3 research outputs found
Any Regular Polyhedron Can Transform to Another by O(1) Refoldings
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
Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid
In this paper, we investigate the common unfolding between regular tetrahedra and Johnson-Zalgaller solids. More precisely, we investigate the sets of all edge developments of Johnson-Zalgaller solids that fold into regular tetrahedra. We show that, among 92 Johnson-Zalgaller solids,only J17 (gyroelongated square dipyramid) and J84 (snub disphenoid) have some edge developments that fold into a regular tetrahedron, and the remaining Johnson-Zalgaller solids do not have any such edge development
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum