On Traces ofd-stresses in the Skeletons of Lower Dimensions of Piecewise-lineard-manifolds

Abstract

We show how a d-stress on a piecewise-linear realization of an oriented (non-simplicial, in general) d-manifold in R d naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The mappings we construct are not linear, but polynomial. In the 1860-70s J. C. Maxwell described an interesting relationship between self-stresses in planar frameworks and vertical projections of polyhedral 2-surfaces. We offer a spatial analog of Maxwell's correspondence based on our polynomial mappings. By applying our main result we derive a class of three-dimensional spider webs similar to the two-dimensional spider webs described by Maxwell. We also conjecture an important property of our mappings that is based on the lower bound theorem (g 2 (d +1) = dim Str ess 2 ≥ 0) for d-pseudomanifolds generically realized in R d+1 [10].

Rigidity and scene analysis

“…For example, for Voronoi diagrams and Delaunay simplicial complexes, the edges of one are perpendicular to facets of the other, in all dimensions. Moreover, for appropriate sphere-like homology, the existence of a reciprocal corresponds to the existence of nontrivial liftings [CW94,ERR01,Ryb99]. Such geometric structures are also related to k-rigidity and to combinatorial proofs of the g-theorem in polyhedral combinatorics [TW00].…”

Perfect Delaunay polytopes and perfect quadratic functions on lattices

“…In geometry of numbers and number theory this approach was pioneered by Hermite (1850). Korkine (Erdahl, 2001), and later reported by Rybnikov (2001) and Erdahl and Rybnikov (2002). Perfect Delaunay polytopes have been classified up to dimension 7 -Dutour (2004) proved that G 6 = 2 21 is the only perfect polytope for d = 6.…”

“…Laurent (1992-1997, in various combinations) found more examples of perfect Delaunay polytopes in dimensions 15, 16, 22, and 23, but all of those seemed to be sporadic. The first construction of infinite sequences of perfect Delaunay polytopes was described at the Conference dedicated to the Seventieth Birthday of Sergei Ryshkov (Erdahl, 2001), and later reported by Rybnikov (2001) and Erdahl and Rybnikov (2002). Perfect Delaunay polytopes have been classified up to dimension 7 -Dutour (2004) proved that G 6 = 2 21 is the only perfect polytope for d = 6.…”

Stresses and Liftings of Cell-Complexes

“…We conjecture that our mapping somehow generalizes a mapping constructed with algebraic methods by Lee in Theorem 6 of [26]. We will explore the connection between stresses on skeletons of different dimensions in a subsequent paper [24].…”

“…A realization of the torus depicted in Fig. 3 demonstrates that in general one cannot drop the condition H 1 (M d , Z 2 ) = 0 (triangles (123), (456) are not loaded; edges (16), (24), (35) have arbitrary tensions, which are resolved by other edges; since lines 16, 24, 35 do not pass through a common point, the torus does not lift).…”