Steffen's polyhedron
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Short description: Flexible polyhedron with 14 triangle faces
A net for Steffen's polyhedron. The solid and dashed lines represent mountain folds and valley folds, respectively.
In geometry, Steffen's polyhedron is a flexible polyhedron discovered (in 1978[1]) by and named after Klaus Steffen (de). It is based on the Bricard octahedron, but unlike the Bricard octahedron its surface does not cross itself.[2] With nine vertices, 21 edges, and 14 triangular faces, it is the simplest possible non-crossing flexible polyhedron.[3] Its faces can be decomposed into three subsets: two six-triangle-patches from a Bricard octahedron, and two more triangles (the central two triangles of the net shown in the illustration) that link these patches together.[4]
It obeys the strong bellows conjecture, meaning that (like the Bricard octahedron on which it is based) its Dehn invariant stays constant as it flexes.[5]
References
- ↑ Optimizing the Steffen flexible polyhedron Lijingjiao et al. 2015
- ↑ Klarner, David A., ed. (1981), "Flexing surfaces", The Mathematical Gardner, Springer, pp. 79–89, doi:10.1007/978-1-4684-6686-7_10, ISBN 978-1-4684-6688-1.
- ↑ "23.2 Flexible polyhedra", Geometric Folding Algorithms: Linkages, origami, polyhedra, Cambridge University Press, Cambridge, 2007, pp. 345–348, doi:10.1017/CBO9780511735172, ISBN 978-0-521-85757-4.
- ↑ Fuchs, Dmitry (2007), Mathematical Omnibus: Thirty lectures on classic mathematics, Providence, RI: American Mathematical Society, p. 354, doi:10.1090/mbk/046, ISBN 978-0-8218-4316-1, https://books.google.com/books?id=IiG9AwAAQBAJ&pg=PA347.
- ↑ Alexandrov, Victor (2010), "The Dehn invariants of the Bricard octahedra", Journal of Geometry 99 (1-2): 1–13, doi:10.1007/s00022-011-0061-7.
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