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In the paper Fair Dice, the author describes "face transitivity" as the ability to transfer one face to the position of another and still have the solid look the same (this will ensure it forms a fair die).

When I look for the definition of transitivity on Wikipedia, it says -

Whenever an element a is related to an element b and b is related to an element c then a is also related to c.

I must be missing something obvious, but how is the above related to the ability to swap faces?

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  • $\begingroup$ A relation is transitive. In the case of dice, the "face transitivity" is a property of the face ? Or it is a property of the positions on the dice ? $\endgroup$ – Mauro ALLEGRANZA Jan 3 '18 at 9:52
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A solid is face-transitive (isohedral) when, for any faces, A and B, of the solid, there exists a symmetry of the entire solid, via some reflections and rotations, that maps A onto B.

Well, if there is such a map of A onto B and such a map of B onto C, then there is also such a map of A onto C.

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  • $\begingroup$ Makes sense. But in the case of solids, it sounds like transitivity is incidental. There could be other operations on the solid defined on the faces that are transitive. So, calling the "swappable by rotation" property "face transitivity" doesn't seem like a good way to put it. Would you agree? $\endgroup$ – Rohit Pandey Jan 3 '18 at 16:12

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