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Received: 8 June 2017 / Revised: 13 January 2018 / Accepted: 22 January 2018

Open Problems on
Configuration Spaces of Tensegrities

Oleg Karpenkov Department of Mathematical Sciences,

University of Liverpool,

Mathematical Sciences Building,

Liverpool L69 7ZL UK

karpenk@liverpool.ac.uk

University of Liverpool,

Mathematical Sciences Building,

Liverpool L69 7ZL UK

karpenk@liverpool.ac.uk

In this small paper we bring together some open problems related to the study
of the configuration spaces of tensegrities, i.e. graphs with stresses on edges.
These problems were announced in Doray et al. (Discrete Comput Geom
43:436–466, [Doray et al.2010]), Karpenkov et al. (ARS Math
Contemp 6:305–322, [Karpenkov et al.2013]), Karpenkov (The
combinatorial geometry of stresses in frameworks. arXiv:1512.02563 [math.MG],
[Karpenkov2017]), and Karpenkov (Geometric
Conditions of Rigidity in Nongeneric settings, [Karpenkov2016]) (by F. Doray, J. Schepers,
B. Servatius, and the author), for more details we refer to the mentioned
articles.

The subject of tensegrities was first considered by J.C. Maxwell in [Maxwell1864], who started to investigate first questions regarding force-loads for frameworks. Nowadays tensegrities are one of the leading directions of study in modern theory of rigidity (see, e.g., [Connelly1993] for further information). Let us recall several standard definitions.

- A configuration is a finite collection $ P$ of $ n$ labeled points $ (p_1,p_2,\ldots,p_n)$, where each point $ p_i$ is in a fixed Euclidean space $ \r^d$.
- The realization of $ G$ with straight edges, induced by mapping $ v_j$ to $ p_j$ is called a tensegrity framework and it is denoted as $ G(P)$. (Here we allow the realization to have self-intersections).
- A stress $ w$ on a framework is an assignment of real scalars $ w_{i,j}$ (called tensions ) to its edges $ p_ip_j$.
- A stress $ w$ is called a self-stress if at every vertex $ p_i$ we have \begin{eqnarray*} \sum\limits_{\{j|j\ne i\}} w_{i,j}(p_j-p_i)=0. \end{eqnarray*}
- A pair $ (G(P),w)$ is a tensegrity if $ w$ is a self-stress for the framework $ G(P)$.
- If $ w_{i,j} {< } 0$ then we call the edge $ p_ip_j$ a cable , if $ w_{i,j} {> } 0$ we call it a strut .

Denote by $ B_d(G)=(\r^d)^n$ the configuration space of all tensegrity frameworks. Let $ W(n)$ denote the linear space with coordinates $ w_{i,j}$ where $ 1\le i,j\le n$. It is clear that $ \dim W(n)=n^2$.

The described equivalence relation gives us a stratification of $ B_d(G)=(\r^d)^n.$ A stratum is by definition a maximal connected component of $ B_d(G)$ with equivalent fibers. In [Doray et al.2010] we prove that all strata are semialgebraic sets.

First we study a particular case $ x_1=0$, which we denote by $ B_1^0(K_3)$. The set $ B_1^0(K_3)$ has the following stratification (see Fig. 1, Left):

- 1 stratum of codimension two: the origin. Here all three vertices coincide and the dimension of the fiber is 3.
- 6 strata of codimension one: some pair of vertices coincide. The dimension of fiber is two.
- 6 connected components of full dimension correspond to triple of distinct vertices. The dimension of a fiber is one.

It is clear that the dimension of the fiber for $ (x_1,x_2,x_3)$ coincides with the dimension of the fiber $ (0,x_2-x_1,x_3-x_1)$ for every $ x_1,x_2,x_3$. Therefore, we have \begin{eqnarray*} B_1(K_3)=B^0_1(K_3)\times \r^1. \end{eqnarray*}

In Example 1.4 above we discussed a stratification of $ B_2(K_n)$. The most important information on the stratification contains its combinatorial structure, namely the list of strata of different dimensions and the adjacency diagram for the strata. For the combinatorial description of $ B_2(K_n)$ as in Example 1.4, for $ n=2,3,4,5$ we refer to [Karpenkov et al.2013]. There is not much known about more complicated configuration spaces. The next simplest and most interesting unstudied cases are listed in the following problem.

As experiments show, for every codimension 1 stratum there exists a certain subgraph that locally identifies the stratum (i.e., for every point $ x$ of the stratum there exists a neighborhood $ B(x)$ such that every configuration in $ B(x)$ has a nonzero self-stress for the subgraph if and only if this point is on the stratum). This observation is valid for all dimensions $ d$.

