Dice & Isohedron

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I enjoyed playing Ludo in my childhood. I believed that the there is only 6 sided dice like these:

Recently, while studying probability, I came to know that we can make dice corresponding to all platonic solids like:

So, I became curious to explore the mathematics involved behind choosing the number of sides of a dice. As usual, Wikipedia was my starting point.  There I discovered a 1989 paper by two Stanford University professors, Persi Diaconis & Joseph B. Keller, titled, “Fair Dice“. Here are few lines which I would like to quote from the paper:

In the beginning they say:

…We shall say that a convex polyhedron is fair by symmetry if and only if it is symmetric with respect to all its faces. This means that any face can be transformed into any other face by a rotation, a reflection, or a combined rotation and reflection, which takes the polyhedron into itself. The collection of all these transformations of a given polyhedron is called its symmetry group….

But towards the end of paper they say:

…There are other fair polyhedra which are not symmetric. To show this we consider, for example, the dual of the n-prism, which is a dipyramid with 2n identical triangular faces. We cut off its two tips with two planes parallel to the base and equidistant from it….  Therefore by continuity there must be cuts for which the two new faces and the 2n old faces have equal probabilities. ….

And the last line reads:

The problem of characterizing all fair dice, not just those which are fair by symmetry or by continuity, is still unsolved.

For the solved part, characterizing all fair dice by symmetry and continuity, we have a specific name for the class of polyhedrons. We call them Isohedron, defined as [source: MathWorld]

An isohedron is a convex polyhedron with symmetries acting transitively on its faces with respect to the center of gravity. Every isohedron has an even number of faces.

There are 30 of isohedrons (including finite solids and infinite classes of solids). All Platonic solids are isohedra.

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http://www.mathpuzzle.com/Fairdice.htm contains a complete list of all possible Isohedrons.

If you are also fascinated by the variety of dice available do check out: World’s Largest Dice Collection (Guinness World Record).

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