Abstract
This paper treats flexible polyhedra in the hyperbolic 3-space ℍ3. It is proved that the geometric characterization of octahedra being infinitesimally flexible of orders 1 or 2 is quite the same as in the Euclidean case. Also Euclidean results concerning continuously flexible octahedra remain valid in hyperbolic geometry: There are at least three types of continuously flexible octahedra in #x210D;3; the line-symmetric Type 1, Type 2 with planar symmetry, and the non-symmetric Type 3 with two flat positions. However, Type 3 can be subdivided into three subclasses according to the type of circles in hyperbolic geometry. The flexibility of Type 3 octahedra can again be argued with the aid of Ivory’s Theorem.
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Stachel, H. (2006). Flexible Octahedra in the Hyperbolic Space. In: Prékopa, A., Molnár, E. (eds) Non-Euclidean Geometries. Mathematics and Its Applications, vol 581. Springer, Boston, MA. https://doi.org/10.1007/0-387-29555-0_11
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DOI: https://doi.org/10.1007/0-387-29555-0_11
Publisher Name: Springer, Boston, MA
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