Geometry of configuration spaces of tensegrities

TL;DR

This paper explores the geometric conditions and stratification of configuration spaces for tensegrities based on underlying graphs, providing new insights into their structure and relations through graph surgeries and examples.

Contribution

It introduces a natural stratification of configuration spaces for tensegrities, investigates graph surgeries relating different strata, and offers a conjecture on the sufficiency of geometric conditions for plane tensegrities.

Findings

Description of geometric conditions for plane tensegrities

Relations between different strata via graph surgeries

Examples of strata for small tensegrity configurations

Abstract

Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in R^d to admit a tensegrity with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in R^d. In particular we find surgeries on graphs that give relations between different strata. Based on numerous examples we give a description of geometric conditions defining the strata for plane tensegrities, we conjecture that the list of such conditions is sufficient to describe any stratum. We conclude the paper with particular examples of strata for tensegrities in the plane with a small number of vertices.