Let us say a few words about two-dimensional tensegrities. In the paper [Doray et al.2010] one can find the classification of all strata of codimension 1 for $ n\le 8$ points. For further examples, see the papers of [White and Whiteley1983], and [de Guzmán and Orden2004], [Guzmán and Orden2006]. In the paper [Karpenkov2017] it was shown how to approach every stratum for the case $ n=9$. The next case which contains unknown strata is $ n=10$ (see also Problem 8 below).

In many cases the strata for different graphs coincide. So it is natural to ask the following question.

Finally the following question remains open.

Here we have only a trivial example of a graph on 2 vertices and one edge in the plane. It has one generic stratum of full dimension and one stratum of codimension 2, corresponding to two coinciding points.

Majority of known examples in the planar case are expressed in terms of Cayley algebra. Recall that the objects of (planar) Cayley algebra are points and lines in the plane. Cayley algebra has two major operations:

- join operation $ \vee$ is defined for a pair of points. The resulting object is the line joining these points.
- meet operation $ \wedge$ is defined for a pair of lines. The meet of two lines is the intersection point of two lines.

It turns out that many geometric conditions can be expressed in terms of Cayley operations. Let us illustrate this with the following example.

In fact in the above example, the property of 6 points to lie on one conic does not depend on the order of these points. Therefore, there are 60 different Cayley algebra systems defining the same stratum. This lead to the following important open problem.

This problem is a kind of a question on finding generators and relations for the set of all conditions.

One of the main long-standing open problems on the Cayley strata description is as follows.

Recently this problem was solved in a weaker settings of extended Cayley algebra in [Karpenkov2017]. Nevertheless it is not clear if it is possible to avoid additional elements involved in the construction of [Karpenkov2017]. Here is an example of a graph for which the systems describing codimension 1 strata are not known.

Currently this example is a strong candidate for a counterexample to Problem 7.

There is almost nothing known in multidimensional case.

We refer to [White and Whiteley1983] for examples of geometric conditions in dimension 3.

[Connelly1993] Connelly, R.: Rigidity. In: Gruber, P.M., Wills, J.M. (eds.)
Handbook of Convex Geometry, vol. A
, pp. 223–271. North-Holland, Amsterdam (1993)

[de Guzmán and Orden2004] de Guzmán, M., Orden, D.: Finding tensegrity structures: Geometric and symbolic aproaches, In: Proceedings of EACA-2004, p 167–172 (2004)

[Guzmán and Orden2006] de Guzmán, M., Orden, D.: From graphs to tensegrity structures: geometric and symbolic approaches. Publ. Mat. 50 , 279–299 (2006)

[Doray et al.2010] Doray, F., Karpenkov, O., Schepers, J.: Geometry of configuration spaces of tensegrities. Discrete Comput. Geom. 43 (2), 436–466 (2010)

[Doubilet et al.1974] Doubilet, P., Rota, G.-C., Stein, J.: On the foundations of combinatorial theory. IX. Combinatorial methods in invariant theory, Studies. Appl. Math. 53 , 185–216 (1974)

[Karpenkov2017] Karpenkov, O.: The combinatorial geometry of stresses in frameworks, preprint (2017), arXiv:1512.02563 [math.MG]

[Karpenkov2016] Karpenkov, O.: Geometric conditions of rigidity in nongeneric settings. In: Sitharam, M., St. John, A., Sidman, J. (eds.) Handbook of Geometric Constraint Systems Principles, chapter 15 (2018) (accepted)

[Karpenkov et al.2013] Karpenkov, O., Schepers, J., Servatius, B.: On stratifications for planar tensegrities with a small number of vertices. ARS Math Contemp 6 (2), 305–322 (2013)

[Li2008] Li, H.: Invariant algebras and geometric reasoning. With a foreword by David Hestenes. World Scientific Publishing, Hackensack (2008)

[Maxwell1864] Maxwell, J.C.: On reciprocal figures and diagrams of forces. Philos. Mag. 4 (27), 250–261 (1864)

[White and McMillan1988] White, N.L., McMillan, T.: Cayley factorization. Symbolic and algebraic computation, Rome, pp. 521–533. Springer, Berlin (1988). (Lecture Notes in Comput. Sci., 358 1989)

[White and Whiteley1983] White, N.L., Whiteley, W.: The algebraic geometry of stresses in frameworks. SIAM J. Alg. Disc. Meth. 4 (4), 481–511 (1983)

[de Guzmán and Orden2004] de Guzmán, M., Orden, D.: Finding tensegrity structures: Geometric and symbolic aproaches, In: Proceedings of EACA-2004, p 167–172 (2004)

