Tensegrities and Global Rigidity by Connelly 2009

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Tensegrities and Global Rigidity

R. Connelly Department of Mathematics, Cornell University

Ithaca, NY 14853, USA

May 8, 2009

Abstract

A tensegrity is nite conguration of points in E d suspended rigidlyby inextendable cables and incompressable struts. Here it is explainedhow a stress-energy function, given by a symmetric stress matrix, canbe used to create tensegrities that are globally rigid in the sense thatthe only congurations that satisfy the cable and strut constraints arecongruent copies.

1 IntroductionIn 1947 a young artist, Kenneth Snelson, was intrigued with a particularstructure that he invented. It was a few sticks that were suspended rigidlyin midair without touching each other. It seemed like a magic trick. Whenhe showed this to the entrepreneur, builder, visionary, and self-styled math-ematician, R. Buckminster Fuller, he was inspired to call it a tensegrity be-cause of its tensional integrity. Fuller talked about them and wrote aboutthem extensively. Snelson went on to build a great variety of fascinatingtensegrity sculptures all over the world including a 60 foot work of art at theHirschhorn museum in Washington, DC. as shown in Figure 1.

Why did these tensegrities hold up? What were the geometric principles?They were often under-braced, and they seemed to need a lot of tension fortheir stability. So Fullers name, tensegrity, is quite appropriate.

Research supported in part by NSF Grant No. DMS0209595 (USA).e-mail: [email protected]

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Figure 1:

My proposal is that there is a very reasonable and pleasant model to de-scribe the stability of most of the tensegrities that Snelson and others havebuilt. There are results that can be used to predict the stability of a tenseg-rity, and there is a calculation that seems to reasonably imply stability, butalso to create tensegrities that are stable. In the following, up to Section 8,there will be a self -contained elementary development of a set of principlesthat can be used to understand many of the Snelson-like tensegrities. Thisrelies on the properties of the stress matrix, discussed in Section 5. Then inSection 8 the properties of the stress matrix are applied to generic congu-rations of bar tensegrities (usually called bar frameworks), where there havebeen a lot of exciting new results recently, and the ideas will be outlined.

The discussion here emphasizes the stress matrix and the stress-energyfunctional, and largely ignores the rst-order theory, about which a lot hasbeen written.

There are several quite interesting applications of the theory of tenseg-

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rities. Of course, there is a natural application to structural engineering,

where the pin-jointed bar-and-joint model is appropriate for an endless col-lection of structures. See [34, 1, 37] for example. See [31, 38, 35, 36, 32] forthe rst-order theory, and see [17] for the more general approach that com-bines the rst-order theory and the stress matrix approach that is developedhere. In computational geometry, there was the carpenters rule conjecture,inspired by a problem in robot arm manipulation. This proposes that a non-intersecting polygonal chain in the plane can be straightened, keeping theedge lengths xed, without creating any self-intersections. The key idea inthat problem uses basic tools in the theory of (rst-order) tensegrity struc-tures. See [14, 13] as well as Subsection 7.8 here. Granular materials of hard spherical disks can be reasonably modeled as tensegrities, where all themembers are struts. Again the theory of tensegrities can be applied to pre-dict behavior and provide the mathematical basis for computer simulationsas well as predict the distribution of internal stresses. See [19].

2 Notation

Formally dene a tensegrity as a nite set of labeled points called nodes,where some pairs of the nodes are connected with inextendable cables, somepairs of nodes are connected with incompressible struts , and some pairs of nodes are connected with inextendable, incompressible bars . The cables,struts, and bars are all called members of the tensegrity. A continuous mo-tion of the nodes, starting at the given conguration of a tensegrity, where themember constraints are satised, is called a ex of the tensegrity. Any con-guration of points has continuous exes, such as rotations, translations, andtheir compositions that are restrictions of congruences of the whole space.These are called trivial exes of the tensegrity. If the tensegrity has onlythe trivial exes that satisfy the member constraints, then it is rigid in E d .Otherwise it is called exible. Note that members can cross one another, in-tersect, and we are not concerned with what materials one might use to builda tensegrity that enforce the member constraints. This is a purely geometric

object. Figure 2 show some examples of rigid and exible tensegrities in theplane and space.The rigid tensegrity in space in Figure 2 is one of the original objects

made by Snelson. It is quite simple but suspends three sticks, the struts,rigidly without any pair of them touching. Indeed Snelson does not like to

