The orbit rigidity matrix of a symmetric framework
TThe orbit rigidity matrix of a symmetricframework
Bernd Schulze ∗ Institute of Mathematics, MA 6-2TU BerlinStrasse des 17. Juni 136,D-10623 Berlin, GermanyandWalter Whiteley † Department of Mathematics and StatisticsYork University4700 Keele StreetToronto, ON M3J1P3, CanadaOctober 25, 2018
Abstract
A number of recent papers have studied when symmetry causes frame-works on a graph to become infinitesimally flexible, or stressed, and whenit has no impact. A number of other recent papers have studied specialclasses of frameworks on generically rigid graphs which are finite mech-anisms. Here we introduce a new tool, the orbit matrix, which connectsthese two areas and provides a matrix representation for fully symmetricinfinitesimal flexes, and fully symmetric stresses of symmetric frameworks.The orbit matrix is a true analog of the standard rigidity matrix for gen-eral frameworks, and its analysis gives important insights into questionsabout the flexibility and rigidity of classes of symmetric frameworks, inall dimensions.With this narrower focus on fully symmetric infinitesimal motions,comes the power to predict symmetry-preserving finite mechanisms - giv-ing a simplified analysis which covers a wide range of the known mech-anisms, and generalizes the classes of known mechanisms. This initialexploration of the properties of the orbit matrix also opens up a numberof new questions and possible extensions of the previous results, includ-ing transfer of symmetry based results from Euclidean space to spherical,hyperbolic, and some other metrics with shared symmetry groups andunderlying projective geometry. ∗ Supported by the DFG Research Unit 565 ‘Polyhedral Surfaces’. † Supported by a grant from NSERC (Canada). a r X i v : . [ m a t h . M G ] J un Introduction
Over the last decade, a substantial theory on the interactions of symmetry andrigidity has been developed [13, 20, 17, 23, 24, 25, 30, 31, 28]. This includesdescriptions of when symmetry changes generically rigid graphs into infinitesi-mally flexible frameworks, and when symmetry does not modify the behavior.These analyses have used tools of representation theory to analyze the stressesand motions of the symmetric realizations of a graph. Some extensions havegone further to describe situations when the symmetry switches a graph intoconfigurations with symmetry-preserving finite flexes [19, 28]. These predictionsof finite symmetric flexes turn out to focus on frameworks with fully symmetricinfinitesimal flexes, and with fully symmetric self-stresses [29].There is a companion, extensive literature on flexible frameworks built ongenerically rigid frameworks, starting with Bricard’s flexible octahedra [7, 35],running though linkages such as Bottema’s mechanism [6] and other finitelyflexible frameworks [11] and Connelly’s flexible sphere [9] to recent work onflexible cross-polytopes in 4-space [36]. Some of this work has looked at creatinganalog examples of finite mechanisms in other metrics such as the spherical andhyperbolic space [1]. In general, it is a difficult task to decide when a specificinfinitesimal flex of a framework on a generically rigid graph extends to a finiteflex. However, on careful examination, many of these known examples havesymmetries and infinitesimal flexes which preserve this symmetry [29, 38]. Itis natural to seek tools and connections that can simplify the creation andgeneration of such examples of finite mechanisms (linkages).In [29], one block of the block decomposition induced by the representationsof the symmetry group was used to study the spaces of fully symmetric mo-tions and fully symmetric self-stresses. This analysis gave some initial resultspredicting finite flexes which remain fully symmetric throughout their path.However, actual generation of this block in the decomposition required substan-tial machinery from representation theory, and the entries in the matrix werenot transparent.In this paper we present the orbit matrix for a symmetric framework as anoriginal, simplifying tool for detecting this whole package of fully symmetricinfinitesimal flexes, fully symmetric self-stresses, and predicting finite flexes forconfigurations which are generic within the symmetry. In our proofs, we willactually show that the orbit matrix is equivalent to the matrix studied in [29],but the construction is transparent, and the entries in the matrix are explicitlyderived. For a symmetric graph, with symmetry group S , this orbit matrix hasa set of columns for each orbit of vertices under the group action, and row foreach orbit of edges under the group action. We will give a detailed constructionfor this matrix in §
5, and show that the kernel is precisely the fully symmetricinfinitesimal motions ( § §
6) and the row dependencies are exactly the fullysymmetric self-stresses ( § § c , the rows r and the dimension of thefully symmetric trivial infinitesimal motions m , we can give some immediatesufficient conditions for the presence of fully symmetric infinitesimal flexes (see § Theorem 7.5
Given a graph which is generically isostatic in -space, and aframework on the graph realized in -space as generically as possible with 2-foldsymmetry with no vertices or edges fixed by the rotation, the framework has afinite flex preserving the symmetry. Because this orbit matrix is a powerful symmetry adapted analog of thestandard rigidity matrix, many of the questions from standard rigidity haveextensions for fully symmetric stresses and motions. Some of these questionsand possible extensions are presented in §
9, along with brief discussions of thepotential for symmetry adapted extensions of the techniques and results. Asone example, it is natural to seek analogs of Laman’s Theorem to characterizenecessary and sufficient conditions for the orbit matrix of a graph and symme-try group to be independent with maximal rank. As a second example, becausekey portions of the point group symmetries and the corresponding counting forthis orbit matrix can be transferred to other metrics (such as the spherical, hy-perbolic and Minkowski spaces), the methods developed here provide a uniformconstruction of mechanisms such as the Bricard octahedron, the flexible cross-polytope, and the Bottema mechanism and its generalizations across multiplemetrics.As a final comment, key results on the global rigidity of generic frameworksdepend on self-stresses and the equivalence of finite flexes and infinitesimal flexesfor generic frameworks. We now have fully-symmetric versions of these tools,and can extract some analogs of the global rigidity results for symmetry-genericframeworks, both as conditions under which they are globally rigid within theclass of fully-symmetric frameworks, and when they are globally rigid within theclass of all frameworks. Still there are additional conjectures and new results tobe explored in this area.We hope that this paper serves as an invitation for the reader to join in thefurther explorations of these many levels of interactions of symmetry, rigidity,and flexibility.
All graphs considered in this paper are finite graphs without loops or multipleedges. The vertex set of a graph G is denoted by V ( G ) and the edge set of G isdenoted by E ( G ).A framework in R d is a pair ( G, p ), where G is a graph and p : V ( G ) → R d is a map such that p ( u ) (cid:54) = p ( v ) for all { u, v } ∈ E ( G ). We also say that ( G, p ) isa d -dimensional realization of the underlying graph G [18, 45]. For v ∈ V ( G ),we say that p ( v ) is the joint of ( G, p ) corresponding to v , and for e ∈ E ( G ), wesay that p ( e ) is the bar of ( G, p ) corresponding to e .3or a framework ( G, p ) whose underlying graph G has the vertex set V ( G ) = { , . . . , n } , we will frequently denote the vector p ( i ) by p i for each i . The k th component of a vector x is denoted by ( x ) k . It is often useful to identify p witha vector in R dn by using the order on V ( G ). In this case we also refer to p as a configuration of n points in R d . Throughout this paper, we do not differentiatebetween an abstract vector and its coordinate column vector relative to thecanonical basis.A framework ( G, p ) in R d with V ( G ) = { , . . . , n } is flexible if there exists acontinuous path, called a finite flex or mechanism , p ( t ) : [0 , → R dn such that(i) p (0) = p ;(ii) (cid:107) p ( t ) i − p ( t ) j (cid:107) = (cid:107) p i − p j (cid:107) for all 0 ≤ t ≤ { i, j } ∈ E ( G );(iii) (cid:107) p ( t ) k − p ( t ) l (cid:107) (cid:54) = (cid:107) p k − p l (cid:107) for all 0 < t ≤ { k, l } of verticesof G .Otherwise ( G, p ) is said to be rigid . For some alternate equivalent definitions ofa rigid and flexible framework see [2, 26], for example.An infinitesimal motion of a framework (
G, p ) in R d with V ( G ) = { , . . . , n } is a function u : V ( G ) → R d such that( p i − p j ) T ( u i − u j ) = 0 for all { i, j } ∈ E ( G ), (1)where u i denotes the column vector u ( i ) for each i .An infinitesimal motion u of ( G, p ) is an infinitesimal rigid motion (or trivialinfinitesimal motion ) if there exists a skew-symmetric matrix S (a rotation) anda vector t (a translation) such that u ( v ) = Sp ( v )+ t for all v ∈ V ( G ). Otherwise u is an infinitesimal flex (or non-trivial infinitesimal motion ) of ( G, p ).(
G, p ) is infinitesimally rigid if every infinitesimal motion of (
G, p ) is aninfinitesimal rigid motion. Otherwise (
G, p ) is said to be infinitesimally flexible [18, 45].The rigidity matrix of (
G, p ) is the | E ( G ) | × dn matrix R ( G, p ) = i j ... { i, j } . . . p i − p j ) T . . . p j − p i ) T . . . ,that is, for each edge { i, j } ∈ E ( G ), R ( G, p ) has the row with ( p i − p j ) , . . . , ( p i − p j ) d in the columns d ( i −
1) + 1 , . . . , di , ( p j − p i ) , . . . , ( p j − p i ) d in the columns d ( j −
1) + 1 , . . . , dj , and 0 elsewhere [18, 45].Note that if we identify an infinitesimal motion u of ( G, p ) with a columnvector in R dn (by using the order on V ( G )), then the equations in (1) can bewritten as R ( G, p ) u = 0. So, the kernel of the rigidity matrix R ( G, p ) is thespace of all infinitesimal motions of (
G, p ). It is well known that a framework(
G, p ) in R d is infinitesimally rigid if and only if either the rank of its associatedrigidity matrix R ( G, p ) is precisely dn − (cid:0) d +12 (cid:1) , or G is a complete graph K n and the points p i , i = 1 , . . . , n , are affinely independent [2].While an infinitesimally rigid framework is always rigid, the converse does4ot hold in general. Asimov and Roth, however, showed that for ‘generic’ con-figurations, infinitesimal rigidity and rigidity are in fact equivalent [2].A self-stress of a framework ( G, p ) with V ( G ) = { , . . . , n } is a function ω : E ( G ) → R such that at each joint p i of ( G, p ) we have (cid:88) j : { i,j }∈ E ( G ) ω ij ( p i − p j ) = 0,where ω ij denotes ω ( { i, j } ) for all { i, j } ∈ E ( G ). Note that if we identify aself-stress ω with a column vector in R | E ( G ) | (by using the order on E ( G )), thenwe have ω T R ( G, p ) = 0. In structural engineering, the self-stresses are alsocalled equilibrium stresses as they record tensions and compressions in the barsbalancing at each vertex.If (
G, p ) has a non-zero self-stress, then (
G, p ) is said to be dependent (sincein this case there exists a linear dependency among the row vectors of R ( G, p )).Otherwise, (
G, p ) is said to be independent . A framework which is both inde-pendent and infinitesimally rigid is called isostatic [15, 42, 45].
