Folding and Unfolding in Computational Geometry

Part III: Polyhedra

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All material in these pages is Copyright 2006 by Erik D. Demaine and Joseph O'Rourke.

Note: The contents list below is out-of-date w.r.t. the final contents, for which see TOC.

18. Introduction and Overview
    • Overview
    • Curvature
    • Gaus-Bonnet Theorem
19. Edge Unfolding of Polyhedra
    • Introduction
      • Problem Features
        • Andrea Mautler's non-simply-connected unfolding
        • Ununfoldable simplicial polyhedra
    • Evidence For Edge Unfoldings
      • Unfoldings of Platonic and Archimedian Polyhedra
      • Hypergami and Javagami
    • Evidence Against Edge Unfoldings
    • Ununfoldable Polyhedra
    • Special Classes of Edge-Unfoldable Polyhedra
    • Vertex Unfoldings
20. Reconstruction of Polyhedra
    • Minkowski's Theorem
    • Cauchy's Rigidity Theorem
      • Cauchy's arm lemma
      • flexible octahedron top
    • Flexible Polyhedra
      • Bricard's Flexible Octahedra
        • Inner flexing of Bricard's octahedron
      • Connelly's Flexible Polyhedron
      • Steffen's Flexible Polyhedron
      • The Bellows Conjecture
    • Alexsandrov's Theorem
      • D-forms
    • Sabitov's Algorithm
21. Shortest Paths
    • Shortest Paths Algorithms
    • Star Unfolding
      • Nonoverlap
      • Cut Locus
    • Geodesics:Lyusternik-Schnirelmann
    • Curve Developement
      • Slice Curves
22. Folding Polygons to Polyhedra
    • Folding a Polygon
      • Metamorphosis of the Cube
        Erik's Description
      • foldings of the square into convex polyhedra
      • Foldings of the Latin Cross
      • rolling tetrahedron
      • asymmetric tetrahedron
      • hexahedron
      • octahedron
      • pentagon
    • Exponential Number of Gluing Trees
    • General Gluing Algorithm
    • The Foldings of the Latin Cross
    • The Foldings of a Square to Convex Polyhedra
    • Reconstructing 3D Shapes
    • Enumerations of Foldings
    • Enumerations of Cuttings
    • Orthogonal Polyhedra
      • NonOrthogonal Polyhedra
23. Unfolding in Higher Dimensions
    • Part I
    • Part II
    • Part III