I believe at least most of the dice that are fair but oddly shaped derive fairness through symmetry; each face is isomorphic to some other face through some spatial rotation or mirroring. I think there's much more to fairness for non-symmetric dice than solid angle from center of gravity. For instance, it's easy to make a polyhedra where a face has a positive solid angle from the center of gravity, yet the center of gravity does not lie over the face when the face is flat (and thus the dice will never be stable on that face). -tom On Wed, Jan 30, 2019 at 1:39 PM Dan Asimov <dasimov@earthlink.net> wrote:
Let F be a face of a bounded convex polyhedron K in R^3. Assume K is solid with constant density. Then:
----- What is the probability
p = p(F; K)
that if K is randomly thrown onto a horizontal surface, it will land on the face F ? -----
My guess back in high school was that
p = (1/4π) * (solid angle(F), subtended from the center of gravity of K).
I recently saw for sale some novelty dice that are irregular hexahedra with planar faces, advertised as having equal probabilities for all faces. So I'm wondering if this is just an issue of that solid angle fraction, or whether there is more to it. Maybe this depends on how "randomly thrown" is defined? A priori it seems that this might be possible, say, if some faces would be unlikely to be landed on if our polyhedral die K were rolling on the horizontal surfaces with enough velocity (or end-over-end angular momentum).
So the random throws might have different statistical outcomes if there are many with high energy.
Or not?
—Dan
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