Maximal origami flip graphs of flat-foldable vertices: properties and algorithms

03/27/2022
by   Thomas C. Hull, et al.
0

Flat origami studies straight line, planar graphs C=(V,E) drawn on a region R⊂ℝ^2 that can act as crease patterns to map, or fold, R into ℝ^2 in a way that is continuous and a piecewise isometry exactly on the faces of C. Associated with such crease pattern graphs are valid mountain-valley (MV) assignments μ:E→{-1,1}, indicating which creases can be mountains (convex) or valleys (concave) to allow R to physically fold flat without self-intersecting. In this paper, we initiate the first study of how valid MV assignments of single-vertex crease patterns are related to one another via face-flips, a concept that emerged from applications of origami in engineering and physics, where flipping a face F means switching the MV parity of all creases of C that border F. Specifically, we study the origami flip graph OFG(C), whose vertices are all valid MV assignments of C and edges connect assignments that differ by only one face flip. We prove that, for the single-vertex crease pattern A_2n whose 2n sector angles around the vertex are all equal, OFG(A_2n) contains as subgraphs all other origami flip graphs of degree-2n flat origami vertex crease patterns. We also prove that OFG(A_2n) is connected and has diameter n by providing two O(n^2) algorithms to traverse between vertices in the graph, and we enumerate the vertices, edges, and degree sequence of OFG(A_2n). We conclude with open questions on the surprising complexity found in origami flip graphs of this type.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset