Paper by Erik D. Demaine

Reference:

Marshall Bern, Erik D. Demaine, David Eppstein, Eric Kuo, Andrea Mantler, and Jack Snoeyink, “Ununfoldable Polyhedra with Convex Faces”, Computational Geometry: Theory and Applications, volume 24, number 2, February 2003, pages 51–62. Special issue of selected papers from the 4th CGC Workshop on Computational Geometry, 1999.

BibTeX

@Article{Ununfoldable,
  AUTHOR        = {Marshall Bern and Erik D. Demaine and David Eppstein and
                   Eric Kuo and Andrea Mantler and Jack Snoeyink},
  TITLE         = {Ununfoldable Polyhedra with Convex Faces},
  JOURNAL       = {Computational Geometry: Theory and Applications},
  journalurl    = {https://www.sciencedirect.com/journal/computational-geometry},
  VOLUME        = 24,
  NUMBER        = 2,
  MONTH         = {February},
  YEAR          = 2003,
  PAGES         = {51--62},
  NOTE          = {Special issue of selected papers from the 4th CGC
                   Workshop on Computational Geometry, 1999.},

  papers        = {CGC99; CCCG99b},
  replaces      = {CGC99; CCCG99b},
  length        = {14 pages},
  doi           = {https://dx.doi.org/10.1016/S0925-7721(02)00091-3},
  dblp          = {https://dblp.org/rec/journals/comgeo/BernDEKMS03},
  comments      = {This paper is also available as
                   <A HREF="http://arXiv.org/abs/cs.CG/9908003">
                   arXiv:cs.CG/9908003</A> of the
                   <A HREF="http://arXiv.org/archive/cs/intro.html">
                   Computing Research Repository (CoRR)</A>.
                   This paper is also available from <A HREF="https://doi.org/10.1016/S0925-7721(02)00091-3">ScienceDirect</A>.},
}

Abstract:

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.

Comments:

This paper is also available as arXiv:cs.CG/9908003 of the Computing Research Repository (CoRR). This paper is also available from ScienceDirect.

Length:

The paper is 14 pages.

Availability:

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Related papers:

CGC99 (Ununfoldable Polyhedra with Triangular Faces)

CCCG99b (Ununfoldable Polyhedra)