Let G be a graph with V ( G ) = { , . . . , n } , and let Aut( G ) denote the auto-morphism group of G . A symmetry operation of a framework ( G, p ) in R d is anisometry x of R d such that for some α ∈ Aut( G ), we have x ( p i ) = p α ( i ) for all i ∈ V ( G ) [21, 30, 28, 31].The set of all symmetry operations of a framework ( G, p ) forms a groupunder composition, called the point group of (
G, p ) [4, 21, 28, 31]. Since trans-lating a framework does not change its rigidity properties, we may assume wlogthat the point group of any framework in this paper is a symmetry group , i.e.,a subgroup of the orthogonal group O ( R d ) [30, 28, 31].We use the Schoenflies notation for the symmetry operations and symmetrygroups considered in this paper, as this is one of the standard notations in theliterature about symmetric structures (see [4, 13, 17, 19, 21, 23, 24, 30, 28, 31],for example). In this notation, the identity transformation is denoted by Id , arotation about a ( d − R d by an angle of πm is denotedby C m , and a reflection in a ( d − R d is denoted by s .While the general results of this paper apply to all symmetry groups, wewill only analyze examples with four types of groups. In the Schoenflies nota-tion, they are denoted by C s , C m , C mv , and C mh . For any dimension d , C s isa symmetry group consisting of the identity Id and a single reflection s , and C m is a cyclic group generated by a rotation C m . The only other possible typeof symmetry group in dimension 2 is the group C mv which is a dihedral groupgenerated by a pair { C m , s } . In dimension d > C mv denotes any symmetrygroup that is generated by a rotation C m and a reflection s whose correspondingmirror contains the rotational axis of C m , whereas a symmetry group C mh isgenerated by a rotation C m and the reflection s whose corresponding mirror isperpendicular to the C m -axis. For further information about the Schoenfliesnotation we refer the reader to [4, 21, 28].Given a symmetry group S in dimension d and a graph G , we let R ( G,S ) denote the set of all d -dimensional realizations of G whose point group is ei-ther equal to S or contains S as a subgroup [30, 28, 31]. In other words, the5et R ( G,S ) consists of all realizations ( G, p ) of G for which there exists a mapΦ : S → Aut( G ) so that x (cid:0) p i (cid:1) = p Φ( x )( i ) for all i ∈ V ( G ) and all x ∈ S . (2)A framework ( G, p ) ∈ R ( G,S ) satisfying the equations in (2) for the mapΦ : S → Aut( G ) is said to be of type Φ, and the set of all realizations in R ( G,S ) which are of type Φ is denoted by R ( G,S, Φ) (see again [30, 28, 31] andFigure 1). p p p p (a) p p p p (b)Figure 1: 2-dimensional realizations of K , in R ( K , , C s ) of different types: theframework in (a) is of type Φ a , where Φ a : C s → Aut( K , ) is the homomorphismdefined by Φ a ( s ) = (1 3)(2)(4) and the framework in (b) is of type Φ b , whereΦ b : C s → Aut( K , ) is the homomorphism defined by Φ b ( s ) = (1 4)(2 3).It is shown in [28, 31] that if the map p of a framework ( G, p ) ∈ R ( G,S ) isinjective, then ( G, p ) is of a unique type Φ and Φ is necessarily also a homomor-phism. For simplicity, we therefore assume that the map p of any framework( G, p ) considered in this paper is injective (i.e., p i (cid:54) = p j if i (cid:54) = j ). In particular,this allows us (with a slight abuse of notation) to use the terms p x ( i ) and p Φ( x )( i ) interchangeably, where i ∈ V ( G ) and x ∈ S . In general, if the type Φ is clearfrom the context, we often simply write x ( i ) instead of Φ( x )( i ).Let ( G, p ) ∈ R ( G,S, Φ) and let x be a symmetry operation in S . Then thejoint p i of ( G, p ) is said to be fixed by x if x ( p i ) = p i (or equivalently, x ( i ) = i ),Let the symmetry element corresponding to x be the linear subspace F x of R d which consists of all points a ∈ R d with x ( a ) = a . Then the joint p i of anyframework ( G, p ) in R ( G,S, Φ) must lie in the linear subspace U ( p i ) = (cid:92) x ∈ S : x ( p i )= p i F x .Note that U ( p i ) (cid:54) = ∅ , because the origin is fixed by every symmetry operationin the symmetry group S . Example 3.1
The joint p of the framework ( K , , p ) ∈ R ( K , , C s , Φ a ) depictedin Figure 1 (a) is fixed by the identity Id ∈ C s , but not by the reflection s ∈ C s ,so that U ( p ) = F Id = R d . The joint p of ( K , , p ) , however, is fixed by boththe identity Id and the reflection s in C s , so that U ( p ) = F Id ∩ F s = F s . Inother words, U ( p ) is the mirror line corresponding to s . O V ( G ) = { , . . . , k } for theorbits S ( i ) = { Φ( x )( i ) | x ∈ S } of vertices of G , then the positions of all jointsof ( G, p ) ∈ R ( G,S, Φ) are uniquely determined by the positions of the joints p , . . . , p k and the symmetry constraints imposed by S and Φ. Thus, any frame-work in R ( G,S, Φ) may be constructed by first choosing positions p i ∈ U ( p i ) foreach i = 1 , . . . , k , and then letting S and Φ determine the positions of the re-maining joints. In particular, by placing the vertices of O V ( G ) into ‘generic’positions within their associated subspaces U ( p i ) we obtain an ( S, Φ) -generic realization of G (i.e., a realization of G that is as ‘generic’ as possible within theset R ( G,S, Φ) ) in this way. For a precise definition of ( S, Φ)-generic, and furtherinformation about ( S, Φ)-generic frameworks, we refer the reader to [28, 31].
An infinitesimal motion u of a framework ( G, p ) ∈ R ( G,S, Φ) is fully ( S, Φ) -symmetric if x (cid:0) u i (cid:1) = u Φ( x )( i ) for all i ∈ V ( G ) and all x ∈ S , (3)i.e., if u is unchanged under all symmetry operations in S (see also Figure 2(a)and (b)). p p p p (a) p p p p (b) p p p p (c)Figure 2: Infinitesimal motions of frameworks in the plane: (a) a fully ( C s , Φ a )-symmetric infinitesimal flex of ( K , , p ) ∈ R ( K , , C s , Φ a ) ; (b) a fully ( C s , Φ b )-symmetric infinitesimal rigid motion of ( K , , p ) ∈ R ( K , , C s , Φ b ) ; (c) an infinites-imal flex of ( K , , p ) ∈ R ( K , , C s , Φ b ) which is not fully ( C s , Φ b )-symmetric.Note that it follows immediately from (3) that if u is a fully ( S, Φ)-symmetric infinitesimal motion of (
G, p ), then u i is an element of U ( p i ) foreach i . Moreover, u is uniquely determined by the velocity vectors u , . . . , u k whenever O V ( G ) = { , . . . , k } is a set of representatives for the vertex orbits S ( i ) = { Φ( x )( i ) | x ∈ S } of G . Example 4.1
Consider the framework shown in Figure 2(a). With p T =( a, b ) , p T = (0 , c ) , p T = ( − a, b ) , and p T = (0 , d ) the rigidity matrix of ( K , , p )7 as the form s (1) 4 { , } ( a, b − c ) ( − a, c − b ) 0 0 0 0 { , } ( a, b − d ) 0 0 0 0 ( − a, d − b ) s ( { , } ) 0 0 ( a, c − b ) ( − a, b − c ) 0 0 s ( { , } ) 0 0 0 0 ( − a, b − d ) ( a, d − b ) This matrix has rank 4, and hence leaves a space of − infinitesimal mo-tions. Thus, there exists a -dimensional space of infinitesimal flexes of ( K , , p ) spanned by u = (cid:0) − ac − b ad − b (cid:1) T . This infinitesimal flexis clearly fully ( C s , Φ a ) -symmetric. Example 4.2
The rigidity matrix of the framework ( K , , p ) shown in Figure2(b,c) with p T = ( a, b ) , p T = ( c, d ) , p T = ( − c, d ) , and p T = ( − a, b ) has the form s (2) 4 = s (1) { , } ( a − c, b − d ) ( c − a, d − b ) 0 0 0 0 { , } (2 a,
0) 0 0 0 0 ( − a, { , } c,
0) ( − c,
0) 0 0 s ( { , } ) 0 0 0 0 ( a − c, d − b ) ( c − a, b − d ) This matrix has again rank 4, and leaves a space of − infinitesimal mo-tions. The -dimensional space of infinitesimal flexes of ( K , , p ) is spanned by u = (cid:16) − − c − a )+ b − dd − b − − c − a )+ b − dd − b (cid:17) T . This infinitesi-mal flex is clearly not fully ( C s , Φ b ) -symmetric. A self-stress ω of a framework ( G, p ) ∈ R ( G,S, Φ) is fully ( S, Φ) -symmetric if( ω ) e = ( ω ) f whenever e and f belong to the same orbit S ( e ) = { Φ( x )( e ) | x ∈ S } of edges of G (see also Figure 3(a)).Note that a fully ( S, Φ)-symmetric self-stress is clearly uniquely determinedby the components ( ω ) , . . . , ( ω ) r , whenever O E ( G ) = { e , . . . , e r } is a set ofrepresentatives for the edge orbits S ( e ) = { Φ( x )( e ) | x ∈ S } of G . α αβδ γγ (a) α − αβ − β − γγδ − δ (b)Figure 3: Self-stressed frameworks in the plane: (a) a fully ( C s , Φ)-symmetricself-stress of ( K , p ) ∈ R ( K , C s , Φ) ; (b) a self-stress of ( G, p ) ∈ R ( G, C s , Ψ) whichis not fully ( C s , Ψ)-symmetric. The types Φ and Ψ are uniquely determined bythe injective realizations [28, 31]. 8t is shown in [24, 30] that the rigidity matrix of a framework (
G, p ) ∈ R ( G,S, Φ) can be transformed into a block-diagonalized form using techniquesfrom group representation theory. In this block-diagonalization of R ( G, p ), thesubmatrix block ˜ R ( G, p ) that corresponds to the trivial irreducible represen-tation of S describes the relationship between external displacement vectors onthe joints and resulting internal distortion vectors in the bars of ( G, p ) that arefully ( S, Φ)-symmetric. So, the submatrix block ˜ R ( G, p ) comprises all the in-formation regarding the fully ( S, Φ)-symmetric infinitesimal rigidity propertiesof (
G, p ). The orbit rigidity matrix of (
G, p ) which we will introduce in the nextsection will have the same properties as the submatrix block ˜ R ( G, p ); how-ever, we will see that the orbit rigidity matrix allows a significantly simplifiedfully ( S, Φ)-symmetric infinitesimal rigidity analysis of (
G, p ), since it can be setup directly without finding the block-diagonalization of R ( G, p ) or using othergroup representation techniques.
To make the general definition of the orbit rigidity matrix more transparent, wefirst consider a few simple examples.
