Platonic Dice Pseudo-Puzzles

[Lepton]

I fulfilled a lifelong dream today by purchasing a set of dice in the shapes of the Platonic solids. The following are some simple questions that came up while I was playing with them.

Tetrahedron: How many different ways can you number the four corners of a tetrahedron?

Cube: Is the cube more stable spinning (quickly) while balanced on a vertex or on a side?

Octahedron: How many different ways can you number an octahedron with the digits 1 to 8 such that (a) the sum of opposite sides is 9, (b) the numbers 1-4 are all adjacent to the same vertex, and (c) all the odd numbers are adjacent to the same vertex?

Dodecahedron: Which has more edges, a dodecahedron or an icosahedron?

Icosahedron: One way to compare the "roundness" of two solids is to find the ratio of the volume to the surface area: ratios closer to 1/3 (a sphere) are "rounder". For an icosahedron and a dodecahedron, each with an edge length of 1, which is more "round"?

[gftt*]

[Scurra]

[bgg1996]

[Scurra]

Surely there would be a point at which the number wouldn't fit into the available space? So it can't be infinite. Maybe 0.99 of infinite?
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[bgg1996]

[Lepton*]

gftt: I didn't say it was a good measurement! In this case, the interplay between the definition and the edge length is what makes this question interesting/surprising for me.

bgg1996: And there are, of course, limitations from quantum information theory if the tetrahedron is physical. Bah, use 1-4.

Scurra: I'm planning to build a model solar system, and I needed something to situate between the heavenly spheres.

[groza528]

For 1) I think it depends on whether we assume that the vertices are indistinguishable (until being labeled, that is.) Because if so then any organic chemist knows immediately that there are only two chiral configurations. If not I guess it would be 4! or 24.

[The Potter]

Cube: Is the cube more stable spinning (quickly) while balanced on a vertex or on a side?

A higher moment of inertia will be more stable. When spinning a block about the center axis, the for the moment of inertia is
(1/12)m(L^2 + D^2)
For spinning about the longest axis it is
(1/6)m(L^2 D^2 + L^2 W^2 + D^2 W^2) / (L^2 + D^2 + W^2)

For both cases L=D=W=s.
Spinning about an axis on a face is (1/6) m s^2
Spinning about the vertex is (1/6) m s^2

It appears there isn't a difference.
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[Zandor]