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I was having some fun looking at different solids to make a die out of (see here ) and came across the gyrate deltoidal icositetrahedron.

This solid is not a Catalan solid, but nevertheless all faces are congruent (see also here for a related question). Furthermore, there is a centre $C$ so that any face can be brought onto another face using a rotation (with centre $C$) or a reflection (whose plane goes through $C$).

Question 1: Can an ideal solid be made out of this die be a fair die [independently of the physical context (such as the surface on which it is thrown and the viscosity of the air) and assuming random initial velocity and momentum]?

Just to recall, when it is possible to send a face to any other face through a rotation and this rotation sends the whole solid into itself, then the die is fair. However this is a priori only a sufficient condition, not a necessary one.

To make the role of the surface clearer, here is a classical argument of continuity (which is only partially true). Look at a triangular prism. The probability of this die falling on a square face depends on its height. Since it is nearly 0 when the height is small, and nearly maximal (1/3) when the height is huge, continuity should tell us, this solid is fair for some height in between. This argument is (unfortunately) false, since the elasticity of the table (or the viscosity of the air) are two additional parameters (in addition to the height). So the die will only be fair for one specific set of conditions. Nonetheless, this does not answer the underlying, more general question either:

Question 2: Can a fair die (on any surface and in any atmospheric conditions) be only made out of a face-uniform solid (Platonic or Catalan solid, bipyramids, trapezohedra, …)?

This question contains also other physical constraints on the fairness of a solid, such as the centre of gravity and the solid angle made by the faces. These two constraints are however satisfied by the gyrate deltoidal icositetrahedron.

[EDIT: Also just to make the randomness of the initial conditions clear, I am not trying to prevent an intentionally biased throw (such as the thrower explicitly trying to spin the die in a certain way). So something like the direction of the velocity and rotation is uniformly chosen at random and while their amplitude follow some quickly decaying law). As pointed out below, even with a rather random start, it could still happen that the geometry of the dice favours some rotations (e.g. after a few bumps).]

[EDIT: You might notice that when the gyrate deltoidal icositetrahedron comes to a rest (i.e. one face is flat on the table) two things can happen: there might be a face on the top which is also lying flat (parallel to the ground, or there might not be such a face. In that sense, it's not the most convenient die possible. But for the sake of this post, assume the number of the face lying on the table is the result of the die.]

[EDIT: this question is also mentioned here but, although the author of the website seems to think this dice is not fair, no answer is given.]

ARG
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  • The only unconditionally fair dice are the five Platonic solids – Narasimham Apr 11 '20 at 13:04
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    I disagree. If you make a triangular bipyramid, with $D_{3h}$ symmetry of course, which faces are favored? – Oscar Lanzi Apr 11 '20 at 13:06
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    on any surface and in any atmospheric conditions – You may want to add rolling conditions to that list, by which I do not mean the specific position of the die, but with how much angular momentum, linear momentum, and angle with respect to the surface you are throwing it. – Wrzlprmft Apr 11 '20 at 13:38
  • @Narasimham I disagree too, the Catalan solids all have symmetry group [which preserves the solid] and is transitive on the faces. Transitivity on the vertices is nice, but not required [for "ideal" fairness]. It could be that platonic solids are more resilient to inhomogeneity (but that would require some proof). – ARG Apr 11 '20 at 18:46
  • @Wrzlprmft Actually, I did not want to include the possibility that the person rolling the die has bad intentions (e.g. that he is intentionally trying to use unevenness of the moment of inertia to favour some faces). But you are right that even if we assume uniform probability at the moment of the throw, there could be a preferred axis after some bumps on the surface... – ARG Apr 11 '20 at 18:50
  • @OscarLanzi: My definition of " fairness " assuming uniform density of dice material is: its mass moment of inertia should be the same for any arbitrary axis through its geometric center. – Narasimham Apr 12 '20 at 10:12
  • It may be tough to calculate given so many edges. – Narasimham Apr 12 '20 at 10:15
  • @Narasimham it's been awhile since I computed inertia moments, but any solid with a large number of symmetries is bound to have a inertia tensor which is a multiple of the identity matrix (because the 3-principal axis can be sent to each other by a symmetry of the solid; and changing the axis of rotation has a trivial effect on the identity matrix [changing the centre would not]). The 4-fold antiprismatic symmetry of the gyrate deltoidal icositetrahedron is definitively rich enough to ensure this. – ARG Apr 12 '20 at 16:51
  • Checked again: the 4-fold antiprismatic is not enough, but its close relation to the deltoidal icositetrahedron could avert the need of a computation. – ARG Apr 12 '20 at 16:58
  • Suspect the assumed axis goes through vertices when required to check an arbitrary axis. – Narasimham Apr 12 '20 at 21:16
  • @Narasimham (a)you don't need the symmetry to be in special position w.r.t. to principal axis. If a diagonal matrix commutes with a large enough group of rotations it has to be the identity (b)you did not provide any reference yet to the fact that the platonic solids are the only one to have this property; in fact, according to this paper the Archimedean solids have this property too (c)even if the Catalan solids were not to have this property (which I doubt),it's not clear to me why it matters (given all faces are equivalent) – ARG Apr 13 '20 at 08:16

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