Example 5.1
Consider the -dimensional framework ( K , , p ) ∈ R ( K , , C , Φ) depicted in Figure 4, where Φ : C → Aut ( K , ) is the homomorphism definedby Φ( C ) = (1 3)(2 4) .If we denote p T = ( a, b ) , p T = ( c, d ) , p T = ( − a, − b ) , and p T = ( − c, − d ) ,then the rigidity matrix of ( K , , p ) is C (1) 4 = C (2) { , } ( a − c, b − d ) ( c − a, d − b ) 0 0 0 0 { , } ( a + c, b + d ) 0 0 0 0 ( − a − c, − b − d ) C { , } c − a, d − b ) ( a − c, b − d ) C { , } a + c, b + d ) ( − a − c, − b − d ) 0 0 This matrix has rank 4, and hence leaves a space of − infinitesimal p p p p center Figure 4: A framework ( K , , p ) ∈ R ( K , , C , Φ) . motions. Thus, there exists a -dimensional space of infinitesimal flexes of ( K , , p ) spanned by u = (cid:0) − x y − x − y (cid:1) T , where x = cd − abad − bc and y = − c − a ad − bc . This infinitesimal flex is clearly fully ( C , Φ) -symmetric.Note that if we are only interested in infinitesimal motions and self-stressesof ( K , , p ) that are fully ( C , Φ) -symmetric, then it suffices to focus on the firsttwo rows of R ( K , , p ) (i.e., the rows corresponding to the representatives { , } nd { , } for the edge orbits S ( e ) = { Φ( x )( e ) | x ∈ C } of K , ). The othertwo rows are redundant in this fully symmetric context. So, the orbit rigiditymatrix for ( K , , p ) will have two rows, one for each representative of the edgeorbits under the action of C . Further, the orbit rigidity matrix will have onlyfour columns, because each of the joints p and p has two degrees of freedom,and the displacement vectors at the joints p = C ( p ) and p = C ( p ) areuniquely determined by the displacement vectors at the joints p and p and thesymmetry constraints given by C and Φ . We write the orbit rigidity matrix of ( K , , p ) as follows: (cid:18) { , } ( p − p ) T ( p − p ) T { , } (cid:0) p − C ( p ) (cid:1) T (cid:0) p − C − ( p ) (cid:1) T (cid:19) = (cid:18) a − c, b − d ) ( c − a, d − b )( a + c, b + d ) ( c + a, d + b ) (cid:19) Example 5.2
The orbit rigidity matrix for the framework ( K , , p ) ∈ R ( K , , C s , Φ a ) in Example 4.1 (Figure 2(a)) has again two rows, since K , hastwo edge orbits (each of size 2) under the action of C s . The vertex orbits arerepresented by the vertices , and , for example. Clearly, the joint p hastwo degrees of freedom, which gives rise to two columns in the orbit matrix.The joints p and p , however, are fixed by the reflection s in C s , so that anyfully ( C s , Φ a ) -symmetric displacement vectors at p and p must lie on the mir-ror corresponding to s (i.e., on the y -axis). Thus, the orbit rigidity matrix of ( K , , p ) has only one column for each of the joints p and p : (cid:18) { , } ( p − p ) T ( p − p ) T (cid:0) (cid:1) { , } ( p − p ) T p − p ) T (cid:0) (cid:1) (cid:19) = (cid:18) a, b − c ) ( c − b ) 0( a, b − d ) 0 ( d − b ) (cid:19) Example 5.3
The orbit rigidity matrix for the framework ( K , , p ) ∈ R ( K , , C s , Φ b ) in Example 4.2 (Figure 2(b)) is a × matrix, since there arethree edge orbits - represented by the edges { , } , { , } , and { , } , for exam-ple - and two vertex orbits - represented by the vertices and , for example,and none of the joints of ( K , , p ) are fixed by the reflection in C s . Note, how-ever, that the end-vertices of the edge { , } lie in the same vertex orbit for under the action of C s and that the end-vertices of the edge { , } lie in the samevertex orbit for under the action of C s . Thus, for the orbit rigidity matrix of ( K , , p ) we write { , } ( p − p ) T ( p − p ) T { , } (cid:0) p − s ( p ) (cid:1) T { , } (cid:0) p − s ( p ) (cid:1) T = a − c, b − d ) ( c − a, d − b )(4 a,
0) 0 00 0 (4 c, We now give the general definition of the orbit rigidity matrix of a symmetricframework. 10 efinition 5.1
Let G be a graph, S be a symmetry group in dimension d , Φ : S → Aut ( G ) be a homomorphism, and ( G, p ) be a framework in R ( G,S, Φ) .Further, let O V ( G ) = { , . . . , k } be a set of representatives for the orbits S ( i ) = { Φ( x )( i ) | x ∈ S } of vertices of G . We construct the orbit rigidity matrix (or in short, orbit matrix ) O ( G, p, S ) of ( G, p ) so that it has exactly one row foreach orbit S ( e ) = { Φ( x )( e ) | x ∈ S } of edges of G and exactly c i = dim (cid:0) U ( p i ) (cid:1) columns for each vertex i ∈ O V ( G ) .Given an edge orbit S ( e ) of G , there are two possibilities for the correspond-ing row in O ( G, p, S ) : Case 1:
The two end-vertices of the edge e lie in distinct vertex orbits. Thenthere exists an edge in S ( e ) that is of the form { a, x ( b ) } for some x ∈ S ,where a, b ∈ O V ( G ) . Let a basis B a for U ( p a ) and a basis B b for U ( p b ) begiven and let M a and M b be the matrices whose columns are the coordinatevectors of B a and B b relative to the canonical basis of R d , respectively.The row we write in O ( G, p, S ) is: (cid:0) a b . . . (cid:0) p a − x ( p b ) (cid:1) T M a . . . (cid:0) p b − x − ( p a ) (cid:1) T M b . . . (cid:1) . Case 2:
The two end-vertices of the edge e lie in the same vertex orbit. Thenthere exists an edge in S ( e ) that is of the form { a, x ( a ) } for some x ∈ S ,where a ∈ O V ( G ) . Let a basis B a for U ( p a ) be given and let M a be thematrix whose columns are the coordinate vectors of B a relative to thecanonical basis of R d . The row we write in O ( G, p, S ) is: (cid:0) a . . . (cid:0) p a − x ( p a ) − x − ( p a ) (cid:1) T M a . . . (cid:1) .In particular, if x ( p a ) = x − ( p a ) , this row becomes (cid:0) a . . . (cid:0) p a − x ( p a ) (cid:1) T M a . . . (cid:1) . Remark 5.1
Note that the rank of the orbit rigidity matrix O ( G, p, S ) is clearlyindependent of the choice of bases for the spaces U ( p a ) (and their correspondingmatrices M a ), a = 1 , . . . , k . Remark 5.2
If none of the joints of (
G, p ) are fixed by any non-trivial sym-metry operation in S , then the orbit rigidity matrix O ( G, p, S ) of (
G, p ) has dk = d | O V ( G ) | columns, and each of the matrices M a and M b may be chosento be the d × d identity matrix. In this case, the matrix O ( G, p, S ) becomesparticularly easy to construct (see Examples 5.1 and 5.3). ( G, p, S ) In this section, we show that the kernel of the orbit rigidity matrix O ( G, p, S ) of asymmetric framework (
G, p ) ∈ R ( G,S, Φ) is the space of all fully ( S, Φ)-symmetricinfinitesimal motions of (
G, p ), restricted to the set O V ( G ) of representatives for11he vertex orbits S ( i ) of G (Theorem 6.1). It follows from this result that we candetect whether ( G, p ) has a fully ( S, Φ)-symmetric infinitesimal flex by simplycomputing the rank of O ( G, p, S ). Theorem 6.1
Let G be a graph, S be a symmetry group in dimension d , Φ : S → Aut ( G ) be a homomorphism, O V ( G ) = { , . . . , k } be a set of representativesfor the orbits S ( i ) = { Φ( x )( i ) | x ∈ S } of vertices of G , and ( G, p ) ∈ R ( G,S, Φ) .Further, for each i = 1 , . . . k , let a basis B i for U ( p i ) be given and let M i bethe d × c i matrix whose columns are the coordinate vectors of B i relative to thecanonical basis of R d . Then ˜ u = ˜ u ... ˜ u k ∈ R c × . . . × R c k lies in the kernel of O ( G, p, S ) if and only if u = M ˜ u ... M k ˜ u k ∈ R dk is the restriction u | O V ( G ) of a fully ( S, Φ) -symmetric infinitesimal motion u of ( G, p ) to O V ( G ) . Proof.
Suppose there exists an edge e = { a, x ( b ) } in G whose two end-verticeslie in distinct vertex orbits (see Case 1 in the definition of the orbit rigidity ma-trix). The row equation of the matrix O ( G, p, S ) for the edge orbit representedby e is then of the form (cid:0) p a − X p b (cid:1) T (cid:0) M a ˜ u a (cid:1) + (cid:0) p b − X − p a (cid:1) T (cid:0) M b ˜ u b (cid:1) = 0,where X is the matrix that represents x with respect to the canonical basisof R d . Since the inner product in the second summand is invariant under theorthogonal transformation x ∈ S , we have (cid:0) p a − X p b (cid:1) T (cid:0) M a ˜ u a (cid:1) + (cid:0) X p b − p a (cid:1) T (cid:0) XM b ˜ u b (cid:1) = 0,which is the row equation of the standard rigidity matrix R ( G, p ) for e = { a, x ( b ) } .Similarly, for any other edge y ( { a, x ( b ) } ), y ∈ S , that lies in the edge orbit S ( e ), we have (cid:16) Y p a − YX p b (cid:17) T (cid:0) YM a ˜ u a (cid:1) + (cid:16) YX p b − Y p a (cid:17) T (cid:0) YXM b ˜ u b (cid:1) = 0,where Y is the matrix that represents y with respect to the canonical basis of R d . This is the standard row equation of R ( G, p ) for the edge y ( { a, x ( b ) } ).Suppose next that there exists a bar e = { a, x ( a ) } in G whose two end-vertices lie in the same vertex orbit (see Case 2 in the definition of the orbitrigidity matrix). The row equation of the matrix O ( G, p, S ) for the edge orbitrepresented by e is then of the form (cid:0) p a − X p a (cid:1) T (cid:0) M a ˜ u a (cid:1) + (cid:0) p a − X − p a (cid:1) T (cid:0) M a ˜ u a (cid:1) = 0.12ince the inner product in the second summand is invariant under the orthogonaltransformation x ∈ S , we have (cid:0) p a − X p a (cid:1) T (cid:0) M a ˜ u a (cid:1) + (cid:0) X p a − p a (cid:1) T (cid:0) XM a ˜ u a (cid:1) = 0,which is the standard row equation of R ( G, p ) for e = { a, x ( a ) } .Similarly, for any other edge y ( { a, x ( a ) } ), y ∈ S , that lies in the edge orbit S ( e ), we have (cid:0) Y p a − YX p a (cid:1) T (cid:0) YM a ˜ u a (cid:1) + (cid:0) YX p a − Y p a (cid:1) T (cid:0) YXM a ˜ u a (cid:1) = 0,which is the standard row equation of R ( G, p ) for the edge y ( { a, x ( a ) } ).It follows that ˜ u lies in the kernel of O ( G, p, S ) if and only if u is the re-striction u | O V ( G ) of a fully ( S, Φ)-symmetric infinitesimal motion u of ( G, p ) to O V ( G ) . (cid:3) Example 6.1
Consider the framework ( K , , p ) ∈ R ( K , , C s , Φ a ) from Examples4.1 and 5.2. The vector ˜ u = (cid:0) − ac − b ad − b (cid:1) T clearly lies in the kernelof O ( K , , p, C s ) , and the vector u = M (cid:0) − (cid:1) M ac − b M ad − b = = Id (cid:0) − (cid:1)(cid:0) (cid:1) ac − b (cid:0) (cid:1) ad − b = = − ac − b ad − b is the restriction of the fully ( C s , Φ a ) -symmetric infinitesimal flex u = (cid:0) − ac − b ad − b (cid:1) T to the set { , , } of representatives forthe vertex orbits C s ( i ) = { Φ a ( x )( i ) | x ∈ C s } . The following extension of the theorem of Asimov and Roth (see [2]) to frame-works that possess non-trivial symmetries was derived in [29] (see also [28]):
Theorem 7.1
Let G be a graph, S be a symmetry group in dimension d , Φ : S → Aut ( G ) be a homomorphism, and ( G, p ) be a framework in R ( G,S, Φ) whosejoints span all of R d . If ( G, p ) is ( S, Φ) -generic and ( G, p ) has a fully ( S, Φ) -symmetric infinitesimal flex, then there also exists a finite flex of ( G, p ) whichpreserves the symmetry of ( G, p ) throughout the path. Remark 7.1
It is also shown in [29] that the condition that (
G, p ) is ( S, Φ)-generic in Theorem 7.1 may be replaced by the weaker condition that the sub-matrix block ˜ R ( G, p ) of the block-diagonalized rigidity matrix ˜ R ( G, p ) whichcorresponds to the trivial irreducible representation of S (or, equivalently, theorbit rigidity matrix O ( G, p, S )) has maximal rank in some neighborhood ofthe configuration p within the space of configurations that satisfy the symmetryconstraints given by S and Φ. In particular, this says that if the rows of the13rbit rigidity matrix O ( G, p, S ) are linearly independent and (
G, p ) has a fully( S, Φ)-symmetric infinitesimal flex, then (
G, p ) also has a symmetry-preservingfinite flex.In combination with Theorem 7.1, Theorem 6.1 gives rise to a simple newmethod for detecting finite flexes in symmetric frameworks. In the followingsubsections, we elaborate on this new method and apply it to a number ofinteresting examples.
