?W H A T I S . . .

a Tensegrity?Robert Connelly

In the late 1940s, a young artist named Ken-neth Snelson showed some of his string-and-sticksculptures to R. Buckminster Fuller. Out of thisinteraction the term “tensegrity” was born. Thesesculptures were quite surprising. The sticks weresuspended in midair and supported by thin wiresthat were almost invisible. Fuller chose the wordtensegrity to describe such structures because oftheir tensional integrity, and it has stuck through-out the years. The following shows a picture ofone of the first small tensegrities that Snelsonmade with just three sticks (taken from his Webpage). Since then, Snelson has made numerousother tensegrity sculptures, many quite large, andthey are on display throughout the world.

But why do these structures hold up? Why arethey rigid? What does it mean to be rigid? How canwe model them mathematically?

The natural model is to define a tensegrity as afinite graph G whose vertices p = (p1, . . . ,pn), theconfiguration, are points in some Euclidean spaceEd with two types of edges labeled cables, corre-sponding to the strings, and struts, correspondingto the sticks. The whole tensegrity is denoted asG(p). The cables are constrained not to get longer,while the struts are constrained not to get shorter.A tensegrity G(p) is defined to be rigid if, for any

Robert Connelly is professor of mathematics at CornellUniversity. His email address is connelly@math.cornell.edu.

Research supported in part by NSF grant No. DMS–0209595(USA).

DOI: http://dx.doi.org/10.1090/noti933

other configuration q in Ed sufficiently close to theconfiguration p and satisfying the cable and strutsconstraints of G(p), q is rigidly congruent to p.Some examples are below. Note that cables andstruts can cross with no effect. A flexible tensegrityis one that is not rigid, and it necessarily has asmooth motion that is not a rigid motion of thewhole tensegrity.

Rigid Flexible

In the plane

Struts are shown as solid line segments andcables as dashed line segments.

Static RigidityThere are two principal methods to show rigidityof a tensegrity. Both methods involve the notion ofa stress in the structure, which, for mathematicalpurposes, is just a scalar ωij =ωji associated toevery cable or strut connecting the ith vertex pito the jth vertex pj . We set ωji = 0 when i, j isnot an edge of G. A stress is proper if ωij ≥ 0 fora cable and ωij ≤ 0 for a strut. The first methodis derived from the linearization of the distanceconstraints. At each vertex pi consider a forcevector Fi , the equilibrium load, such that theseforces do not have any linear or angular momentumon the configuration. This means, summing overall i, that ∑

iFi = 0 and

∑i

Fi ∧ pi = 0.

In dimension 2 or 3 the wedge product can bereplaced by the usual cross product. A tensegrityG(p) is statically rigid if every equilibrium loadF = (F1, . . . ,Fn) can be resolved by a proper stress

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ω = (. . . ,ωij , . . . ) in the sense that, for each i,ωij =ωji and

Fi +∑jωij(pj − pi) = 0.

This condition is then equivalent to there beinga solution to a system of linear equations withlinear inequality constraints on the stress ω,namely, a linear programming feasibility problem.In particular, if there are too few cables and struts,there will always be an equilibrium force thatcannot be resolved by a proper stress. Supposea tensegrity with n vertices and m cables andstruts is statically rigid in dimension d. Thereare dn coordinate variables, and d(d + 1)/2 is thedimension of the equilibrium loads (assuming theconfiguration does not lie in a lower-dimensionalsubspace). Then m ≥ dn− d(d + 1)/2+ 1. (The +1arises due to inequality constraints.) In dimension2, m ≥ 2n− 2, and in dimension 3, m ≥ 3n− 5. Itis a basic result that, for a tensegrity, static rigidityis sufficient but not necessary for rigidity.

Prestress StabilityFor example, the Snelson tensegrity in the firstfigure has n = 6 and m = 12 < 3n − 5, so itis not statically rigid. Nevertheless, it is rigid.Indeed, many of the tensegrities constructed byartists are statically underbraced. Although, froman engineering perspective, nonstatically rigidstructures are often considered too soft to bedependable, we still want to detect and analyzesuch tensegrities. Basically, we can ensure that thetensegrity is rigid if the configuration minimizesan underlying energy function, which dependsonly on the lengths of the cables and struts, andif the minimizing configuration is unique up torigid motions of the whole space. For example,static rigidity can be regarded as achieving a localminimum for reasonable choices of an energyfunction. Although the Snelson tensegrity andothers like it have external loads that cannot beresolved at the given configuration, the tensegritydeforms slightly to resolve them. Indeed, even astatically rigid structure will deform under anynonzero load. When an energy function at thegiven configuration is at a local minimum due tothe second derivative test, as in analysis, then thetensegrity is said to be prestress stable.

A self-stress for a tensegrity is a stress thatresolves the zero load. Suppose that ω is aproper self stress for a tensegrity G(p). Fix ω.Define a quadratic form E(p) on the space of allconfigurations p in Ed by

E(p) =∑i<jωij |pi − pj |2.

