Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

  • Zachary Abel
  • Erik D. Demaine
  • Martin L. Demaine
  • David Eppstein
  • Anna Lubiw
  • Ryuhei Uehara
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)


When can a plane graph with prescribed edge lengths and prescribed angles (from among {0,180°, 360°}) 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°, 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.


Edge Length Outgoing Edge Simple Cycle Level Planarity Folding Problem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Zachary Abel
    • 1
  • Erik D. Demaine
    • 2
  • Martin L. Demaine
    • 2
  • David Eppstein
    • 3
  • Anna Lubiw
    • 4
  • Ryuhei Uehara
    • 5
  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.Computer Science and Artificial Intelligence Lab.MITCambridgeUSA
  3. 3.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  4. 4.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada
  5. 5.School of Information ScienceJapan Advanced Institute of Science and TechnologyIshikawaJapan

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