Thank you, Michael and Stephen! So, my first intuition, that the number of faces converges, is false, but my weaker one, that if it does grow without bound, it does so very slowly, is borne out by Dr. Hueter's work. Andy, no, there's no particular reason to prefer three dimensions. My motivation, which I didn't share in my original post, had to do with the statistics of "random" dice, which is what suggested D=3; then that original question got eclipsed when I realized that I didn't know much more basic things about these random solids. A remaining question: Keith pointed out early on that with probability 1 all the faces are triangles. This implies that the expected degree of a vertex is 6 - 12/V. but I don't have any decent intuition for the actual distribution of these degrees. Are they really almost all 6, or are there lots of 5s and 7s too? How does this change with N? Are there big patches of degree-6 triangular lattice like in a geodesic dome? On Mon, Mar 11, 2019 at 2:56 PM Andy Latto <andy.latto@pobox.com> wrote:
Is there a reason to investigate this in three dimensions first? A lot of the questions (number of vertices, perimeter, area) make sense in two dimensions, and might be more tractable there.
Andy
On Fri, Mar 8, 2019 at 11:08 AM Allan Wechsler <acwacw@gmail.com> wrote:
Suppose we have a random point generator, that generates points in R^3
with
all three coordinates normally distributed around 0, say with a standard deviation of 1, though it doesn't really matter.
Take N random points in this manner, and form the convex closure. This will be a polyhedron. What are the expected numbers of faces, vertices, and edges, in terms of N? Do they approach finite limits as N increases? I imagine that the surface area grows without bound. What about the expected number of faces that have 3, 4, 5, 6 edges, and so on?
I can think of many more questions of this sort, and am wondering if anybody here knows what's known. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
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