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 at most 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.
Description
[2109.03997] Any Regular Polyhedron Can Transform to Another by O(1) Refoldings
%0 Generic
%1 demaine2021regular
%A Demaine, Erik D.
%A Demaine, Martin L.
%A Diomidov, Yevhenii
%A Kamata, Tonan
%A Uehara, Ryuhei
%A Zhang, Hanyu Alice
%D 2021
%K 2021 geometry mit polygon
%T Any Regular Polyhedron Can Transform to Another by O(1) Refoldings
%U http://arxiv.org/abs/2109.03997
%X 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 at most 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.
@misc{demaine2021regular,
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 at most 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.},
added-at = {2021-09-11T05:35:36.000+0200},
author = {Demaine, Erik D. and Demaine, Martin L. and Diomidov, Yevhenii and Kamata, Tonan and Uehara, Ryuhei and Zhang, Hanyu Alice},
biburl = {https://www.bibsonomy.org/bibtex/2d46fe6d7cb894b4b2f421e4d6a5ceb3d/analyst},
description = {[2109.03997] Any Regular Polyhedron Can Transform to Another by O(1) Refoldings},
interhash = {e0fe8116c5c722f5460621b7a7ac95d0},
intrahash = {d46fe6d7cb894b4b2f421e4d6a5ceb3d},
keywords = {2021 geometry mit polygon},
note = {cite arxiv:2109.03997},
timestamp = {2021-09-11T05:35:36.000+0200},
title = {Any Regular Polyhedron Can Transform to Another by O(1) Refoldings},
url = {http://arxiv.org/abs/2109.03997},
year = 2021
}