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FlexibleRigid

In space

In the plane

Figure 2: Nodes are denoted by small round points, cables by dashed linesegments, struts by solid line segments, and bars by thin line segments.

call an object made of cables and struts a tensegrity unless all the struts arecompletely disjoint, even at their nodes. If a tensegrity, by the denitionhere, is such that the struts are disjoint, while all the other members arecables, it will be called a pure tensegrity.

One can build many of the rigid tensegrities shown here with rubber (orplastic) bands for cables, and dowel rods with a slot at their ends serving asstruts or bars.

In what follows, there will be some discussion of techniques for computingthe rigidity of tensegrties. As a by-product of this analysis global rigidity,dened in Section 3, will emerge naturally.

Let G denote the underlying tensegrity graph of nodes, where the edgesof G, the members, are each labeled as cables, struts or bars. Let p =(p 1 , . . . , p n ) denote the conguration of nodes, where each p i is a vectorin E d . The whole tensegrity is denoted as G(p ). I regard this notation assomewhat bezarre, but it is helpful to distinguish between the congurationp and the way the pairs of nodes are connected with the three types of

members. There are occasions, where the graph G is not needed, and theconguration can stand alone by itself, and there are other times when onthe graph G is relevant, and the conguration is put in the background.

An important concept is the notion of a stress associated to a tensegrity,which is a scalar ij = ji associated to each member {i, j }of G. Call the

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vector = ( . . . , ij , . . . ), the stress . We can suppress the role of G here

by simply requiring that ij = 0 for any non-member {i, j }of G. A stress = ( . . . , ij , . . . ) is proper if ij 0 for a cable {i, j }and ij 0 for a strut{i, j }. There is no condition when {i, j }is a bar. We say a proper stress is strict if each ij = 0 when {i, j }is a cable or strut. One should be carefulhere, since in the paper [32], a proper stress is called what is strict and properhere. I prefer the denition here since it is convenient to not necessarily insistthat all proper stresses are strict. One should also be careful not to confusethe notion of stress here with that used in structure analysis, in physics or inengineering. There stress is dened as a force per cross-sectional area. In theset-up here there are no cross-sections; the scalar ij is better interpreted asa force per unit length.

Let = ( . . . , ij , . . . ) be a proper stress for a tensegrity graph G. Forany conguration p of nodes in E d , dene the stress-energy associated to as

E (p ) =i


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Taking derivatives and evaluating at t = 0, we get:

ddt

E (p + tp )|t =0 = 2i


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If the conguration p is in equilibrium with respect to the stress , then

so is any affine transformation q of p , as is seen by the following calculation:

j

ij (q j q i ) = j

ij (Ap j + b Ap i b ) = A j

ij (p j p i ) = 0 .So our stress-energy functional E cant see affine transformations, at leastat critical points. Of course we know that when something is globally rigid,it cannot exclude rigid congruences, but the group of affine transformationsare more than we would like. Notice that even projections, which are singu-lar affine transformations, also preserve equilibrium congurations. Indeed,the equilibrium formula (4) is true if and only if it is true for each coordi-nate, which is the same as being true for orthogonal projections onto each

coordinate axis.This brings us to the question, for a tensegrity G(p ) in E d , when is therean affine transformation that preserves the member constraints (2)? It isclear that the matrix A is the only relevant part. For us, it will turn out thatwe also only need to consider when the members are bars. If {i, j }is a barof G, then the matrix A determines a transformation that preserves that barlength if and only if the following holds:

(p i p j )2 = ( q i q j )

2

= ( Ap i Ap j )2

= [A(p i

p j )]T A(p i

p j )

= ( p i p j )T AT A(p i p j ),or equivalently,

(p i p j )T (AT A I d )(p i p j ) = 0 (5)where () T is the transpose operation, I d is the d-by-d identity matrix, andvectors are regarded as column vectors in this calculation. If Equation (5)holds for all bars in G(p ), we say it has a bar preserving affine image , whichis non-trival if A is not orthogonal. Similarly, G(p ) has a non-trivial affine ex if there is a continuous family of d-by-d matrices At , where A0 = I d , fort in some interval containing 0 such that each At satises Equation (5) for tin the interval.

This suggests the following denition. If v = {v 1 , . . . , v k}is a collectionof vectors in E d , we say that they lie on a quadric at innity if there is anon-zero symmetric d-by-d matrix Q such that for all v iv

v T i Qv i = 0 . (6)

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The reason for this terminology is that real projective space RP d 1 can be

regarded as the set of lines through the origin inE d

, and equation (6) is thedenition of a quadric in RP d 1 .Notice that since the denition of an orthogonal matrix A is that AT A I d = 0 , the affine transformation denes a quadric at innity if and only if

the affine transformation is not a congruence.Call the bar directions of a bar tensegrity the set {p i p j }, for {i, j }a barof G. With this terminology, Equation (6) says that if the member directions

of a bar tensegrity under an affine transformation A satisfy (5), they lie ona quadric at innity. Conversely suppose that the member directions of abar tensegrity G(p ) lie on a quadric at innity in E d given by a non-zerosymmetric matrix Q. By the spectral theorem for symmetric matrices, weknow that there is an orthogonal d-by-d matrix X = ( X T ) 1 such that:

X T QX =

1 0 0 00 2 0 00 0 3 0... ... ... . . . ...0 0 0 d

.

Let be the smallest i , and let + be the largest i . Note 1/


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affine image if and only if it has a non-trivial bar preserving affine ex if and

only if the bar directions lie on a quadric at innity.It is useful to consider when bar tensegrities have the bar directions that

lie on a quadric at innity. In E 2 , the quadric at innity consists of twodistinct directions. So a parallelogram or a grid of parallelograms have anon-trivial affine ex. In E 3 it is more interesting. The quadric at innityis a conic in RP 2 , the projective plane, and such a conic is determined by 5points. An interesting example is the bar tensegrity in Figure 5. The surface

(a) (b)

Figure 5: Figure (a) is the ruled hyperboloid given by x2 + y2 z2 = 1.Figure (b) is the attened version after an affine ex.

is obtained by taking the line ( x, 1, x) and rotating it about the z-axis. Thiscreates a ruling of the surface by disjoint lines. Similarly ( x, 1, x) createsanother ruling. Each line in one ruling intersects each line in the other rulingor they are parallel. A bar tensegrity is obtained by placing nodes where aline on one ruling intersects a line on the other ruling, and bars such thatthey join every pair of nodes that lie on same line on either ruling.

Consider the diagonal matrix Q with diagonal entries 1 = 2 = 1 , 3 =

1. When one node of each bar is translated to a single point, they alllie on a circle at innity given by Q. The ex given by Formula (7) exesthe conguration until the nodes lie on a line when t = 1 / + = 1 becausetwo of the eigenvalues for Q vanish for that value of t, and in the otherdirection, when t = 1 / = 1, the nodes lie in a plane. There is a pleasantdescription of this motion in [23] at the end of Chapter 1. This structure iseasy to build with dowel rods and rubber bands securing the joints wherethe rulings intersect.