[Guzmán and Orden2006] de Guzmán, M., Orden, D.: From graphs to tensegrity structures: geometric and symbolic approaches. Publ. Mat. 50 , 279–299 (2006)

[Doray et al.2010] Doray, F., Karpenkov, O., Schepers, J.: Geometry of configuration spaces of tensegrities. Discrete Comput. Geom. 43 (2), 436–466 (2010)

[Doubilet et al.1974] Doubilet, P., Rota, G.-C., Stein, J.: On the foundations of combinatorial theory. IX. Combinatorial methods in invariant theory, Studies. Appl. Math. 53 , 185–216 (1974)

[Karpenkov2017] Karpenkov, O.: The combinatorial geometry of stresses in frameworks, preprint (2017), arXiv:1512.02563 [math.MG]

[Karpenkov2016] Karpenkov, O.: Geometric conditions of rigidity in nongeneric settings. In: Sitharam, M., St. John, A., Sidman, J. (eds.) Handbook of Geometric Constraint Systems Principles, chapter 15 (2018) (accepted)

[Karpenkov et al.2013] Karpenkov, O., Schepers, J., Servatius, B.: On stratifications for planar tensegrities with a small number of vertices. ARS Math Contemp 6 (2), 305–322 (2013)

[Li2008] Li, H.: Invariant algebras and geometric reasoning. With a foreword by David Hestenes. World Scientific Publishing, Hackensack (2008)

[Maxwell1864] Maxwell, J.C.: On reciprocal figures and diagrams of forces. Philos. Mag. 4 (27), 250–261 (1864)

[White and McMillan1988] White, N.L., McMillan, T.: Cayley factorization. Symbolic and algebraic computation, Rome, pp. 521–533. Springer, Berlin (1988). (Lecture Notes in Comput. Sci., 358 1989)

[White and Whiteley1983] White, N.L., Whiteley, W.: The algebraic geometry of stresses in frameworks. SIAM J. Alg. Disc. Meth. 4 (4), 481–511 (1983)

- 00
- × Connelly, R.: Rigidity. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A , pp. 223–271. North-Holland, Amsterdam (1993)
- × de Guzmán, M., Orden, D.: Finding tensegrity structures: Geometric and symbolic aproaches, In: Proceedings of EACA-2004, p 167–172 (2004)
- × de Guzmán, M., Orden, D.: From graphs to tensegrity structures: geometric and symbolic approaches. Publ. Mat. 50 , 279–299 (2006)
- × Doray, F., Karpenkov, O., Schepers, J.: Geometry of configuration spaces of tensegrities. Discrete Comput. Geom. 43 (2), 436–466 (2010)
- × Doubilet, P., Rota, G.-C., Stein, J.: On the foundations of combinatorial theory. IX. Combinatorial methods in invariant theory, Studies. Appl. Math. 53 , 185–216 (1974)
- × Karpenkov, O.: The combinatorial geometry of stresses in frameworks, preprint (2017), arXiv:1512.02563 [math.MG]
- × Karpenkov, O.: Geometric conditions of rigidity in nongeneric settings. In: Sitharam, M., St. John, A., Sidman, J. (eds.) Handbook of Geometric Constraint Systems Principles, chapter 15 (2018) (accepted)
- × Karpenkov, O., Schepers, J., Servatius, B.: On stratifications for planar tensegrities with a small number of vertices. ARS Math Contemp 6 (2), 305–322 (2013)
- × Li, H.: Invariant algebras and geometric reasoning. With a foreword by David Hestenes. World Scientific Publishing, Hackensack (2008)
- × Maxwell, J.C.: On reciprocal figures and diagrams of forces. Philos. Mag. 4 (27), 250–261 (1864)
- × White, N.L., McMillan, T.: Cayley factorization. Symbolic and algebraic computation, Rome, pp. 521–533. Springer, Berlin (1988). (Lecture Notes in Comput. Sci., 358 1989)
- × White, N.L., Whiteley, W.: The algebraic geometry of stresses in frameworks. SIAM J. Alg. Disc. Meth. 4 (4), 481–511 (1983)

1.1. Tensegrities

2. Combinatoric Properties of Stratification
Definition 1.1

1.2. Configuration Space of Tensegrities and its
Stratification
Definition 1.2 Definition
1.3 Example
1.4 Figure 1 Example
1.5

Problem 1 Remark
2.1 Problem 2
Problem 3
Problem 4
Problem 5

3. Geometric Conditions Defining Strata in $ \r^2$
Figure 2 Problem 6
Problem 7
Problem 8
Problem 9