Flexible cross-polytopes in spaces of constant curvature

Abstract

Abstract. We construct self-intersected flexible cross-polytopes in the spaces of constant curvature, that is, Euclidean spaces E n , spheres S n , and Lobachevsky spaces Λ n of all dimensions n. In dimensions n ≥ 5, these are the first examples of flexible polyhedra. Moreover, we classify all flexible cross-polytopes in each of the spaces E n , S n , and Λ n . For each type of flexible cross-polytopes, we provide an explicit parametrization of the flexion by either rational or elliptic functions.

Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry

“…A higher-dimensional analog of the octahedron is called cross-polytope. Recently, flexible cross-polytopes in X d were classified by Gaifullin [18].…”

Flexible polyhedra and their volumes

“…In [16] the author constructed examples of flexible cross-polytopes in the Euclidean spaces R n , the Lobachevsky spaces Λ n , and the round spheres S n of all dimensions, and obtained a complete classification of all flexible cross-polytopes.…”

A sufficient condition for a polyhedron to be rigid

“…(d) during the flex, they necessarily keep unaltered the Dehn invariants, see [21]; (e) the notion of a flexible polyhedron can be introduced in all spaces of constant curvature of dimension 3 and above (see [36], [37], [16], [17], [19], as well as in pseudo-Euclidean spaces of dimension 3 and above (see [3]); moreover, it is known that in many of these spaces flexible polyhedra do exist and possess properties similar to properties (a)-(d) (see [24], [14], [15], [18], [20]).…”