Allan Wechsler <acwacw@gmail.com> wrote:
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?
I too look forward to an answer to that question. And to other questions, such as what the odds are that the origin is outside the polyhedron. Intuitively, that one ought to have a closed-form solution. Consider the case N=4. The origin will be outside the tetrahedron iff there's a plane going through the origin such that all four points are on the same side of it. My first thought was that since the X, Y, and Z coordinates are statistically independent, the answer must be 1 - (7/8)^3, which is exactly 0.330078125. But then I realized that the points can all be on the same side even if they're bipolar in each of X, Y, and Z. For instance if all the points were on the plane X+Y+Z = 1.