Geometric Folding Algorithms:
Linkages, Origami, Polyhedra

The book contains many open problems (see the Index under Open Problems, p. 468). Inevitably, some will be resolved as time passes and before we can release a Second Edition. Here we will maintain a list of those Open Problems in the book which have been closed.

• Open Problem 23.2: Practical Algorithm for Cauchy Rigidity (p. 357)

  In a break-through paper [1], Bobenko and Izmestiev defined a procedure that can reconstruct a convex polyhedron from its faces by numerically solving a differential equation. And indeed they implemented it and the software is publically available. For a high-level description of the Bobenko-Izmestiev algorithm, see [3]. Their original algorithm had no bound, but recently Kane et al. [2] showed that it can be implemented to run in pseudopolynomial time.

• Open Problem 23.1: D-forms and Pita-Forms (p. 354)

  This problem was settled in a recent paper [4] that shows that D-forms have no creases but pita-forms may have one crease. A new open problem here is to determine whether or not a pita form might have no crease, as all examples have one.

1. Alexander I. Bobenko and Ivan Izmestiev. "Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes," Annales de l'institut Fourier, 58 No. 2 (2008), p. 447-505.
2. Daniel Kane, Gregory N. Price, Erik D. Demaine, "A Pseudopolynomial Algorithm for Alexandrov's Theorem," arXiv:0812.5030v1 [cs.CG], 2008.
3. Joseph O'Rourke, "Computational Geometry Column 49," SIGACT News, Vol. 38, No. 2, Issue 143, June 2007, pp. 55-59.
4. Gregory N. Price and Erik D. Demaine, "Generalized D-forms have no spurious creases, Discrete & Comptutational Geometry, 43(1) 179-186.