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The space of d-by-d symmetric matrices is of dimension d + ( d2 d)/ 2 =d(d +1) / 2. So if the vector directions of a tensegrity are less than d(d + 1) / 2,then it is possible to nd a non-zero d-by-d symmetric matrix that satisesEquation (6), and then ex it into a lower dimensional subspace. This provesthe following, due to Barvinok in [2].

Theorem 2. If G(p ) is a bar framework in E D with less than d(d + 1) / 2bars, then it has a realization in E d with the same bar lengths.

This suggests the following denition: If a (bar) graph G is such that anyrealization G(p ), for a conguration p in some E D , implies that G have arealization in E d with the same bar lengths, then we say G is d-realizable.Note that this is a property of the graph G, and in order to qualify forbeing d-realizable, one has to be able to push a realization in E D down toa realization E d for ALL realizations in E D . For example, the 1-realizablegraphs are forests, graphs with no cycles. In particular, a triangle is not1-realizable.

This is inspired from a problem in nuclear magnetic resonance (NMR)spectroscopy. The atoms of a protein are tagged and some of the pairwisedistances are known. The problem is to identify a conguration in E 3 thatsatises those distance constraints. Finding such a conguration in E D , forsome large D is computationally feasible, and if G is 3-realizable, one canexpect to nd another conguration in E 3 that satises the distance con-

straints.For any graph G, another graph H is called a minor of G if it can be

obtained by edge contractions or deletions. In particular if a minor of agraph G is not d-realizable, then G itself is not d-realizable. It is easy tosee that a graph is 1-realizable if and only if it does not have a triangle as aminor. In other words, the triangle is a complete list of forbidden minors for1-realizability. It is not too hard to show that the graph K 4 , the tetrahedron,is also a complete list of forbidden minors for 2-realizability. In [4, 3] M. Belkand I show the following:

Theorem 3. A complete list of forbidden minors for 3-realizability is the set of two graphs, K 5 and edge graph of the regular octahedron.

This implies that there is a reasonable algorithm to detect 3-realizablityfor an abstract graph and, when the edge lengths are given, to nd a real-ization in E 3 . Tensegrity techniques are used in a signicant way in [3].

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5 The stress matrix

The stress-energy function E dened by (1) of Section 2 is really a quadraticform. It is an easy matter to compute the (symmetric) matrix associated tothat quadratic form. Note that the square of the difference of two variables,x1 and x2 is

(x1 x2 )2 = x1 x2 +1 1

1 +1x1x2 .

For each i = j , dene an n-by-n matrix ( i, j ), where the ( i, i ) and ( j,j )entries are 1, the ( i, j ) and ( j,i ) entries are 1, while are the other entriesare 0. Then

x1 x2 xni


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It is also convenient to rewrite the equilibrium condition (4) in terms of

matrices. Dene the conguration matrix for the conguration p as

P = p 1 p 2 p n1 1 1,

a (d + 1)-by- n matrix, and the equilibrium condition (4) is equivalent to

P = 0 .

Each coordinate of P as a row vector multiplied on the right by representsthe equilibrium condition in that coordinate. The last row of ones of P represent the condition that the column sums (and therefore the row sums)of are 0 . It is also easy to see that the linear rank of P is the same as thedimension of the affine span of p 1 , . . . , p n in E d .

Suppose that we add rows to P until all the rows span the co-kernel of . The corresponding conguration p will be called a universal conguration for (or equivalently ).

Proposition 4. If p is a universal conguration for , any other congura-tion q which is in equilibrium with respect to is an affine image of p .

Proof. Let Q be the conguration matrix for q . Since the rows of P area basis for the co-kernel of , and the rows of Q are, by denition, in theco-kernel of , there is a (d + 1)-by-( d + 1) matrix A such that AP = Q.Since P and Q share the last row of ones, we know that A takes the form

A = A0 b0 1 ,

where A0 is a d-by-d matrix, b is a 1-by-d matrix (a vector in E d ), and thelast row is all 0s except for the 1 in the lower right hand entry. Then we seethat for each i = 1 , . . . , n , q i = A0 p i + b , as desired.

