Re: Seven-sided die (was Re: This just in)

From: Mark E. Shoulson (
Date: Wed Jan 13 2010 - 12:52:25 CST

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    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.


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