[extropy-chat] alt dot fair dice

[Spike]

We usually think of a cube when someone mentions
a gaming die, but of course any of the five platonic
solids can make a fair die.  By thought experiment,
we can verify that each face is the same shape and
the CG is the same distance from the table with
any face downward.

Can other shapes be made such that there is
equal probability of any face downward?  I can
think of one: a five sided pyramid shaped
solid (four triangular faces and one square
face).  If the pyramid is tall and skinny, it
is less likely to land on the square face.  If
it is short and flat, the square face is more
likely to end downward.  So (I think) the
intermediate value theorem demands that there
is an aspect ratio somewhere between short and
tall that would make the square face equally 
likely to land downward, even if the surface
area of the square face is different from
the triangular.  Also the CG height is different
with the square face down than with the triangular
faces down.  (Is it?)

Actually that suggests a class of non-platonic
fair dice.  A six sided "pyramid" with five
triangles and a pentagonal base also would
hafta have an intermediate base to height ratio
that would fair-ize it.  Right?  Wouldn't that
argument apply to arbitrarily many triangular
sides?  

One could even make a three-sided fair die, if 
one did not demand the "sides" be flat planes: two 
curved kinda triangular surfaces with a base that 
looks sorta like an ellipse but with pointy ends.

A two sided fair die is a coin, but that suggests
another three sided die: a cylinder, like a
really fat coin, equally likely to land on edge
as on either side.  For that matter one could 
grind arbitrarily many flat sides on a cylinder,
so that the cylinder gets longer and thinner
as the number of flat sides gets larger.

Other than machining a bunch of these oddball
shapes, is there any way to mathematically
prove that they would have equal probabilities 
of landing on any face?

spike