The stress matrix plays a central role in what follows. Note that whenthe conguration p is universal, with affine span all of E d , for the stress ,the dimension of the co-kernel (which is the dimension of the kernel) of isd, and the rank of is n d 1. But even when the conguration p is notuniversal for , it is the projection of a universal conguration, and so therank n d 1.

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dimension i and d i such that their intersection is a common point, whichis a relative interior point of each simplex. They can also be used to denea super stable tensegrity as well. Write d+2k =1 k p k = 0 , where

d+2k =1 k = 0,

while k > 0 for k = 1 , . . . , i + 1, and k < 0 for k = i + 2 , . . . , d + 2. Thenthe following rank one matrix is the stress matrix

=

12...

d+2

1 2 d+2 ,

since for the conguration matrix P ,

P

12...

d+2

= 0 .

So the stress ij = i j . The edges of i and d i are struts, while all the

other members are cables. Since the rank is d + 2 d 1 = 1, and ispositive semi-denite, the tensegrity is super stable. Figure 7 shows the twoexamples in the plane and in three-space.

Figure 7:

This tensegrity has been described in [7].

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Figure 9: The prismic tensegrity P (6, 1).

7.6 Highly symmetric tensegritiesMany of the tensegrities created by artists such as Snelson have the superstable property discussed here. They need the stress for their stability. Theirtensional integrity is part of their stability. Symmetry seems to a natural partof art, so I thought it would be interesting to see what symmetric tensegritieswere super stable. It turns out that the symmetry simplies the calculationof the rank and deniteness of the stress matrix. In addition the theoryof the representations of nite groups is a natural tool that can be used todecompose the stress matrix. With Allen Back and later Robert Terrell,we created a website, where one can view and rotate the pictures of these

tensegrities. This is available at [40]. See also [11] for an explanation of thegroup theory and rigidity theory.The tensegrity graph G is chosen so that there is an underlying nite

group acting on the tensegrity such that the action of takes cables tocables and struts to struts, and the following conditions hold:

i.) The group acts transitively and freely on the nodes. In other words,for each p i , p j nodes, there is a unique element g such that gp i = p j .

ii.) There is one transitivity class of struts. In other words, if {p i , p j }and{p k , p l}are struts, then there is g such that {gp i , gp j }= {p k , p l}as sets.

iii.) There are exactly two transitivity classes of cables. In other words,all the cables are partitioned into two sets, where permutes the theelements of each set transitively, but no group element takes a cablefrom one partition to the other.

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The user must choose the abstract group, the group elements that correspond

to the cables, the group element that corresponds to the struts, and the ratioof the stresses on the two classes of cables. Then the tensegrity is rendered.Figure 10 shows a typical picture from the catalog.

Figure 10: A super stable tensegrity from the catalog [40]. In the catalog, thestruts are colored green, one cable transitivity class is colored red, and theother blue. In this example, the cables lie on the convex hull of the nodes,and struts are inside.

7.7 Compound tensegrities

The sum of positive semi-denite matrices is positive semi-denite. So we canglue two super stable tensegrities along some common nodes, and maintain

Condition 1.). Condition 3.) is no problem. But the rank Condition 2.) maybe violated. But even if Condition 2.) does not hold, each of the individualtensegrities will remain globally rigid, even if some of the stresses vanish onoverlapping members.

One example of this process is the delta-Y transformation. If one super

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stable tensegrity has a triangle of cables in it, one can add a tensegrity of the

form in the upper right of Figure 7 so the stresses on the overlap of the threestruts exactly cancel with the three cable stresses in the other tensegrity. Sothe three triangle cables are replaced the three other cables joined to a newnode inside the triangle. In this case the resulting tensegrity is still superstable since the radon tensegrity is planar and using Condition 3.). Figure 11shows how this might work for the top triangle of the Snelson tensegrity of Figure 2. Taken from [39] Figure 12 shows this replacement on both triangles.