It turns out that the matrix of E is the Kroneckerproduct of d copies of an n-by-n matrix Ω with

identity, where n is the number of vertices of thegraph G, since the energy decouples into the valueon each coordinate separately. For i 6= j , the i, jthentry of Ω is −ωij , while the diagonal entries aresuch that the row and column sums of Ω are 0.

The stress energy E and the associated stressmatrix Ω have some interesting properties.

1) If ω is a proper self-stress for the tenseg-rity G(p) and the vertices of p span ad-dimensional (affine) linear subspace ofEd , then the kernel of the associated stressmatrix Ω is at least (d + 1)-dimensional.

2) If ω is a proper self-stress for the tensegrityG(p) in Ed , the kernel of the associated stressmatrix Ω is (d + 1)-dimensional, and G(q) isanother tensegrity for another configurationq with the same self stress ω, then q is anaffine image of p.

3) If ω is a proper self-stress for the tensegrityG(p) in Ed , the kernel of the associated stressmatrixΩ is (d+1)-dimensional,Ω is positivesemidefinite, and G(q) is a tensegrity for aconfiguration q with cables no longer andstruts no shorter, then q is an affine imageof p.

Note that Property 3 does not quite say that thetensegrity is rigid, just that it is rigid up to affinemotions, and they must preserve the lengths of theedges i, j, where ωij 6= 0. Nevertheless, this isclose enough in many circumstances. For a tenseg-rity G(p) with a self-stress ω, its stressed edgedirections are lines through the origin determinedby the vectors pi − pj , where ωij 6= 0. Think ofthese lines as points in the projective space RPd−1

of dimension d − 1. We say that the stressed edgedirections of a tensegrity G(p) with self-stress ωlie on a conic at infinity if, as points in RPd−1, theylie on a conic. Then it is a pleasant exercise to showthat, if the stressed edge directions of a tensegrityG(p) do not lie on a conic at infinity, then there isno affine map of the configuration p that satisfiesthe cable and strut constraints on the stressededges other than a rigid motion of the whole spaceEd , a congruence.

Putting this all together, we get the followingbasic result.

Theorem 1. Suppose that G(p) is a tensegrity inEd with n vertices and a proper self-stress ω withassociated stress matrix Ω, and the following hold:

1) The rank of Ω is n− d − 1.2) Ω is positive semidefinite.3) The stressed directions of G(p) do not lie on

a conic at infinity.

If q is another configuration in any ED ⊃ Ed satis-fying the cable and strut constraints of G(p), thenq is congruent to p.

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I call any tensegrity that satisfies the threeconditions of Theorem 1 superstable, and this sortof stability is a strong example of a structure beingprestress stable. Quite a few of the tensegrities ofSnelson and other artists are superstable. From ageometric point of view, the very strong conclusionof Theorem 1 about global reconfigurations is alsointeresting.

Global RigidityAn edge of a tensegrity can also be regarded asboth a cable and a strut, in which case it is calleda bar ; and if all the edges are bars, it is calleda bar framework. In other words, bars are notpermitted to change their lengths at all. Froman engineering perspective, a tensegrity is justa bar framework where some bars can supportonly tension—these are the cables—and others cansupport only compression—these are the struts.A bar framework, or tensegrity, G(p) is calledglobally rigid in Ed if any other configurationq = (q1, . . . ,qn) in Ed that satisfies the constraintsof G(p) is congruent to p. While Theorem 1 givesconditions for a tensegrity to be globally rigid inall dimensions, the following result (see [3]) is forbar frameworks in a fixed Ed . A configuration pin Ed is generic, which means it is typical, if thecoordinates of all of the points do not satisfy anynonzero polynomial with integer coefficients.

Theorem 2. A bar framework G(p) with p =(p1, . . . ,pn) generic is globally rigid in Ed if andonly if G is the complete graph (all vertices con-nected to all others) on d + 1 or fewer vertices or ithas a self-stress ω and corresponding stress matrixΩ of rank n− d − 1.

There has been a lot of activity applying theideas here to packing, granular materials, andpoint location in computational geometry, forexample. One instance is that packings of sphericaldisks in a polyhedral container can be regarded astensegrities with all struts connecting centers oftouching disks to each other and the boundary ofthe container. Some of the results described herecan be found in [1], [2], [3].

References1. R. Connelly and A. Back, Mathematics and tensegrity,

American Scientist 86 (1998), no. 2, 142–151.2. Robert Connelly and Walter Whiteley, Second-order

rigidity and prestress stability for tensegrity frame-works, SIAM J. Discrete Math. 9 (1996), no. 3, 453–491.MR1402190 (97e:52037)

3. Steven J. Gortler, Alexander D. Healy, andDylan P. Thurston, Characterizing generic globalrigidity, Amer. J. Math. 132 (2010), no. 4, 897–939.MR2663644 (2011i:52041)

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