Mathematics in Computer Science

, Volume 10, Issue 1, pp 115–141

Geodesic Universal Molecules



The first phase of TreeMaker, a well-known method for origami design, decomposes a planar polygon (the “paper”) into regions. If some region is not convex, TreeMaker indicates it with an error message and stops. Otherwise, a second phases is invoked which computes a crease pattern called a “universal molecule”. In this paper we introduce and study geodesic universal molecules, which also work with non-convex polygons and thus extend the applicability of the TreeMaker method. We characterize the family of disk-like surfaces, crease patterns and folded states produced by our generalized algorithm. They include non-convex polygons drawn on the surface of an intrinsically flat piecewise-linear surface which have self-overlap when laid open flat, as well as surfaces with negative curvature at a boundary vertex.


Algorithmic origami Planar subdivision Metric tree Non-convex polygon 

Mathematics Subject Classification

68U05 (Computer graphics; computational geometry) 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceJames Madison UniversityHarrisonburgUSA
  2. 2.Department of Computer ScienceSmith CollegeNorthamptonUSA

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