Flat foldings of plane graphs with prescribed angles and edge lengths

Authors

  • Zachary Abel Department of Mathematics, Massachusetts Inst. of Technology
  • Erik D. Demaine Computer Science and Artificial Intelligence Lab., Massachusetts Inst. of Technology
  • Martin L. Demaine Computer Science and Artificial Intelligence Lab., Massachusetts Inst. of Technology
  • David Eppstein University of California, Irvine
  • Anna Lubiw David R. Cheriton School of Computer Science, University of Waterloo
  • Ryuhei Uehara School of Information Science, Japan Advanced Institute of Science and Technology

DOI:

https://doi.org/10.20382/jocg.v9i1a3

Abstract

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ,
360^\circ\}$) be folded flat to lie in an infinitesimally thick line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.

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Author Biography

David Eppstein, University of California, Irvine

Professor of Computer Science in the Donald Bren School of Information & Computer Sciences at the University of California, Irvine

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Published

2018-02-27

Issue

Vol. 9 No. 1 (2018)

Section

Articles

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