This very interesting question of Jim's, below, seems closely related to John Conway's 1997 question about whether there exists a "holyhedron" (revised phrasing as of 2000): "The Holyhedron Problem. Is there a polyhedron in e-dimensional Euclidean space that has only finitely many plane faces, each of which is a closed connected subset of the appropriate plane whose relative interior in that plane is multiply connected?" Jade Vinson found the first example about two years after the problem was posed; it has over 78 million faces. Don Hatch found an example with only 492 faces in about 2003. This seems to be the current record for fewest faces. Conway offered a prize of $10,000 divided by the number of faces for an example of a holyhedron with a record low number of faces. It is unclear whether this offer still stands. ((( Varying the definition slightly to allow a) a noncompact polyhedron in R^3, there is an easy example consisting of a Z^3-translated collection of interpenetrating solid tetrahedra. One then takes the boundary of their union. and b) a holyhedron in S^3, based on 24 likewise interpenetrating spherical tetrahedra. ))) —Dan
On Apr 25, 2016, at 8:36 AM, James Propp <jamespropp@gmail.com> wrote:
Given convex polytopes P and Q in R^n, say P "pierces" Q if Q\P is connected but not simply connected.
Are there convex polyhedra P,Q in R^3 that pierce each other? (I don't think so but no proof leaps to mind.)
Are there cubes P,Q,R in R^3 such that P pierces Q, Q pierces R, and R pierces P?
(One might conceive of "mutually ruperting" cubes subject to some rules that flout, in various mathematically specific ways, the fact that two physical objects cannot occupy the same space at the same time.)
Perhaps Pechenik and Shultz already consider such questions?