First, we consider situations where knowledge of only the size of the orbit rigiditymatrix already allows us to detect finite flexes in symmetric frameworks.The following result is an immediate consequence of Theorem 6.1
Theorem 7.2
Let G be a graph, S be a symmetry group in dimension d , Φ : S → Aut ( G ) be a homomorphism, and ( G, p ) be a framework in R ( G,S, Φ) .Further, let r and c denote the number of rows and columns of the orbit rigiditymatrix O ( G, p, S ) , respectively, and let m denote the dimension of the space offully ( S, Φ) -symmetric infinitesimal rigid motions of ( G, p ) . If r < c − m , (4) then ( G, p ) has a fully ( S, Φ) -symmetric infinitesimal flex. Recall from Section 5 that the number of rows, r , and the number of columns, c , of the orbit rigidity matrix O ( G, p, S ) of (
G, p ) do not depend on the config-uration p , but only on the graph G and the prescribed symmetry constraintsgiven by S and Φ. As shown in [28], the dimension m of the space of fully( S, Φ)-symmetric infinitesimal rigid motions of (
G, p ) is also independent of p ,provided that the joints of ( G, p ) span all of R d . So, suppose the set R ( G,S, Φ) contains a framework whose joints span all of R d . Then, as shown in [31], thejoints of all ( S, Φ)-generic realizations of G also span all of R d . Thus, if (4)holds, then all ( S, Φ)-generic realizations of G have a fully ( S, Φ)-symmetricinfinitesimal flex, and hence, by Theorem 7.1, also a finite symmetry-preservingflex.The dimension m of the space of fully ( S, Φ)-symmetric infinitesimal rigidmotions of (
G, p ) can easily be computed using the techniques described in[30, 28]. In particular, in dimension 2 and 3, m can be deduced immediatelyfrom the character tables given in [13]. Thus, in order to check condition (4) itis only left to determine the size of the orbit rigidity matrix O ( G, p, S ) whichbasically requires only a simple count of the vertex orbits and edge orbits of thegraph G .Alternatively, the values of r and c in (4) can also be found by expressing thecharacters of the ‘internal’ and ‘external’ matrix representation for the group S (see [17, 24, 30, 31], for example) as linear combinations of the charactersof the irreducible representations of S : the numbers r and c are the respectivecoefficients corresponding to the trivial irreducible representation in these lin-ear combinations (see [28, 29] for details). However, our new ‘orbit approach’ is14uch simpler than this method of computing characters, since it allows us to de-termine r and c directly without using any techniques from group representationtheory. Let’s apply our new method to the symmetric quadrilaterals we discussed inSection 5.We first consider the quadrilateral with point group C from Example 5.1(see Figure 4). There are two vertex orbits, represented by the vertices 1 and 2,for example, and we have c i = dim (cid:0) U ( p i ) (cid:1) = 2 for i = 1 ,
2. Further, there aretwo edge orbits, and we have m = 1, since the only infinitesimal rigid motionsthat are fully symmetric with respect to this half-turn symmetry are the onesthat correspond to rotations about the origin (see [28] for details). Thus, wehave r = 2 < c − m .So, we may conclude that ( C , Φ)-generic realizations of K , have a symmetry-preserving mechanism (see also Figure 5(a)).This result can easily be generalized to predict the flexibility of a whole classof 2-dimensional frameworks with half-turn symmetry.Recall from Section 3 that a joint p i of ( G, p ) ∈ R ( G,S, Φ) is said to be fixed by x ∈ S if x ( p i ) = p i . Similarly, we say that a bar { p i , p j } of ( G, p ) is fixed by x if either x ( p i ) = p i and x ( p j ) = p j or x ( p i ) = p j and x ( p j ) = p i . Thenumber of joints and bars of ( G, p ) that are fixed by x are denoted by j x and b x , respectively. Theorem 7.3
Let G be a graph with | E ( G ) | = 2 | V ( G ) | − , C = { Id, C } be the half-turn symmetry group in dimension , and Φ : C → Aut ( G ) be ahomomorphism. If j C = b C = 0 , then ( C , Φ) -generic realizations of G have asymmetry-preserving mechanism. Proof.
Since j C = 0 we have c = 2 | V ( G ) | = | V ( G ) | , and since b C = 0 we have r = | E ( G ) | = | V ( G ) | −
2. As mentioned above, we have m = 1, so that r = | V ( G ) | − < | V ( G ) | − c − m .Thus, by Theorem 6.1 and 7.1, ( C , Φ)-generic realizations of G have asymmetry-preserving mechanism. (cid:3) Next, we consider the quadrilateral with point group C s from Example 4.1(see Figure 2(a)). There are three vertex orbits, represented by the vertices1, 2, and 4, for example, and we have c i = dim (cid:0) U ( p i ) (cid:1) = 1 for i = 2 , c = dim (cid:0) U ( p ) (cid:1) = 2. Further, there are two edge orbits, and we have m = 1,since the only infinitesimal rigid motions that are fully symmetric with respectto this mirror symmetry are the ones that correspond to translations along themirror line [28]. Thus, we have r = 2 < c − m .So, we may conclude that ( C s , Φ a )-generic realizations of K , have a symmetry-preserving mechanism (see also Figure 5(b)).The following theorem provides a generalized version of this result.15 a) (b) Figure 5: Symmetry-preserving mechanisms of the quadrilaterals from Examples5.1 (a) and 4.1 (b).
Theorem 7.4
Let G be a graph with | E ( G ) | = 2 | V ( G ) | − , C s = { Id, s } be a‘reflectional’ symmetry group in dimension , and Φ : C s → Aut ( G ) be a homo-morphism. If b s = 0 , then ( C s , Φ) -generic realizations of G have a symmetry-preserving mechanism. Proof.
Let | V (cid:48) | be the number of vertex orbits of size 2 and | E (cid:48) | be the numberof edge orbits of size 2. Then we have | E ( G ) | = 2 | V ( G ) | − | E (cid:48) | + b s = 2(2 | V (cid:48) | + j s ) − | E (cid:48) | + 12 b s = 2 | V (cid:48) | + j s − | E (cid:48) | + b s = 2 | V (cid:48) | + j s − b s − r = c − m + 12 b s − b s < b s = 0, since b s = 1 contradicts the count | E ( G ) | = 2 | V ( G ) | −
4) then r < c − m . The result now follows from Theorems6.1 and 7.1. (cid:3) Similar results to Theorems 7.3 and 7.4 can of course also be established forother symmetry groups in dimension 2.Note that the orbit count for the quadrilateral with mirror symmetry fromExample 4.2 (see Figure 2(b,c)) is r = 3 = c − m ,and by computing the rank of the corresponding orbit rigidity matrix explic-itly, it is easy to verify that this quadrilateral does in fact not have any fully( C s , Φ b )-symmetric infinitesimal flex, let alone a symmetry-preserving mecha-nism. However, it does have a mechanism that breaks the mirror symmetry. The Bricard octahedra [7] are famous examples of flexible frameworks in 3-space. While it follows from Cauchy’s Theorem ([8]) that convex realizations of16he octahedral graph are isostatic, the French engineer R. Bricard found threetypes of octahedra with self-intersecting faces whose realizations as frameworksare flexible. Two of these three types of Bricard octahedra possess non-trivialsymmetries: Bricard octahedra of the first type have a half-turn symmetry andBricard octahedra of the second type have a mirror symmetry. In the following,we use our new ‘orbit approach’ to not only show that both of these typesof frameworks are flexible, but also that they possess a ‘symmetry-preserving’finite flex. Various other treatments of the Bricard octahedra can be found in[3, 29, 35], for example. R. Connelly’s celebrated counterexample to Euler’srigidity conjecture from 1776 is also based on a flexible Bricard octahedron(of the first type) [9]. However, since the flexible Connelly sphere - as well asSteffen’s modified Connelly sphere - break the half-turn symmetry as they flex,our methods do not apply to these particular examples.We let G be the graph of the octahedron, C be a ‘half-turn’ symmetry groupin dimension 3, and Φ a : C → Aut( G ) be the homomorphism defined byΦ a ( Id ) = id Φ a ( C ) = (1 3)(2 4)(5 6).Then there are three vertex orbits - represented by the vertices 1, 2, and 5,for example (see also Figure 6(a)). Since none of the joints p , p , and p arefixed by the half-turn C , the number of columns of the orbit rigidity matrix O ( G, p, C ) is c = 3 · r = 6.Finally, as shown in [28], we have m = 2,since the fully ( C , Φ a )-symmetric infinitesimal rigid motions are those that arisefrom translations along the C -axis and rotations about the C -axis. It followsthat r = 6 < c − m ,so that we may conclude that ( C , Φ a )-generic realizations of the octahedralgraph possess a symmetry-preserving finite flex.This result can be generalized as follows. Theorem 7.5
Let G be a graph with | E ( G ) | = 3 | V ( G ) | − , C = { Id, C } be a half-turn symmetry group in dimension , and Φ : C → Aut ( G ) be ahomomorphism. If j C = b C = 0 , then ( C , Φ) -generic realizations of G have asymmetry-preserving mechanism. Proof.
Since j C = 0 we have c = 3 | V ( G ) | , and since b C = 0 we have r = | E ( G ) | .As mentioned above, we have m = 2, and hence r = 3 | V ( G ) | − < | V ( G ) | − c − m .17 p p p p p (a) p p p p p p (b)Figure 6: Flexible Bricard octahedra with point group C (a) and C s (b).So, by Theorems 6.1 and 7.1, ( C , Φ)-generic realizations of G have a symmetry-preserving mechanism. (cid:3) Next, let G again be the graph of the octahedron, C s be a ‘reflectional’symmetry group in dimension 3, and Φ b : C s → Aut( G ) be the homomorphismdefined by Φ b ( Id ) = id Φ b ( s ) = (1 3)(2)(4)(5 6).Then there are four vertex orbits - represented by the vertices 1, 2, 4, and 5,for example (see also Figure 6(b)). Since the joints p and p are fixed by thereflection s , and the joints p and p are not, the number of columns of the orbitrigidity matrix O ( G, p, C s ) is c = 2 · · r = 6.Finally, as shown in [28], we have m = 3,since the fully ( C s , Φ b )-symmetric infinitesimal rigid motions are those that arisefrom translations along the mirror plane and rotations about the axis throughthe origin which is perpendicular to the mirror [28]. It follows that r = 6 < c − m ,so that we may conclude that ( C s , Φ b )-generic realizations of the octahedralgraph also possess a symmetry-preserving finite flex.More generally, we have the following result.18 heorem 7.6 Let G be a graph with | E ( G ) | = 3 | V ( G ) | − , C s = { Id, s } be areflectional symmetry group in dimension , and Φ : C s → Aut ( G ) be a homo-morphism. If j s > b s , then ( C s , Φ) -generic realizations of G have a symmetry-preserving mechanism. Proof.
