Experimental Polyhedral D9: The Donny

Uploaded at the request of an inquisitive redditor, this 9 sided polyhedron is created with the intended use as a basis for a 9 sided die. This die has not yet been tested for fairness, and there is no mathematical proof of fairness, so I make no promises. That said, the design does possess some physical motivations.

The Thomson problem is a physics problem where one asks what the minimum energy configuration is of N equally charged particles constrained to move on the surface of a sphere. While not solved definitively for 9 points, the best known configuration consists of three equilateral triangles arranged in three equally spaced parallel planes with alternating orientations. Paramaterizing this configuration based on the polar angle of the points on one of the outer, non-equatorial triangles, the lowest energy was found when the polar angle θ is equal to the root of the following function near θ=.79. 

Using these points as the vertices of a polyhedron with 9 vertexes and 14 faces, I then took the geometric dual to produce the 9 face polyhedron above, consisting of 3 rhomboidal faces and 6 pentagonal faces. The rhomboidal faces are slightly larger in terms of area, while the pentagonal faces are larger in terms of angular size as viewed from the center of the die. The center of mass always sits precisely the same height off the ground no matter which face it rests on. It possesses D_3h symmetry.

Numbering this die would be strange. 6 values would need to be indicated on the vertexes opposite the pentagonal faces, as on a D4, while 3 values would need to be written on the edges opposite the rhomboidal faces. If the die is not truly fair, and the rhomboidal faces are slightly favored or slightly disfavored, I'd recommend numbering the edges opposite them 2, 5, and 8, as this would produce the overall flattest and most fair possible probability distribution for results. 

My 3D printer is currently inoperable, so I can't actually print this myself, but if anyone feels like printing one or several of these out and testing them for fairness, please tell me your results! I'd love to hear. If anyone is curious about the name, it's named after somebody I am very close to.