Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Martin L. Demaine, Jenny Diomidova, Tonan Kamata, Ryuhei Uehara, and Hanyu Alice Zhang, “Any Regular Polyhedron Can Transform to Another by O(1) Refoldings”, in Proceedings of the 33rd Canadian Conference in Computational Geometry (CCCG 2021), Halifax, Nova Scotia, Canada, August 10–12, 2021, pages 332–342.

BibTeX

@InProceedings{Refolding_CCCG2021,
  AUTHOR        = {Erik D. Demaine and Martin L. Demaine and Jenny Diomidova and Tonan Kamata and Ryuhei Uehara and Hanyu Alice Zhang},
  authororig    = {Erik D. Demaine and Martin L. Demaine and Yevhenii Diomidov and Tonan Kamata and Ryuhei Uehara and Hanyu Alice Zhang},
  TITLE         = {Any Regular Polyhedron Can Transform to Another by {$O(1)$} Refoldings},
  BOOKTITLE     = {Proceedings of the 33rd Canadian Conference in Computational Geometry (CCCG 2021)},
  bookurl       = {https://projects.cs.dal.ca/cccg2021/},
  ADDRESS       = {Halifax, Nova Scotia, Canada},
  MONTH         = {August 10--12},
  YEAR          = 2021,
  PAGES         = {332--342},

  withstudent   = 1,
  unrefereed    = 1,
  dblp          = {https://dblp.org/rec/conf/cccg/DemaineDDKUZ21},
  comments      = {This paper is also available from the <A HREF="https://projects.cs.dal.ca/cccg2021/wordpress/wp-content/uploads/2021/08/CCCG2021.pdf#page=342">electronic proceedings</A> and as <A HREF="https://arXiv.org/abs/2109.03997">arXiv:2109.03997</A>.},
  papers        = {Refolding_CGTA},
}

Abstract:

We show that several classes of polyhedra are joined by a sequence of O(1) refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface area, we prove that (1) any two tetramonohedra are refoldable to each other, (2) any doubly covered triangle is refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and doubly covered regular polygon is refoldable to a tetramonohedron, (4) any tetrahedron has a 3-step refolding sequence to a tetramonohedron, and (5) the regular dodecahedron has a 4-step refolding sequence to a tetramonohedron. In particular, we obtain a ≤ 6-step refolding sequence between any pair of Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all other Platonic solids. As far as the authors know, this is the first result about common unfolding involving the regular dodecahedron.

Comments:

This paper is also available from the electronic proceedings and as arXiv:2109.03997.

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Related papers:

Refolding_CGTA (Any Platonic solid can transform to another by O(1) refoldings)