From: Mark E. Shoulson (mark@kli.org)
Date: Wed Jan 13 2010 - 12:52:25 CST
It is well-known that the only "geometrically obviously fair" dice
possible are (a) the five Platonic solids, (b) the 24 Catalan solids
(the duals of the Archimedean solids), (c) bipyramids and trapezohedra
(pyramids of the same base stuck base-to-base (for 4n sides) or shifted
half a face over with quadrilateral sides, like a 10-sided die, for 4n+2
sides), or (d) basically n curved "faces" like slices of a sphere or
something meeting at two points, so there's essentially a rod with n-gon
cross-section terminated by vertices. In practice, this can be
approximated by an n-gon prism that's so long it is unlikely to land on
its ends.
I call these "geometrically obviously fair" because they are plainly
fair by their geometric construction: all the sides are exactly the
same. There are 7-sided and 5-sided dice out there which are *claimed*
to be fair, and probably can be fair within tolerance, but aren't as
plainly so. Basically if you take a triangular prism that's really
long, it will almost always land on one of its sides. If you take one
that's really short and flat, it will almost always land on one of its
ends. By intermediate-value theorem, there ought to be a length such
that the odds of landing on an end is exactly the same as the odds of
landing on one of the sides. And this is what the dice-manufacturers
have claimed to have made. You could probably do some experiments to
judge just how fair they are; it might depend on how they are thrown, etc...
(There are 100-sided dice out there too, which are essentially spheres
with 100 flat facets sliced off. It isn't going to land on the curved
parts in between, but one could possibly show that uneven probability on
the curves could lead to uneven distribution on the faces. But with so
many faces it would take a lot of testing to get an answer either way.
They also have little "things" inside them, like sand or little balls or
something, to help steal energy away from the die as it rolls, so it
will come to a stop relatively quickly and not go rolling off the table
all the time.)
Believe me. I own examples of all these dice, having gone several times
through big obsessions with weird and different kinds of dice. I've
read up a lot about all these.
~mark
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