Let | V (cid:48) | be the number of vertex orbits of size 2 and | E (cid:48) | be the numberof edge orbits of size 2. Then we have | E ( G ) | = 3 | V ( G ) | − | E (cid:48) | + b s = 3(2 | V (cid:48) | + j s ) − | E (cid:48) | + 12 b s = 3 | V (cid:48) | + 32 j s − | E (cid:48) | + b s = 3 | V (cid:48) | + 2 j s + 12 ( b s − j s ) − r = c + 12 ( b s − j s ) − m .So if j s > b s , then r < c − m . Thus, by Theorems 6.1 and 7.1, ( C s , Φ)-genericrealizations of G have a symmetry-preserving mechanism. (cid:3) The flexibility of the Bricard octahedra shown in Figures 6(a) and (b) followimmediately from Theorems 7.5 and 7.6.Note that Theorems 7.5 and 7.6 also prove the existence of a symmetry-preserving finite flex in a number of other famous and interesting frameworks in3-space. For example, Theorem 7.5 applies to symmetric ‘double-suspensions’which are frameworks that consist of an arbitrary 2 n -gon and two ‘cone-vertices’that are linked to each of the joints of the 2 n -gon [11] and to some symmetricring structures and reticulated cylinder structures like the ones analyzed in [19],[38], and [28]. Similarly, Theorem 7.6 applies to the famous ‘double-banana’ (see[18], for example) with mirror symmetry (with the two connecting vertices ofthe two ‘bananas’ lying on the mirror), to various bipartite frameworks (such as3-dimensional realizations of K , with mirror symmetry, where all the joints ofeither one of the partite sets lie in the mirror [28]), and to some other symmetricring structures and reticulated cylinder structures.Similar results to Theorems 7.5 and 7.6 can of course also be established forsome other symmetry groups in dimension 3. In particular, it can be shownthat realizations of the octahedral graph which are generic with respect to thedihedral symmetry arising from the half-turn symmetry and the mirror sym-metry defined in the examples above also possess a finite flex which preservesthe dihedral symmetry throughout the path (see also [28]). Notice that theconfigurations which are symmetry generic for the dihedral symmetry are notsymmetry generic for the half-turn, or the mirror, alone, so this separate analysisis needed. d > all dimensions, our newmethod becomes particularly useful in analyzing symmetric structures in higherdimensional space. We demonstrate this by giving a very simple proof for theflexibility of the 4-dimensional cross-polytope described in [36]. As we will19iscuss in Section 9.3, this example can also immediately be transferred tovarious other metrics.For the graph G of the 4-dimensional cross-polytope, we have | V ( G ) | = 8and | E ( G ) | = 24, and hence | E ( G ) | − (4 | V ( G ) | −
10) = 24 − (4 · −
10) = 2.Therefore, there will always be at least two linearly independent self-stresses inany 4-dimensional realization of G . However, it turns out that certain symmet-ric G with dihedral symmetry C v which are constructed by placingtwo joints in each of the two perpendicular mirrors that correspond to the tworeflections in C v , and adding their mirror reflections, and then connecting eachof these eight vertices to all other vertices of G except its own mirror image.This gives rise to four vertex orbits - each of size 2. Since each mirror is a3-dimensional hyperplane, we have c = 4 · r = 8 edge orbits (4 orbits of size 4and 4 orbits of size 2) and that m = 3 (the fully symmetric infinitesimal rigidmotions are the ones that arise from translations along the symmetry element(the ‘plane of rotation’) of C and rotations about the plane perpendicular tothe symmetry element of C ). It follows that r = 8 < c − m ,which, by Theorems 7.5 and 7.6, implies that 4-dimensional cross-polytopeswhich are generic with respect to this type of dihedral symmetry possess asymmetry-preserving finite flex.Next, we provide some general counting results for frameworks with pointgroups C and C s in dimension d > | E ( G ) | = d | V ( G ) | − (cid:0) d +12 (cid:1) to be generically isostatic in dimension d . Using the techniques described in [28] it is easy to show that for a d -dimensional framework ( G, p ) with point group C ( C s ), the space of fully sym-metric infinitesimal translations has dimension ( d −
2) ( d −
1, respectively) andthe space of fully symmetric infinitesimal rotations has dimension (cid:0) d − (cid:1) + 1( (cid:0) d − (cid:1) , respectively), so that the dimension m of fully symmetric infinitesimalrigid motions is equal to d − (cid:0) d − (cid:1) + 1 = 1 + (cid:0) d − (cid:1) ( d − (cid:0) d − (cid:1) = (cid:0) d (cid:1) ,respectively), provided that the joints of ( G, p ) span all of R d . Alternatively,this can be shown by computing the dimension of the kernel of the correspond-ing orbit rigidity matrix O ( K n , p, S ), where K n is the complete graph on thevertex set of G . Theorem 7.7
Let G be a graph with | E ( G ) | = d | V ( G ) | − (cid:0) d +12 (cid:1) , C = { Id, C } be a half-turn symmetry group in dimension d , and Φ : C → Aut ( G ) be ahomomorphism.(i) If d = 4 and b C = 0 , then ( C , Φ) -generic realizations of G have asymmetry-preserving mechanism;(ii) if d > and j C > b C d − + d ( d − d − , then ( C , Φ) -generic realizations of G have a symmetry-preserving mechanism. roof. Let | V (cid:48) | be the number of vertex orbits of size 2 and | E (cid:48) | be the numberof edge orbits of size 2.(i) If d = 4, then with b C = 0 we have r = 2 | V ( G ) | − < | V ( G ) | − | V (cid:48) | + j s ) − c − m .(ii) If d >
4, then | E ( G ) | = d | V ( G ) | − (cid:18) d + 12 (cid:19) | E (cid:48) | + b C = d (2 | V (cid:48) | + j C ) − (cid:18) d + 12 (cid:19) | E (cid:48) | + b C = d | V (cid:48) | + d j C + 12 b C − d ( d + 1)4 r = (cid:0) d | V (cid:48) | + ( d − j C (cid:1) − d − j C + 12 b C − d ( d + 1)4 r = c − d − j C + 12 b C − d ( d + 1)4 r = c + (cid:16) b C − d − j C + d ( d − (cid:17) − (cid:16) (cid:18) d − (cid:19)(cid:17) r = c + (cid:16) b C − d − j C + d ( d − (cid:17) − m .It is now easy to verify that if j C > b C d − + d ( d − d − , then r < c − m . The resultnow follows from Theorems 6.1 and 7.1. (cid:3) In the formula (cid:16) b C − d − j C + d ( d − +2 (cid:17) , the case d = 4, b C = 0 matchespart (i). For d = 5, the formula becomes ( b C − j C − j C > b C guarantees the existence of a symmetry-preserving finite flex in a( C , Φ)-generic realization of G . Theorem 7.8
Let G be a graph with | E ( G ) | = d | V ( G ) | − (cid:0) d +12 (cid:1) , C s = { Id, s } be a reflectional symmetry group in dimension d > , and Φ : C s → Aut ( G ) bea homomorphism. If j s > b s d − + d ( d − d − , then ( C s , Φ) -generic realizations of G have a symmetry-preserving mechanism. Proof.
Let | V (cid:48) | be the number of vertex orbits of size 2 and | E (cid:48) | be the numberof edge orbits of size 2. Then, analogously to the proof of Theorem 7.7, we have | E ( G ) | = d | V ( G ) | − (cid:18) d + 12 (cid:19) | E (cid:48) | + b s = d | V (cid:48) | + d j s + 12 b s − d ( d + 1)4 r = (cid:0) d | V (cid:48) | + ( d − j s (cid:1) − d − j s + 12 b s − d ( d + 1)4 r = c + (cid:16) b s − d − j s + d ( d − (cid:17) − (cid:16) d − (cid:18) d − (cid:19)(cid:17) r = c + (cid:16) b s − d − j s + d ( d − (cid:17) − m .21t is now easy to verify that if j s > b s d − + d ( d − d − , then r < c − m . The resultnow follows from Theorems 6.1 and 7.1. (cid:3) d ≥ d , to thecounts for a finite flex of the cone in dimension d +1 [41]. It is natural to considerhow this impacts the counts of the orbit matrix. First - the symmetry-coningwill transfer the symmetry group by adding the new vertex on the normal tothe origin into the new dimension, extending the axes of any rotations, and themirrors of any reflections in the symmetry group into the larger space. Withthis placement, the cone vertex is fixed by the entire symmetry group, the prioredges and vertices have the same orbits, and the new edges from the cone vertexto the prior vertices have orbits corresponding to their end points - that is onefor each of the prior vertex orbits. This process transfers the counts whichguaranteed symmetry-preserving finite flexes of the original graph in dimension d to counts on the cone which guarantee symmetry-preserving finite flexes ofthe cone graph in dimension d + 1. Combined, the orbit matrices and coningprovide a powerful tool to construct flexible polytopes in all dimensions.Consider, for example, the graph G of the octahedron which is genericallyisostatic in dimension 3. If we cone G (i.e., we add a new vertex to G andconnect it to each of the original vertices of G ), then we obtain a new graph G ∗ which is generically isostatic in dimension 4. If we now realize G ∗ ‘generically’with half-turn symmetry in 4-space so that no bar is fixed by the half-turn, andthe cone vertex is the only vertex that lies on the (2-dimensional) half-turn axis,then the resulting framework possesses a symmetry-preserving mechanism (theorbit counts are r = 6 + 3 = 9 < c − m = 3 · − d >
3, and we repeatedly cone the octahedral graph ( d −
3) times, then the resulting graph G ∗ is generically isostatic in dimension d .However, if we realize G ∗ ‘generically’ with half-turn symmetry in dimension d so that the ( d −
3) cone vertices all lie on the ( d − r = 6 + 3 · ( d −
3) + (cid:0) d − (cid:1) (for the 6 edge orbits of G , the( d −
3) connections from the cone vertices to each of the vertices of G , and the (cid:0) d − (cid:1) edges in the half-turn axis for the complete graph on the cone vertices), c = d · d − d − m = 1 + (cid:0) d − (cid:1) , and hence r = ( c − m ) −
1. Thus,we obtain flexible polytopes with half-turn symmetry in all dimensions in thisway.Analogously, based on the realization of the octahedron with point group C s in Figure 6(b), we may construct flexible polytopes with mirror symmetry in alldimensions. Of course we may also symmetrically cone other flexible polytopes(e.g., the cross-polytope) to produce new flexible polytopes in the next higherdimension (see also Section 9.3). 22 .2 Detection of finite flexes from the rank of the orbitrigidity matrix We have seen that the count r ≥ c − m is a necessary condition for a symmetricframework ( G, p ) ∈ R ( G,S, Φ) to have no fully ( S, Φ)-symmetric infinitesimalflex (Theorem 7.2). However, it is not a sufficient condition, so that if (
