From: Marcin 'Qrczak' Kowalczyk (qrczak@knm.org.pl)
Date: Sat Jan 09 2010 - 16:12:36 CST
In practice it is quite possible to have a symmetric dice with 7 sides. Make
a pyramid with 7 triangular sides, letting its base be something that the
dice won't stay on, like half of sphere.
On Jan 9, 2010 10:38 PM, "William J Poser" <wjposer@ldc.upenn.edu> wrote:
A seven-sided die is impossible if it is required to be fully symmetric
like a six-sided die. There is no seven-sided regular polyhedron.
The five convex regular polyhedra (aka the Platonic solids) have
4, 6, 8, 12, and 20 sides respectively. See the MathWorld article:
http://mathworld.wolfram.com/RegularPolyhedron.html
The seven-sided dice that a Google search turns up are not fully symmetric.
They have five square faces orthogonal to a pair of pentagonal faces.
I don't know whether asymmetry necessarily means that the dice are
not fair. I can imagine that one could make them fair by varying the
density of the material as a function of location within the die
in a suitable manner, but I haven't thought/calculated enough to see
if that would really work, and I don't know enough about the materials
and manufacturing processes to know how easily such a thing could be
implemented even if mathematically possible.
Bill
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