1999/02/05 by Erdahl, Robert M., Rybnikov, Konstantine, Ryshkov, Sergeĭ S.
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.math/9902160
We show how a d-stress on a piecewise-linear realization of an oriented (non-simplicial, in general) d-manifold in \rd 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 constructed mappings are not linear, but polynomial. In 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 partial analog of Maxwell correspondence for self-stresses in spatial frameworks and vertical projections of 3-dimensional surfaces based on our construction of polynomial mappings. Applying this theorem we derive a class of three-dimensional spider webs similar to the family of two-dimensional spider webs described by Maxwell. In addition, we conjecture an important property of our mappings which is supported by a heuristic count based on the lower bound theorem (g2(d+1)=dim Stress2 ≥ 0) for d-pseudomanifolds generically realized in \Rd+1 (Fogelsanger).