G, p )satisfies the count r ≥ c − m , we need to compute the actual rank of theorbit rigidity matrix O ( G, p, S ) to determine whether (
G, p ) has a fully ( S, Φ)-symmetric infinitesimal flex.Alternatively, one could use group representation theory to block-diagonalizethe rigidity matrix R ( G, p ) as described in [24, 30], and then compute the rank ofthe submatrix block which corresponds to the trivial irreducible representationof S . This approach, however, requires significantly more work than simplyfinding the rank of the orbit rigidity matrix.In the following, we demonstrate the simplicity of our new method for de-tecting finite flexes via the rank of the orbit rigidity matrix with the help ofsome examples. K , with partitesets { , , , } and { , , , } . Note that the graph K , is generically rigid indimension 2. Moreover, any 2-dimensional realization of K , has three linearlyindependent self-stresses since | E ( K , ) | − (2 | V ( K , ) | −
3) = 16 − (2 · −
3) = 3.However, as we will see, under certain symmetry conditions, 2-dimensional re-alizations of K , become flexible.Let C v = { Id, C , s h , s v } be the dihedral symmetry group in dimension 2generated by the reflections s h and s v whose corresponding mirror lines are the x -axis and y -axis, respectively, and let Φ : C v → Aut( K , ) be the homomor-phism defined by Φ( Id ) = id Φ( C ) = (1 3)(2 4)(5 7)(6 8)Φ( s h ) = (1 4)(2 3)(5 8)(6 7)Φ( s v ) = (1 2)(3 4)(5 6)(7 8).A framework ( K , , p ) in the set R ( K , , C v , Φ) is depicted in Figure 7. Let’sfirst compute the size of the orbit rigidity matrix O ( K , , p, C v ). There aretwo vertex orbits - represented by the vertices 1 and 5, for example - and alsofour edge orbits - represented by the edges { , } , { , } , { , } , and { , } ,for example. Since m is clearly equal to zero, and c = dim (cid:0) U ( p ) (cid:1) = 2 and c = dim (cid:0) U ( p ) (cid:1) = 2, we have r = 4 = c − m .So, to determine whether ( K , , p ) possesses a fully ( C v , Φ)-symmetric infinites-imal flex, we need to set up the matrix O ( K , , p, C v ) explicitly. If we denote23 p p p p p p p s v s h Figure 7: A fully ( C v , Φ)-symmetric infinitesimal flex of a framework in R ( K , , C v , Φ) .( p ) T = ( a, b ) and ( p ) T = ( c, d ), then we have O ( K , , p, C v ) = { , } (cid:0) p − p (cid:1) T (cid:0) p − p (cid:1) T { , } (cid:0) p − s v ( p ) (cid:1) T (cid:0) p − s − v ( p ) (cid:1) T { , } (cid:0) p − C ( p ) (cid:1) T (cid:0) p − C − ( p ) (cid:1) T { , } (cid:0) p − s h ( p ) (cid:1) T (cid:0) p − s − h ( p ) (cid:1) T = ( a − c, b − d ) ( c − a, d − b )( a + c, b − d ) ( c + a, d − b )( a + c, b + d ) ( c + a, d + b )( a − c, b + d ) ( c − a, d + b ) .It is now easy to see that for any choice of a, b, c , and d , the rows of O ( K , , p, C v ) are linearly dependent (the sum of the first and third row vectorminus the sum of the second and fourth row vector is equal to zero). Thus,the kernel of O ( K , , p, C v ) is non-trivial, and since m = 0, it follows that anyrealization of K , in R ( K , , C v , Φ) possesses a fully ( C v , Φ)-symmetric infinites-imal flex. (By computing an element in the kernel of O ( K , , p, C v ) explicitly,it can be seen that all the velocity vectors of this infinitesimal flex are orthog-onal to the conic determined by the joints of ( K , , p ) (see also Figure 7)). ByTheorems 7.5 and 7.6, it follows that ( C v , Φ)-generic realizations of K , pos-sess a symmetry-preserving finite flex. This flex is also known as ‘Bottema’s!mechanism [6].Figure 8 shows another type of realization of K , in the plane with dihedralsymmetry. This framework is an element of the set R ( K , , C v , Ψ) , where Ψ : C v → Aut( K , ) is the homomorphism defined byΨ( Id ) = id Ψ( C ) = (1 4)(2 3)(5 8)(6 7)Ψ( s h ) = (1)(2)(3)(4)(5 8)(6 7)Ψ( s v ) = (1 4)(2 3)(5)(6)(7)(8).24 p p p p p p p s v s h Figure 8: A fully ( C v , Ψ)-symmetric infinitesimal flex of a framework in R ( K , , C v , Ψ) .The vertex orbits are represented by the set { , , , } , for example, and we have c i = dim (cid:0) U ( p i ) (cid:1) = 1 for i = 1 , , ,
6. Further, the edge orbits are representedby the set (cid:8) { , } , { , } , { , } , { , } (cid:9) , for example. Thus, the orbit count isagain r = 4 = c − m .To determine whether ( K , , p ) possesses a fully ( C v , Ψ)-symmetric infinitesimalflex, we set up the orbit matrix O ( K , , p, C v ). With ( p ) T = ( a, p ) T =( b, p ) T = (0 , c ), and ( p ) T = (0 , d ), we have O ( K , , p, C v )= { , } (cid:0) p − p (cid:1) T (cid:0) (cid:1) (cid:0) p − p (cid:1) T (cid:0) (cid:1) { , } (cid:0) p − p (cid:1) T (cid:0) (cid:1) (cid:0) p − p (cid:1) T (cid:0) (cid:1) { , } (cid:0) p − p (cid:1) T (cid:0) (cid:1) (cid:0) p − p (cid:1) T (cid:0) (cid:1) { , } (cid:0) p − p (cid:1) T (cid:0) (cid:1) (cid:0) p − p (cid:1) T (cid:0) (cid:1) = a c a d b c b d .Clearly, the rows of O ( K , , p, C v ) are linearly dependent. Thus, analogouslyas above, we may conclude that ( C v , Ψ)-generic realizations of K , also possessa symmetry-preserving finite flex. d > K , with partite sets { , . . . , } and { , . . . , } . This graph is generically rigid in dimension 3. Moreover, every3-dimensional realization of K , possesses at least 6 linearly independent self-stresses, because | E ( K , ) | − (3 | V ( K , ) | −
6) = 36 − (3 · −
6) = 6. However,with the help of the orbit rigidity matrix it is easy to see that certain symmetric K , become flexible.Let C h be the symmetry group in dimension 3 which is generated by thereflection s whose corresponding mirror plane is the xy -plane and the 3-foldrotation C whose corresponding rotational axis is the z -axis. Further, we letΦ : C h → Aut( K , ) be the homomorphism defined byΦ( C ) = (1 2 3)(4 5 6)(7 8 9)(10 11 12)Φ( s ) = (1 4)(2 5)(3 6)(7 10)(8 11)(9 12),and let ( K , , p ) be a ( C h , Φ)-generic realization of K , (see also Figure 9). p p p p p p p p p p p p Figure 9: A framework ( K , , p ) in R ( K , , C h , Φ) .We first compute the size of the orbit rigidity matrix O ( K , , p, C h ). Thereare two vertex orbits - represented by the vertices 1 and 7, for example - and sixedge orbits - represented by the edges { , i } , i = 7 , . . . ,
12, for example. Sincerotations about the C -axis are clearly the only infinitesimal rigid motions thatare fully ( C h , Φ)-symmetric, we have m = 1. Since we also have c = c = 3, itfollows that r = 6 > c − m .So we detect a fully ( C h , Φ)-symmetric self-stress, but no fully ( C h , Φ)-symmetric infinitesimal flex of ( K , , p ) with this count. If we want to showthat ( K , , p ) possesses a fully ( C h , Φ)-symmetric infinitesimal flex, we needto prove that the rank of O ( K , , p, C h ) is at most 4. We assume wlog that26 p ) T = ( √ , ,
1) and ( p ) T = ( a, b, d ). Then we have O ( K , , p, C h )= { , } (cid:0) p − p (cid:1) T (cid:0) p − p (cid:1) T { , C (7) } (cid:0) p − C ( p ) (cid:1) T (cid:0) p − C ( p ) (cid:1) T { , C (7) } (cid:0) p − C ( p ) (cid:1) T (cid:0) p − C ( p ) (cid:1) T { , s (7) } (cid:0) p − s ( p ) (cid:1) T (cid:0) p − s ( p ) (cid:1) T { , sC (7) } (cid:0) p − sC ( p ) (cid:1) T (cid:0) p − sC ( p ) (cid:1) T { , sC (7) } (cid:0) p − sC ( p ) (cid:1) T (cid:0) p − sC ( p ) (cid:1) T = ( √ − a, − b, − c ) ( a − √ , b, c − √ a + √ b , −√ a − b , − c ) ( a + √ , b + , c − √ a −√ b , √ a − b , − c ) ( a + √ , b − , c − √ − a, − b, c ) ( a − √ , − b, c + 1)( √ a + √ b , −√ a − b , c ) ( a + √ , b + , c + 1)( √ a −√ b , √ a − b , c ) ( a + √ , b − , c + 1) .Clearly, the equation ω T O ( K , , p, C h ) = 0is satisfied for the linearly independent vectors ω T = (cid:0) − − (cid:1) and ω T = (cid:0) − − (cid:1) . Thus, ( C h , Φ)-generic realizations of K , indeed possess a symmetry-preserving finite flex.In general, for any dimension d >
3, we may construct d -dimensional realiza-tions of the complete bipartite graph K d, d with point group C dh by choosingone vertex from each of the two partite sets of K d, d as representatives for thevertex orbits and placing them ‘generically’ off the mirror plane correspondingto the reflection in C dh and also off the rotational axis corresponding to the d -foldrotation in C dh . This gives rise to two vertex orbits (each of size 2 d ) and 2 d edgeorbits. Since the infinitesimal rigid motions corresponding to rotations aboutthe d -fold axis will always be fully-symmetric with respect to this C dh symmetry,the orbit counts will always detect a fully-symmetric self-stress, but no fully-symmetric infinitesimal flex for these frameworks. However, these frameworkscan be shown to possess a symmetry-preserving finite flex analogously as aboveby computing the actual ranks of the corresponding orbit matrices.Note that for these types of realizations of K d, d with C dh symmetry, thegeometry of quadric surfaces (see [5, 43], for example) can be used to predictthe existence of a fully-symmetric infinitesimal flex and therefore also the rankproperties of the corresponding orbit matrices. ( G, p, S ) T In this section, we show that for a framework (
G, p ) ∈ R ( G,S, Φ) , the cokernel ofthe orbit rigidity matrix O ( G, p, S ) is the space of all fully ( S, Φ)-symmetric self-27tresses of (
G, p ), restricted to the corresponding set O E ( G ) of representativesfor the edge orbits S ( e ) = { Φ( x )( e ) | x ∈ S } of G (Theorem 8.3). To prove thisresult, we need the following two lemmas: Lemma 8.1
Let G be a graph, S be a symmetry group in dimension d , Φ : S → Aut ( G ) be a homomorphism, and ( G, p ) ∈ R ( G,S, Φ) . Further, let S ( e ) = { Φ( x )( e ) | x ∈ S } be an edge orbit of G whose representative e = { a, x ( b ) } isan edge whose end-vertices lie in distinct vertex orbits of G . Then there existrespective bases B a and B b for U ( p a ) and U ( p b ) (whose coordinate columnvectors relative to the canonical basis form the d × c a matrix M a and the d × c b matrix M b , respectively), a scalar α e ∈ R , and two invertible d × d matrices A and B such that (cid:88) j : { a,j }∈ S ( e ) ( p a − p j ) T = 1 α e (cid:16)(cid:0) p a − x ( p b ) (cid:1) T M a , , . . . , (cid:17) A (5) and (cid:88) j : { b,j }∈ S ( e ) ( p b − p j ) T = 1 α e (cid:16)(cid:0) p b − x − ( p a ) (cid:1) T M b , , . . . , (cid:17) B . (6) Proof.
