Configuration Spaces of Tensegrities — Problem 9

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a simple graph with labeled vertices $1,\ldots,n$. A configuration is $P=(p_1,\ldots,p_n)\in(\mathbb R^d)^n$; coincident vertices are allowed. A self-stress is a real edge assignment $w_{ij}=w_{ji}$, set to zero on missing edges, satisfying equilibrium at every vertex: \[\sum_{j\ne i}w_{ij}(p_j-p_i)=0\quad(1\leq i\leq n).\] Let $W(G,P)$ be the vector space of all such self-stresses. Negative and positive edge values are the source’s cable and strut signs. The base $B_d(G)$ is the full labeled configuration space $(\mathbb R^d)^n$, stratified by the following fiber equivalence: $W(G,P)$ and $W(G,Q)$ are equivalent if a homeomorphism between them preserves the sign of every edge coordinate of every stress. A stratum is a maximal connected set of configurations with equivalent fibers; these strata are semialgebraic. This base is not quotiented by Euclidean or affine motions. Write $K_n$ for the complete graph on $n$ labeled vertices. In the planar model, the ordered geometric variables lie in the real projective plane $\mathbb{RP}^2=\mathbb R^2\cup\ell_\infty$, which adds a line of directions so that parallel affine lines meet at a point at infinity. Elementary conditions are coincidence of two points, collinearity of three points, and the five-point intersection relation $p_i=[p_j,p_{j\prime};p_k,p_{k\prime}]$. The last is the closure in $(\mathbb{RP}^2)^5$ of the relation $p_i=(p_jp_{j\prime})\cap(p_kp_{k\prime})$ where the two joining lines are well-defined and distinct. This closed relation also covers degenerate endpoint pairs where a joining line is undefined. For example, when both joining lines coincide, $p_i$ may be any point on their common line. A finite system may introduce auxiliary points in $\mathbb{RP}^2$; its conditional solutions among the original points are those for which suitable projective auxiliary points exist. For a graph $G$ and $k\geq1$, the source calls $\{P:\dim W(G,P)\geq k\}$ a $(G,k)$-stratum (possibly a union of base strata). It is geometric if it is a finite union of conditional-solution sets of such geometric systems. These planar descriptions motivate the higher-dimensional question.

Statement recorded in the supplied source.

Develop a theory of geometric conditions describing tensegrity strata in dimensions $d\geq3$; in particular, find elementary geometric conditions in three and higher dimensions analogous to the planar incidence conditions and their conditional systems. [S2]

Short English statement

Develop higher-dimensional geometric incidence descriptions of tensegrity strata, extending the planar theory.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6900009 (AMR-068-0009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Franck Doray, Oleg Karpenkov and Jan Schepers, Geometry of configuration spaces of tensegrities, arXiv:0806.4976v2 (2008), Sections 2 and 5.2--5.3, Definitions 2.1 and 2.6, Theorem 2.8, Definitions 5.1 and 5.4 and Problem 3, pp. 3--6 and 17--18. https://arxiv.org/pdf/0806.4976v2.