Pentagonal bipyramids lead to the smallest flexible embedded polyhedron: https://arxiv.org/abs/2410.13811, new preprint by
Matteo Gallet, Georg Grasegger, Jan Legerský, and Josef Schicho
Convex polyhedra are rigid, but some special non-convex polyhedra have at least one continuous degree of freedom. (More than one degree of freedom can be obtained in a trivial way by gluing such polyhedra.) The Bricard octahedra (https://en.wikipedia.org/wiki/Bricard_octahedron) have the same combinatorial structure as a regular octahedron, and can sort of flex, but only if you allow faces that can cross through each other. You can avoid the crossings, for instance by making little gussets near where they would happen, at the expense of making the shape more complicated.
It was believed that Steffen's polyhedron, with 9 vertices, 21 edges, and 14 triangular faces, was the simplest possible non-self-intersecting flexible polyhedron. For instance, Demaine and O'Rourke's book _Geometric Folding Algorithms_ (p. 347) writes "As it was later proven that all triangulated polyhedra of eight vertices are rigid, Steffen’s example is minimal in this sense."
But it was apparently not proven, and the new preprint claims a better example: an 8-vertex polyhedron formed from a pentagonal bipyramid by subdividing one triangle into three. This seems a bizarre choice because this subdivision does not affect flexibility. But it can act as a gusset pulling two parts of the boundary away from each other, changing a self-crossing bipyramid into a non-self-crossing subdivided bipyramid.
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2024-10-20 07:32:44
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"content": "Pentagonal bipyramids lead to the smallest flexible embedded polyhedron: https://arxiv.org/abs/2410.13811, new preprint by \nMatteo Gallet, Georg Grasegger, Jan Legerský, and Josef Schicho\n\nConvex polyhedra are rigid, but some special non-convex polyhedra have at least one continuous degree of freedom. (More than one degree of freedom can be obtained in a trivial way by gluing such polyhedra.) The Bricard octahedra (https://en.wikipedia.org/wiki/Bricard_octahedron) have the same combinatorial structure as a regular octahedron, and can sort of flex, but only if you allow faces that can cross through each other. You can avoid the crossings, for instance by making little gussets near where they would happen, at the expense of making the shape more complicated.\n\nIt was believed that Steffen's polyhedron, with 9 vertices, 21 edges, and 14 triangular faces, was the simplest possible non-self-intersecting flexible polyhedron. For instance, Demaine and O'Rourke's book _Geometric Folding Algorithms_ (p. 347) writes \"As it was later proven that all triangulated polyhedra of eight vertices are rigid, Steffen’s example is minimal in this sense.\"\n\nBut it was apparently not proven, and the new preprint claims a better example: an 8-vertex polyhedron formed from a pentagonal bipyramid by subdividing one triangle into three. This seems a bizarre choice because this subdivision does not affect flexibility. But it can act as a gusset pulling two parts of the boundary away from each other, changing a self-crossing bipyramid into a non-self-crossing subdivided bipyramid.",
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