Let { Id = y , y , . . . , y t } be the stabilizer S p a = { x ∈ S | x ( p a ) = p a } of p a , Y l be the matrix that represents y l with respect to the canonical basis of R d for each l , and α e = | S p a ∩ S x ( p b ) | . Then we have (cid:88) j : { a,j }∈ S ( e ) ( p a − p j ) T = 1 α e t (cid:88) l =0 (cid:16) Y l (cid:0) p a − x ( p b ) (cid:1)(cid:17) T ,because (cid:16) y l ( { p a , x ( p b ) } ) (cid:17) l =0 ,...,t is the family of those bars of ( G, p ) whose corresponding edges lie in S ( e ) andare incident with a , and because p a − x ( p b ) = y l ( p a − x ( p b )) if and only if y l isan element of the coset S p a ∩ S x ( p b ) of S p a .Consider the matrix representation H : S p a → GL ( R d ) that assigns to each y l ∈ S p a the d × d matrix Y l which represents y l with respect to the canonicalbasis of R d . By definition, the H -invariant subspace V of R d corresponding tothe trivial irreducible representation of H is the space U ( p a ). Thus, by theGreat Orthogonality Theorem, there exists an orthogonal d × d matrix of basistransformation M (i.e., M − = M T ) such that t (cid:88) l =0 M − Y l M = M − (cid:16) t (cid:88) l =0 Y l (cid:17) M = t . . . t ,where the first c a column vectors of M are the coordinate vectors of a basis for V = U ( p a ) relative to the canonical basis of R d . We let M a be such that M = 1 t ... ... M a ∗ . . . ∗ ... ... .28hen, for A = M T , we have α e (cid:88) j : { a,j }∈ S ( e ) ( p a − p j ) T = (cid:16)(cid:0) t (cid:88) l =0 Y l (cid:1) ( p a − x ( p b )) (cid:17) T = (cid:16) M t . . . t M T ( p a − x ( p b )) (cid:17) T = (cid:16) M M Ta . . . . . . ... . . . . . . ( p a − x ( p b )) (cid:17) T = (cid:16)(cid:0) p a − x ( p b ) (cid:1) T M a , , . . . , (cid:17) A .This proves (5).Note that if we can show that | S p a ∩ S x ( p b ) | = | S p b ∩ S x − ( p a ) | , then the proofof (6) proceeds completely analogously to the one of (5).Consider the map ψ : (cid:26) S p a ∩ S x ( p b ) → S p b ∩ S x − ( p a ) y (cid:55)→ x − yx .We show that ψ is well-defined. Note that S x ( p b ) = xS p b x − and S x − ( p a ) = x − S p a x . Thus, y ∈ S p a ∩ S x ( p b ) if and only if y ∈ S p a and y = x ˆ yx − withˆ y ∈ S p b . We have ψ ( y ) = x − yx = x − ( x ˆ yx − ) x = ˆ y ∈ S p b , and hence ψ ( y ) ∈ S p b ∩ S x − ( p a ) . Since ψ is clearly bijective, we indeed have | S p a ∩ S x ( p b ) | = | S p b ∩ S x − ( p a ) | . This gives the result. (cid:3) Lemma 8.2
Let G be a graph, S be a symmetry group in dimension d , Φ : S → Aut ( G ) be a homomorphism, and ( G, p ) ∈ R ( G,S, Φ) . Further, let S ( e ) be an edgeorbit of G whose representative e = { a, x ( a ) } is an edge whose end-vertices liein the same vertex orbit of G . Then there exists a basis B a for U ( p a ) (whosecoordinate column vectors relative to the canonical basis form the d × c a matrix M a ), a scalar α e ∈ R , and an invertible d × d matrix A such that (cid:88) j : { a,j }∈ S ( e ) ( p a − p j ) T = 1 α e (cid:16)(cid:0) p a − x ( p a ) − x − ( p a ) (cid:1) T M a , , . . . , (cid:17) A . (7) Proof.
Let { Id = y , y , . . . , y t } be the stabilizer S p a = { x ∈ S | x ( p a ) = p a } of p a , and let F and F be the families of bars of ( G, p ) defined by F = (cid:16) y l ( { p a , x ( p a ) } ) (cid:17) l =0 ,...,t F = (cid:16) y l ( { p a , x − ( p a ) } ) (cid:17) l =0 ,...,t The range of the families F and F are the bars that correspond to the sum-mands in the left hand side of equation (7). Note that we either have F = F F ∩ F = ∅ ( F = F if and only if there exists y l ∈ S p a such that y l ( x ( p a )) = x − ( p a )). Moreover, we have | S p a ∩ S x ( p a ) | = | S p a ∩ S x − ( p a ) | since,by a similar argument as in the proof of Lemma 8.1, the map ψ : (cid:26) S p a ∩ S x ( p a ) → S p a ∩ S x − ( p a ) y (cid:55)→ x − yx is well-defined and bijective.Suppose first that F ∩ F = ∅ . Then we have (cid:88) j : { a,j }∈ S ( e ) ( p a − p j ) T = 1 α e (cid:16) t (cid:88) l =0 (cid:16) Y l (cid:0) p a − x ( p a ) (cid:1)(cid:17) T + t (cid:88) l =0 (cid:16) Y l (cid:0) p a − x − ( p a ) (cid:1)(cid:17) T (cid:17) = 1 α e (cid:16)(cid:0) t (cid:88) l =0 Y l (cid:1)(cid:0) p a − x ( p a ) − x − ( p a ) (cid:1)(cid:17) T , (8)where Y l is the matrix that represents y l with respect to the canonical basis of R d for each l , and α e = | S p a ∩ S x ( p a ) | = | S p a ∩ S x − ( p a ) | .Suppose next that F = F . Then t (cid:88) l =0 (cid:16) Y l (cid:0) p a − x ( p a ) (cid:1)(cid:17) T = t (cid:88) l =0 (cid:16) Y l (cid:0) p a − x − ( p a ) (cid:1)(cid:17) T ,and hence (cid:88) j : { a,j }∈ S ( e ) ( p a − p j ) T = 1 α e (cid:16)(cid:0) t (cid:88) l =0 Y l (cid:1)(cid:0) p a − x ( p a ) − x − ( p a ) (cid:1)(cid:17) T , (9)where α e = 2 | S p a ∩ S x ( p a ) | .Now, by the same argument as in the proof of Lemma 8.1, it follows fromequations (8) and (9) that for the scalars α e defined above and the matrices M a and A defined in Lemma 8.1, equation (7) holds. (cid:3) Theorem 8.3
Let G be a graph with V ( G ) = { , . . . , n } , S be a symmetrygroup in dimension d , Φ : S → Aut ( G ) be a homomorphism, O V ( G ) = { , . . . , k } and O E ( G ) = { e , . . . , e r } be sets of representatives for the vertex orbits S ( i ) = { Φ( x )( i ) | x ∈ S } and edge orbits S ( e ) = { Φ( x )( e ) | x ∈ S } of G , respectively, and ( G, p ) ∈ R ( G,S, Φ) . If the scalars α e i , i = 1 , . . . , r , and the bases for the spaces U ( p i ) , i = 1 , . . . , k , are defined as in Lemmas 8.1 and 8.2, then ˜ ω ∈ R r is anelement of the kernel of O ( G, p, S ) T if and only if ω = α e (˜ ω ) ... α e r (˜ ω ) r is the restriction ω | O E ( G ) of a fully ( S, Φ) -symmetric self-stress ω of ( G, p ) to O E ( G ) . roof. We let O i,a be the c a -dimensional row vector which consists of thosecomponents of the i th row of O ( G, p, S ) that correspond to the vertex a ∈ V ( G ).We further let O i,a be the d -dimensional row vector ( O i,a , , . . . , ω is the restriction ω | O E ( G ) of a fully ( S, Φ)-symmetricself-stress ω of ( G, p ). Then for every vertex a = 1 , . . . , k , we have r (cid:88) i =1 (cid:88) j : { a,j }∈ S ( e i ) ( ω ) i ( p a − p j ) T = 0 T .By (5), (6), and (7), for every vertex a = 1 , . . . , k , we have r (cid:88) i =1 (cid:88) j : { a,j }∈ S ( e i ) ( ω ) i ( p a − p j ) T = r (cid:88) i =1 ( ω ) i (cid:88) j : { a,j }∈ S ( e i ) ( p a − p j ) T = r (cid:88) i =1 (˜ ω ) i (cid:0) O i,a A (cid:1) = (cid:0) r (cid:88) i =1 (˜ ω ) i O i,a (cid:1) A ,where A is defined as in Lemmas 8.1 and 8.2. (In particular, if p a is not fixedby any non-trivial symmetry operation in S , then A is the d × d identity matrixand O i,a = O i,a .) Since A is invertible, it follows that r (cid:88) i =1 (˜ ω ) i O i,a = 0 T ,and hence r (cid:88) i =1 (˜ ω ) i O i,a = 0 T .Conversely, if ˜ ω is an element of the kernel of O ( G, p, S ) T , then for everyvertex a = 1 , . . . , k , we have r (cid:88) i =1 (˜ ω ) i O i,a = 0 T .and hence, by the same argument as above, r (cid:88) i =1 (cid:88) j : { a,j }∈ S ( e i ) ( ω ) i ( p a − p j ) T = 0 T .Moreover, for every x ∈ S , we have r (cid:88) i =1 (cid:88) j : { a,j }∈ S ( e i ) ( ω ) i (cid:0) X ( p a − p j ) (cid:1) T = (cid:0) r (cid:88) i =1 (cid:88) j : { a,j }∈ S ( e i ) ( ω ) i ( p a − p j ) T (cid:1) X T = 0 T X T = 0 T ,where X is the matrix that represents x with respect to the canonical basis of R d . This completes the proof. (cid:3) .2 Fully symmetric tensegrities It is natural to investigate how stressed symmetric frameworks can convertto tensegrity frameworks, with cables (members that can get shorter but notlonger), struts (members that can get longer but not shorter) as well as bars(whose length is fixed) [26]. A number of the classical tensegrity frameworksare based on symmetric frameworks, and the Robert Connelly’s web site [10]permits an interactive exploration of a range of examples of symmetric tensegrityframeworks.We give a few basic definitions and translate some standard results to thesymmetric setting.A tensegrity graph ˆ G has a partition of the edges of G into three disjointparts E ( G ) = E + ( G ) ∪ E − ( G ) ∪ E ( G ). E + ( G ) are the edges that are cables , E − ( G ) are the struts and E ( G ) are the bars . For a tensegrity framework ( ˆ G, p ),a proper self-stress is a self-stress on the underlying framework (
G, p ) with theadded condition that ω ij ≥ , { i, j } ∈ E + , ω ij ≤ , { i, j } ∈ E − [26].Given a symmetric framework ( G, p ) ∈ R ( G,S, Φ) , it is possible to use a fully( S, Φ)-symmetric self-stress on the bar and joint framework (
G, p ) to investi-gate both the infinitesimal rigidity of (
G, p ), and the infinitesimal rigidity ofan associated fully symmetric tensegrity framework ( ˆ
G, p ) (i.e., the edges of anedge orbit are either all cables, or all struts, or all bars), with all members with ω ij > ω ij < Theorem 8.4 (Roth, Whiteley [26])
A tensegrity framework ( ˆ
G, p ) is in-finitesimally rigid if and only if the underlying bar framework ( G, p ) is infinites-imally rigid as a bar and joint framework and ( G, p ) has a self-stress which has ω ij > on cables and ω ij < on struts. Translated in terms of the orbit matrix for a symmetric framework, this says:
Corollary 8.5
A fully symmetric tensegrity framework ( ˆ
G, p ) is infinitesimallyrigid if and only if the underlying bar framework ( G, p ) ∈ R ( G,S, Φ) is infinitesi-mally rigid as a bar and joint framework and the orbit matrix O ( G, p, S ) has aself-stress which has ω ij > on cables and ω ij < on struts. Often, tensegrity frameworks are built which are rigid, but not infinitesimallyrigid [12, 10]. Clearly, the underlying framework (
G, p ) is not generic (whererigidity is equivalent to infinitesimal rigidity), so (
G, p ) has some self-stress. Theresults of Connelly [12] tell us that ( ˆ
G, p ) has a non-zero proper self-stress.
Theorem 8.6 (Connelly [12] Theorem 3)
Let ( ˆ
G, p ) be a rigid tensegrityframework with a cable or strut. Then there is a proper self-stress in the tenseg-rity framework (with ω ij > on cables and ω ij < on struts). Given a fully symmetric rigid tensegrity framework ( ˆ
G, p ), we can show thatthe guaranteed self-stress can be chosen to be fully symmetric.
Corollary 8.7
Let ( ˆ
G, p ) be a fully symmetric rigid tensegrity framework witha cable or strut whose underlying bar framework lies in R ( G,S, Φ) . Then there is a) (b) (c) Figure 10: We give three plane configurations for the edge graph of a cube. In(a) there is C symmetry, in (b) there is C v symmetry, and in (c) there is C v symmetry. (a) has a symmetry-preserving finite flex, (b) has a finite flex whichbreaks the mirror symmetry, and (c) has a fully symmetric self-stress whichmakes it rigid. a fully ( S, Φ) -symmetric non-zero proper self-stress in the tensegrity framework(with ω ij > on cables and ω ij < on struts). Proof.
