Vol. 8, No. 3, 2015

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The zipper foldings of the diamond

Erin W. Chambers, Di Fang, Kyle A. Sykes, Cynthia M. Traub and Philip Trettenero

Vol. 8 (2015), No. 3, 521–534
Abstract

In this paper, we classify and compute the convex foldings of a particular rhombus that are obtained via a zipper folding along the boundary of the shape. In the process, we explore computational aspects of this problem; in particular, we outline several useful techniques for computing both the edge set of the final polyhedron and its three-dimensional coordinates. We partition the set of possible zipper starting points into subintervals representing equivalence classes induced by these edge sets. In addition, we explore nonconvex foldings of this shape which are obtained by using a zipper starting point outside of the interval corresponding to a set of edges where the polygon folds to a convex polyhedron; surprisingly, this results in multiple families of nonconvex and easily computable polyhedra.

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Keywords
computational geometry, folding algorithms, combinatorial geometry
Mathematical Subject Classification 2010
Primary: 68U05
Milestones
Received: 31 January 2014
Revised: 20 March 2014
Accepted: 1 July 2014
Published: 5 June 2015

Communicated by Kenneth S. Berenhaut
Authors
Erin W. Chambers
Department of Mathematics and Computer Science
Saint Louis University
St. Louis, MO 63103
United States
Di Fang
Department of Mathematics and Computer Science
Saint Louis University
St. Louis, MO 63103
United States
Kyle A. Sykes
Department of Mathematics and Computer Science
Saint Louis University
St. Louis, MO 63103
United States
Cynthia M. Traub
Department of Mathematics and Statistics
Southern Illinois University Edwardsville
Edwardsville, IL 62026
United States
Philip Trettenero
University of Illinois
Urbana, IL 61801
United States