=+

Figure 11:

Figure 12:

If the replacement as in Figure 12 is done for a polygon of with four ormore vertices, the resulting tensegrity may not be super stable or even rigid,but in [39] it is shown that if the polygons have an odd number of vertices andthe struts are placed as far away from the vertical cables as possible, then the

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resulting tensegrity is super stable. In other words, if the star construction is

done on P (2k + 1 , k) as in Figure 13 the resulting tensegrity is super stable.

Figure 13:

It is also possible to put two (or more) super stable together on a com-mon polygon to create a tensegrity with a stress matrix that satises Con-dition 1.) while the universal conguration is 4-dimensional instead of 3-dimensional. But each of the original pieces are universally globally rigid.The 4-dimensional realization has an affine ex around the 2-dimensionalpolygon used to glue the two pieces together. So the tensegrity has two non-congruent congurations in E 3 as one piece rotates about the other in E 4 .

Meanwhile struts and cable stresses can be arranged to cancel, and thus thosemembers are not needed in the compound tensegrity. Figure 14 shows thiswith two Snelson tensegrities combined along a planar hexagonal tensegrity.This is similar to the situation in [26] by T. Jordan and Z. Szabadka, wheresome pairs of nodes do not change their distances for other non-congruentrealizations.

This is something like the start of the Snelson tower of Figure 1, buthexagonal polygon in the middle is planar, which seems a bit surprising.This tensegrity is unyielding and rigid, but not super stable. But possiblyto create more stability Snelson includes more cables from one unit to theother, and this destroys the planarity of the hexagon.

There are many different ways to combine super stable units, possiblyerasing some of the members in the basic units to get similar rigid tensegrities.

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Figure 14:

7.8 Pure and exible examples

Recall that a pure tensegrity is one that has only cables and struts andthe struts are all disjoint. We have seen several examples in E 3 of puretensegrities, the simplest being Snelsons original as in Figure 11 on theleft. But what about the plane? One might be tempted to think that thetensegrity of Figure 15 is rigid, but it isnt. Indeed, there are no pure rigid

Figure 15:

tensegrities in the plane. This follows from the main theorem in [14], theproof of the carpenters rule property. This theorem says that any chainof non-overlaping edges in the plane can be continuously expanded (exed)

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until it is straight. This result also allows for disjoint edges, and at least,

for a short time, the expansion can be run backwards to be a contraction,keeping the struts at a xed length. See [13] for a discussion of expansiveexes from a tensegrity point of view.

8 Generic global rigidity

The congurations in previous sections must be constructed carefully. Whatabout a bar framework where the conguration is more general? It turns outthat the problem of determining when a bar framework is globally rigid isequivalent to a long list of problems known to be hard. See [33] for exam-

ple. The problem of whether a cyclic chain of edges in the line has anotherrealization with the same bar lengths, is equivalent to the uniqueness of asolution of the knapsack problem. This is one of the many problems on thelist of NP complete problems.

One way to avoid this difficulty, is to assume that the congurationscoordinates are generic. This means that the coordinates of p in E d are alge-braically independent over the rational numbers , which means that there is nonon-zero polynomial with rational coordinates satised by the coordinates of p . This implies, among other things, that no d +2 nodes lie in a hyperplane,for example, and a lot more. In [12] I proved the following:

Theorem 6. If p = ( p 1 , . . . , p n ) in E d

is generic and G(p ) is a rigid bar tensegrity in E d with a non-zero stress matrix of rank n d1, then G(p )is globally rigid in E d .Notice that the hypothesis includes Conditions 2 and 3 of Theorem 5.

The idea of the proof is to show that since the conguration p is generic,if G(q ) has the same bar lengths as G(p ), then they should have the samestresses. Then Proposition 1 applies.

