Single-Vertex Origami and Spherical Expansive Motions

  • Ileana Streinu
  • Walter Whiteley
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)


We prove that all single-vertex origami shapes are reachable from the open flat state via simple, non-crossing motions. We also consider conical paper, where the total sum of the cone angles centered at the origami vertex is not 2π. For an angle sum less than 2π, the configuration space of origami shapes compatible with the given metric has two components, and within each component, a shape can always be reconfigured via simple (non-crossing) motions. Such a reconfiguration may not always be possible for an angle sum larger than 2π.

The proofs rely on natural extensions to the sphere of planar Euclidean rigidity results regarding the existence and combinatorial characterization of expansive motions. In particular, we extend the concept of a pseudo-triangulation from the Euclidean to the spherical case. As a consequence, we formulate a set of necessary conditions that must be satisfied by three-dimensional generalizations of pointed pseudo-triangulations.


Great Circle Planar Framework Antipodal Point Alignment Event Polygonal Path 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ileana Streinu
    • 1
  • Walter Whiteley
    • 2
  1. 1.Computer Science DepartmentSmith CollegeNorthamptonUSA
  2. 2.Department of MathematicsYork UniversityTorontoCanada

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