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 sevensided
dice. The conclusion: if you want a fair die, you have no choice
but to use something barrelshaped.
 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 the the
Colonel has a
Perl program on his web site that will calculate "all pairs of
nsided dice that give the same sums as standard
nsided 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.
 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 righthanded 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 valence4
vertices and one valence5 vertex, and eight have one valence4
vertex and two valence5 vertices. These latter eight form the belt
of the antiprism.
 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 upperleft to lowerright, or from
upperright to lowerleft. But it doesn't matter; you get the
same thing either way.
 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 upsidedown 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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