Then recently in [20] the converse of Theorem 6 was shown as follows:

Theorem 7. If p = ( p 1 , . . . , p n ) in E d is generic and G(p ) is a globally rigid bar tensegrity in E d , then either G(p ) is a bar simplex or there is stressmatrix for G(p ) with rank n d 1.

The idea here, very roughly, is to show that a map from an appropriatequotient of an appropriate portion of the space of all congurations has eventopological degree when mapped into the space of edge lengths.

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As pointed out in [20], using these results it is possible to nd a polynomial

time numerical (probablistic) algorithm that calculates whether a given graphis generically globally rigid in E d , and that the property of being globally rigidis a generic property. In other words, if G(p ) is globally rigid in E d at onegeneric conguration p , it is globally rigid at all generic congurations.

Interestingly, it is also shown in [20] that if p is generic in E d , and G(q )has the same bar lengths in G(p ) in E d , then G(p ) can be exed to G(q ) inE d+1 , similar to the tensegrity mentioned in Subsection 7.7 of compoundedSnelson tensegrities.

A bar graph G is dened to be generically redundantly rigid in E d if G(p ) is rigid at a generic conguration p , and it remains rigid after theremoval of any bar. A graph is vertex k-connected if it takes the removal of at least k vertices to disconnect the rest of the vertices of G. The followingtheorem of Hendrickson [22], provides two necessary conditions for genericglobal rigidity.

Theorem 8. If p is a generic conguration in E d , and the bar tensegrity G(p ) is globally rigid in E d , then

a.) G is vertex (d + 1) -connected, and

b.) G(p ) is redundantly rigid in E d .

Condition a.) on vertex connectivity is clear since otherwise it is possibleto reect one component of G about the hyperplane determined by some d orfewer vertices. Condition b.) on redundant rigidity is natural since if, after abar {p i , p j }is removed, G(p ) is exible, one watches as the distance betweenp i and p j changes during the ex, and waits until the distance comes backto it original length. If p is generic to start with, the new conguration willbe not congruent to the original conguration.

Hendrickson conjectured that Conditions a.) and b.) were also sufficientfor generic global rigidity, but it turns out in [10] that the complete bipartitegraph K 5 ,5 in E 3 is a counterexample. This is easy to see as follows.

Similar to the analysis in Subsection 7.3 for each of the nodes for the

two partitions of K 5 ,5 consider the affine linear dependency5

i =1 i p i =0 , 5i =1 i = 0 and10i =6 i p i = 0 ,

10i =6 i = 0, where ( p 1 , . . . , p 5 ) and

(p 6 , . . . , p 10 ) are the two partitions of K 5 ,5 . According to [8], when the con-guration p = ( p 1 , . . . , p 10 ) is generic in E 3 , then, up to a scaling factor, the

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stress matrix for K 5 ,5 (p ) is

=

0 6...

10 1 5

1...

56 10 0

.

But the rank of is 2 < 10 3 1 = 6, while rank 6 is needed for genericglobal rigidity in this case by Theorem 7.In

E 3

, K 5 ,5 is the only counterexample to Hendricksons conjecture that Iknow of. On the other hand in [18] it is shown that a graph G is genericallyglobally rigid E d if an only if the cone over G is generically globally rigidin E d+1 . This gives more examples in dimensions greater than 3, and thereare some other bipartite graphs as well in higher dimensions by an argumentsimilar to the one here.

Meanwhile, the situation in the plane is better. Suppose G is a graph and

{i, j }is an edge of G, determined by nodes i and j . Remove this edge, addanother node k and join k to i, j , and d 1 distinct other nodes not i or j .This is called a Henneberg operation or sometimes edge splitting . Figure 16shows this operation on a tensegrity (although we are concerned here withbar frameworks).

b.)a.)

Figure 16: Graph a.) is split along the lower right inside edge, to get thegraph b.).

