Bumpy Pyramid Folding

We investigate folding problems for a class of petal polygons P, which have an n-polygonal base B surrounded by a sequence of triangles. We give linear time algorithms using constant precision to determine if P can fold to a pyramid with flat base B, and to determine a triangulation of B (crease pattern) that allows folding into a convex (triangulated) polyhedron. By Alexandrov's theorem, the crease pattern is unique if it exists, but the general algorithm known for this theorem is pseudo-polynomial, with very large running time; ours is the first efficient algorithm for Alexandrov's theorem for a special class of polyhedra. We also give a polynomial time algorithm that finds the crease pattern to produce the maximum volume triangulated polyhedron

Ghost chimneys

A planar point set S is an (i, t) set of ghost chimneys if there exist lines H0,H1, ..., Ht-1 such that the orthogonal projection of S onto Hj consists of exactly i + j distinct points. We give upper and lower bounds on the maximum value of t in an (i, t) set of ghost chimneys, showing that it is linear in i

Toward Unfolding Doubly Covered nn-Stars

info:eu-repo/semantics/publishe

Toward Unfolding Doubly Covered n-Stars

We present nonoverlapping general unfoldings of two infinite families of nonconvex polyhedra, or more specifically, zero-volume polyhedra formed by double-covering an n-pointed star polygon whose triangular points have base angle α. Specifically, we construct general unfoldings when n∈ { 3, 4, 5, 6, 8, 9, 10, 12 } (no matter the value of α ), and we construct general unfoldings when α< 60∘(1 + 1 / n) (i.e. when the points are shorter than equilateral, no matter the value of n, or slightly larger than equilateral, especially when n is small). Whether all doubly covered star polygons, or more broadly arbitrary nonconvex polyhedra, have general unfoldings remains open.SCOPUS: cp.kinfo:eu-repo/semantics/publishe

the date of receipt and acceptance should be inserted later

be inserted by the editor