Tailoring for Every Body: Reshaping Convex Polyhedra
share
Given any two convex polyhedra P and Q, we prove as one of our main results that the surface of P can be reshaped to a homothet of Q by a finite sequence of "tailoring" steps. Each tailoring excises a digon surrounding a single vertex and sutures the digon closed. One phrasing of this result is that, if Q can be "sculpted" from P by a series of slices with planes, then Q can be tailored from P. And there is a sense in which tailoring is finer than sculpting in that P may be tailored to polyhedra that are not achievable by sculpting P. It is an easy corollary that, if S is the surface of any convex body, then any convex polyhedron P may be tailored to approximate a homothet of S as closely as desired. So P can be "whittled" to e.g., a sphere S. Another main result achieves the same reshaping, but by excising more complicated shapes we call "crests," still each enclosing one vertex. Reversing either digon-tailoring or crest-tailoring leads to proofs that any Q inside P can be enlarged to P by cutting Q and inserting and sealing surface patches. One surprising corollary of these results is that, for Q a subset of P, we can cut-up Q into pieces and paste them non-overlapping onto an isometric subset of P. This can be viewed as a form of "unfolding" Q onto P. All our proofs are constructive, and lead to polynomial-time algorithms.
READ FULL TEXT