It is not hard to show that edge splitting, as in Figure 16 preserves genericglobal rigidity in E d . When the added node lies in the relative interior of the

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line segment of the bar that is being split, there is a natural stress for the new

bar tensegrity, and the subdivided tensegrity is also universal with respectto the new stress. If the original conguration is generically rigid, a smallperturbation of the new conguration to a generic one will not change therank of the stress matrix. Thus generic global rigidity is preserved underedge splitting. In [5, 24], A. Berg and T. Jordan and later B. Jackson andT. Jordan solve a conjecture of mine with the following:

Theorem 9. If a graph G is vertex 3-connected (Condition a.) for d = 2 )and is generically redundantly rigid in the plane (Condtion b.) for d = 2 ),then G can be obtained from the graph K 4 , by a sequence of edge splits (asin Figure 16) and insertions of additional bars.

Thus Hendricksons conjecture, that Condition a.) and Condtion b.)are sufficient as well as necessary for generic global rigidity in the plane, issufficient as well as necessary. This also gives an efficient non-probablisticpolynomial-time algorithm for determining generic global rigidity in the plane.See also [25, 28, 30, 27] for other results and applications.

References

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[3] Maria Belk : Realizability of graphs in three dimensions. DiscreteComput. Geom. 37 (2007), no. 2, 139162.

[4] Maria Belk; Robert Connelly : Realizability of graphs, DiscreteComput. Geom. 37 (2007), no. 2, 125137.

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[6] K aroly Bezdek; Robert Connelly Stress Matrices and M Matri-ces. Oberwolfach Reports. Vol. 3, No. 1 (2006), 678680.

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tions from Euclidean space. Discrete geometry and rigidity (Budapest,1999). Period. Math. Hungar. 39 (1999), no. 1-3, 185200.

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[10] R. Connelly : On generic global rigidity, in Applied Geometry andDiscrete Mathematics , DIMACS Ser. Discrete Math, Theoret. Com-put. Scie 4 , AMS, 1991, pp 147155.

[11] R. Connelly; A. Back : Mathematics and tensegrity, in AmericanScientist , Vol. 86, 2 , (March-April 1998), 142151.

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[17] R. Connelly; W. Whiteley : Second-order rigidity and prestressstability for tensegrity frameworks. SIAM J. Discrete Math. 9 (1996),no. 3, 453491.

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inger; Robert Connelly : A linear programming algorithm to testfor jamming in hard-sphere packings. J. Comput. Phys. 197 (2004), no.1, 139166.

[20] S. Gortler, A. Healy, and D. Thurston : Characterizing genericglobal rigidity, arXiv:0710.0907v1. (2007)

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Direction-Length rigidity Matroid, J. Combinatorial Theory B (2009)doi:1016/j.jotb.2009.03.004

[26] Bill Jackson; Tibor Jord an; Zolt an Szabadka : Globally linkedpairs of vertices in equivalent realizations of graphs. Discrete Comput.Geom. 35 (2006), no. 3, 493512.

[27] Bill Jackson; Tibor Jord an : A sufficient connectivity conditionfor generic rigidity in the plane. Discrete Discrete Applied Mathematics157 (2009) 1965-1968.

[28] T. Jord an; Z. Sabadka : Operations preserving the global rigidity of graphs and frameworks in the plane, Computational Geometry, (2009),doi:10:1016/j.comgeo.2008.09.007

[29] L aszl o Lov asz : Steinitz representations of polyhedra and the Colin deVerdiere number. J. Combin. Theory Ser. B 82 (2001), no. 2, 223236.

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[38] W. Whiteley : Rigidity and Scene Analysis ; Handbook of Discreteand Computational Geometry , J. Goodman and J. ORourke (eds.),(second edition), 2004, pp 13271354

[39] J. Y. Zhang: Simon D. Guest; Makoto Ohsaki; Robert Con-nelly : Dihedral Star Tensegrity Structures, (submitted).

[40] http://www.math.cornell.edu%7Etens/

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Tensegrities and Global Rigidity by Connelly 2009