargames to express the strength of "regular" troops, and the standard {1, 2, 3, 4, 5, 6} die to express the strength of "irregular" troops. This makes the outcome of battles involving regular forces more predictable than those involving irregular forces.

• Regarding deltahedra and the snub disphenoid
  • Several people proposed alternative constructions for the snub disphenoid.
    1. Brooks Moses suggested the following construction: Take a square antiprism, squash the top square into a rhombus, and insert a strut along the short diagonal of the rhombus. Then squash and strut the bottom square similarly.

       It seems, when you think about this, that there are two ways to do the squashing. Suppose you squash the bottom square horizontally in all cases. The top square is turned 45° relative to the bottom (because it's an antiprism) and so you can squash it along the -45° diagonal or along the +45° diagonal, obtaining a left- and a right-handed version of the final solid. But if you do this, you find that the two solids are the same, under a 90° rotation.

       This construction, incidentally, is equivalent to the one I described in the previous article: I said you should take two rhombuses and connect corresponding vertices. I had a paragraph that read:

       But this is where I started to get it wrong. The two wings have between them eight edges, and I had imagined that you could glue a rhombic antiprism in between them. . . .

       But no, I was right; you can do exactly this, and you get a snub disphenoid. What fooled me was that when you are looking at the snub disphenoid, it is very difficult to see where the belt of eight triangles from the antiprism got to. It winds around the polyhedron in a strange way. There is a much more obvious belt of triangles around the middle, which is not suitable for an antiprism, being shaped not like a straight line but more like the letter W, if the letter W were written on a cylinder and had its two ends identified. I was focusing on this belt, but the other one is there, if you know how to see it.

       The snub disphenoid has four vertices with valence 4 and four with valence 5. Of its 12 triangular faces, four have two valence-4 vertices and one valence-5 vertex, and eight have one valence-4 vertex and two valence-5 vertices. These latter eight form the belt of the antiprism.

    2. M. Moses also suggested taking a triaugmented triangular prism, which you will recall is a triangular prism with a square pyramid erected on each of its three square faces, removing one of the three pyramids, and then squashing the exposed square face into a rhombus shape, adding a new strut on the diagonal. This one gives me even less intuition about what is going on, and it seems even more strongly that it shou,ld matter whether you put in the extra strut from upper-left to lower-right, or from upper-right to lower-left. But it doesn't matter; you get the same thing either way.

    3. Jacob Fugal pointed out that you can make a snub disphenoid as follows: take a pentagonal dipyramid, and replace one of the equatorial *----*----* figures with a rhombus. This is simple, but unfortunately gives very little intuition for what the disphenoid is like. It is obvious from the construction that there must be pentagons on the front and back, left over from the dipyramid. But it is not at all clear that there are now two new upside-down pentagons on the left and right sides, or that the disphenoid has a vertical symmetry.

  • A few people asked me where John Batzel got they magnet toy that I was using to construct the models. It costs only $5! John gave me his set, and I bought three more, and I now have a beautiful set of convex deltahedra and a stellated dodecahedron on my desk. (Actually, it is not precisely a stellated dodecahedron, since the star faces are not quite planar, but it is very close. If anyone knows the name of this thing, which has 32 vertices, 90 edges, and 60 equilateral triangular faces, I would be pleased to hear about it.) Also I brought my daughter Katara into my office a few weekends ago to show her the stella octangula ("I wanna see the stella octangula, Daddy! Show me the stella octangula!") which she enjoyed; she then stomped on it, and then we built another one together.

  • [ Addendum 20070908: More about deltahedra. ]

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