Generalizing the notion of average to a continuous distribution is not easy. Ideally, all calculations should be checked by alternative integration procedures. Here's another one: Excluding a measure zero set with the plane through one edge, the configuration space can be written as a R=3x1x3 rectangular volume, with a unit-length central edge, between left and right facets, each with three more unit-edge boundaries. Once the three points are chosen, the two edges across adjacent facets are known, and a plane is determined to find the others. With A for area and P for perimeter, let the averages be: Avg(A) = Int_R A*dV, Avg(P) = Int_R P*dV. Are these correct, and how accurately can they be calculated? I am actually interested to work it out and see what happens, but unfortunately I am falling behind on another project and another. Cheers --Brad PS. The same tactic works for the dodecahedron, with R=4x1x4 on pentagonal edges, and similar for other regular polyhedra because their symmetry groups act transitively on the edges and faces. On Wed, Aug 7, 2019 at 12:27 PM James Propp <jamespropp@gmail.com> wrote:
Keith’s 2.8819 seems awfully far off the mark