By Theorem 8.2, there is a non-zero proper self-stress in ( ˆ
G, p ). Wewant to symmetrize this self-stress. For each element of the group x ∈ S , andeach edge { i, j } , we have the coefficient ω x ( i,j ) of the corresponding element ofthe orbit. If we add over all elements of the group, this is a finite sum, and wehave a combined coefficient ω S ( i,j ) . It is a direct computation to confirm thatthese coefficients are a self-stress (the sum of self-stresses is a self-stress) andthat they form a fully symmetric self-stress. Since the original stress was properon a fully symmetric tensegrity framework, all the ω x ( i,j ) for a given edge havethe same sign, so there is no cancelation. We conclude that this is the requirednon-zero proper fully symmetric self-stress. (cid:3) The following example illustrates how these pieces fit together in the layersof symmetry-preserving finite flexes in symmetry generic configurations, non-symmetric finite flexes for symmetry generic configurations for a larger group,and fully symmetric stresses giving rigidity for an even larger group.
Example 8.1
Consider the graph and frameworks illustrated in Figure 10.1. Figure 10(a) shows the graph G realized at a generic configuration with C symmetry. The counts for the rank of the orbit matrix are: r = 6 , c = 8 and m = 1 . This guarantees a symmetry-preserving finite flex. While it isnot immediate, the standard result for such planar graphs [16] shows thatthis framework only has a self-stress if it is the projection of a plane facedpolyhedron - which this is not (there is no consistent line of intersection ofthe outside quadrilateral and the inside face). The symmetry-preservingfinite flex is also the flex guaranteed by the basic generic counts: | E ( G ) | =12 , | V ( G ) | = 16 and | E ( G ) | = 12 < − | V ( G ) | − .2. Figure 10(b) shows a symmetry generic configuration for C v . The newcounts are r = 4 , c = 4 and m = 0 . The corresponding orbit matrix countsto be independent - and in fact the framework still has no self-stress. The ramework is still not the projection of a plane faced polyhedron. Thereis a finite flex, but it is not symmetry-preserving for this C v symmetry(only for C ).3. Figure 10(c) shows a symmetry generic configuration for C v . The revisedcounts are: r = 3 , c = 2 and m = 0 . We are guaranteed a fully symmetricself-stress. (One can also see this as the projection of a plane faced cube-line polyhedron). It is now possible that this is rigid (and remains rigidwith cables and struts following the signs of the self-stress). With cableson the interior of the framework, this is a spider web, and the approach of[12] just works to confirm that these are rigid (though not infinitesimallyrigid). As the example illustrates, and the many structures on [10] confirm, a fullysymmetric self-stress can be the way of forming a rigid tensegrity frameworkwhich is too undercounted to be infinitesimally rigid.We conjecture that a further analog of Connelly’s Theorem also holds, andthat the basic proof can be symmetry adapted:
Conjecture 8.8
Let ( ˆ
G, p ) be a fully symmetric tensegrity framework with acable or strut which has no symmetry-preserving finite flex. Then there is afully symmetric non-zero proper self-stress in the tensegrity framework (with ω ij > on cables and ω ij < on struts). As mentioned in the introduction, the analysis of the orbit matrix opens up anumber of questions which are analogs of the previous work for the standardrigidity matrix. The following samples are not exhaustive, and we find newpossibilities keep opening up for us as we continue to work with the tools andreflect on the possibilities.
An important question for the standard rigidity matrix has been deriving nec-essary and sufficient conditions on the graph for the rigidity matrix to be of fullrank (generic rigidity), or independent, or to have a self-stress. The most famousexample is Laman’s Theorem characterizing generic rigidity in the plane [22].Within the context of symmetric frameworks, there are generalizations for keyplane groups ( C , C s , and C ) presented in [28, 32]. With these combinatorialcalculations come fast algorithms for verifying the generic rigidity.It is natural to seek necessary and sufficient conditions for the orbit matrixof ( G, p ) ∈ R ( G,S, Φ) to be of full rank (i.e., for ( G, p ) to have only trivial fully( S, Φ)-symmetric infinitesimal motions) for a symmetry generic p , or to be in-dependent (i.e., for ( G, p ) to have no fully ( S, Φ)-symmetric self-stresses). Ofcourse, given a symmetric framework (
G, p ) ∈ R ( G,S, Φ) which is independentand infinitesimally rigid with the usual rigidity matrix, its orbit matrix will alsobe independent and of maximal rank. However, we have seen that there are34rameworks which are dependent but the lack of a fully ( S, Φ)-symmetric selfstress means that the orbit matrix is independent, as well as frameworks whichhave infinitesimal flexes but the lack of a fully ( S, Φ)-symmetric infinitesimalflex means that the orbit matrix is of full rank. So we are seeking new resultsand will need new techniques.The Fully Symmetric Maxwell’s Rule ( r = c − m ) gives the standard neces-sary counts on G , S , and Φ for independence and full rank of an orbit matrixwith c columns, r rows, and a space of trivial fully ( S, Φ)-symmetric infinites-imal motions (kernel of the orbit matrix for the complete graph) of dimension m . As usual, there are some added necessary conditions for independence of therows which come from subgraphs G (cid:48) of the graph G :1. If the rows of the orbit matrix are independent, then for each fully sym-metric subgraph G (cid:48) (generating r (cid:48) rows and c (cid:48) columns, as well as m (cid:48) trivial infinitesimal motions for these columns), we have r (cid:48) ≤ c (cid:48) − m (cid:48) ;2. If H is a subgraph of G such that H and x ( H ) are disjoint for each x ∈ S ,then | E ( H ) | ≤ d | V ( H ) | − (cid:0) d +12 (cid:1) , where the framework is in dimension d ,with | V ( H ) | ≥ d .Notice that we do not add special conditions for ‘small’ subgraphs in part 1above. The reference to m (cid:48) actually codes for all those special cases.How could we generate sufficient conditions? One traditional way for thestandard rigidity matrix has been to start with minimal examples, and use in-ductive techniques which preserve the independence and full rank of the rigiditymatrix. These techniques include versions of vertex addition, edge splitting,and vertex splitting. This has been extended to fully symmetric inductive tech-niques, still with the standard rigidity matrix, in [28, 32]. Transferred to theorbit matrix, such fully symmetric techniques will still preserve the indepen-dence and the full rank of the orbit matrix. However, there are many moreinductive techniques which preserve the full rank of the orbit matrix - but wouldnot preserve the full rank of the original rigidity matrix, since they would leaveinfinitesimal flexes which are not fully symmetric. For example, simply addinga vertex along the axis of a 2-fold rotation in 3-space (which adds one column)will only require one added edge orbit - which could be one edge (along the axis)or two edges (the orbit of a single edge) and this would definitely not generatean infinitesimally rigid framework in 3-space!It is unclear whether there are symmetry groups for which the full charac-terization is accessible. When we find such a characterization, we will have afully symmetrized version of the pebble game, for the orbit multi-graph. For standard rigidity, there has been an algebraic geometric exploration of whena specific configuration p makes a generically rigid graph G into an infinites-imally flexible framework ( G, p ). The conditions are expressed in terms of apolynomial pure condition in the coordinates of p which is = 0 if and only if( G, p ) is infinitesimally flexible. There will be a comparable theory for when con-35 a) (b) (c)(d) (e)
Figure 11: The graph in (a) is generically rigid, and is also symmetry genericallyrigid for 3-fold rotation (b). The geometric condition for a non-trivial infinitesi-mal motion is three collinear induced points (c), which can also be achieved with3-fold symmetry (d). Only the configuration (e), with parallelograms
ABCA (cid:48) have symmetry-preserving finite flexes.figurations lower the symmetry generic rank of the orbit matrix. We illustratethe layers of this for a specific plane example with C symmetry. Example 9.1
Consider the framework illustrated in Figure 11(a). The graphis generically rigid, and the pure condition for a lower rank of the rigidity matrixcan be simplified to: make any of the four triangles collinear or make the inducedpoints in Figure 11(c) collinear.Symmetry generic realizations with C symmetry are still infinitesimallyrigid (Figure 11(b)). Assuming C symmetry, the condition for an infinites-imal flex is that the three collinear points lie at infinity - or equivalently that thepairs of lines AA (cid:48) , BC are parallel (Figure 11(d)).This is not enough for a fully symmetric infinitesimal flex - or equivalentlyfor a drop in the rank of the orbit matrix. A direct geometric analysis verifiesthat the geometric condition for a fully symmetric infinitesimal flex (i.e. a dropin the rank of the orbit matrix) is that the three congruent faces A, B, C, A (cid:48) areparallelograms (Figure 11(e)). From the geometric theory of such structures ofparallelograms and triangles, it is known that this infinitesimal flex is a finite,symmetry-preserving flex. Thus, we can express the condition on a configu-ration lowering the rank of the orbit matrix in terms of a polynomial in therepresentative vertices
A, B, C and image of A under C . This example suggests that there is some interesting algebraic geometry toexplore here. In previous work [39] the polynomial conditions were extracted36y using ’tie-downs’ (equivalently striking out some columns) to square up thematrix. This approach is still relevant, but rules for which tie-downs or pinnedvertices remove all fully symmetric trivial motions are more complex.
The paper [27] presents results about the transfer of first-order rigidity proper-ties (essentially all properties of the rigidity matrix) among frameworks whichrealize a given graph, on the same projective configuration, in the metric spaces E d , S d and H d . What about a transfer of the orbit matrix for a symmetry groupin E d to the other metrics with the same symmetry groups?In E d , all groups of isometries for a framework are point groups (there is afixed point). These other spaces also share these same point groups - a connec-tion that can be seen by coning up a dimension and then slicing the cone alonga corresponding unit sphere. S d and H d have additional groups of isometrieswhich do not fix a point and these can vary from space to space.For simplicity, consider a point group in E d and a sphere S d tangent to theEuclidean space at the central point of the group. It is not hard to give acorrespondence to a point group in the spherical space as well as a correspon-dence between symmetry generic frameworks in the two spaces. This correspon-dence will conserve fully symmetric infinitesimal flexes, fully symmetric trivialinfinitesimal motions, and fully symmetric self-stresses. In short, the orbit ma-trices of the two configurations in the two metrics will have a simple invertiblecorrespondence generated by multiplication on the right and left by appropriateinvertible matrices [33].Underlying this transfer is the operation of symmetric coning - with a newvertex in the next dimension, which is on the normal to the lower dimensionand extends the axes and mirrors in the lower space in a way that conserves thegroup, and preserves symmetry, including finite flexes. A particular byproductof this is the observation that repeated coning of the flexible octahedron or theflexible cross-polytope will generate flexible polytopes in every dimension [33].A similar process transfers orbit matrices and the predictions of finite flexesamong E d , S d , and H d . This transfer gives a simple derivation of prior results onthe flexibility of classes of Bricard octahedra and cross-polytopes in the sphericaland hyperbolic metrics [1]. It is unusual for flexibility to transfer - so symmetryis a special situation. This transfer extends to other spaces with the sameunderlying projective geometry, such as the Minkowskian metric, provided thatthe point group is also realized as isometries in this metric. The full explorationof this transfer is the topic of continuing exploration, and further details andresults will be presented in [33].These other spaces such as S d have additional symmetry groups which arenot point groups (do not fix any point, or pair of antipodal points) and hence donot correspond to the symmetries in E d . There will be orbit matrices for thesegroups as well, and hence we can study these cases using a direct extension ofthe methods presented in this paper. These connections will be further exploredin [33]. 37 .4 Extensions to body-bar frameworks One now standard extension of bar and joint frameworks are the body-barframeworks [40, 14]. These are a special class of frameworks, which in dimen-sions 3 and higher have a complete characterization for the multi-graphs whichare generically isostatic (rigid, independent). The basic analysis of symmetryadapted rigidity matrices for these structures has been presented in [20].It is clear that there are corresponding orbit matrices for body-bar frame-works, since they have bar and joint models, and the desired orbit matrix can,in principle, be extracted from that. The counting of columns and rows can alsobe adapted - though it would be helpful to give this in full detail.A further extension studies body-hinge frameworks, with an emphasis onmolecular models, where bodies (atoms) are connected by bonds (sets of 5bars). The molecular models also have bar and joint models, so in principlethere are corresponding orbit matrices, and counts to predict finite flexes. Theclassical ‘boat and chair’ configurations of cyclohexane in chemistry (a ring ofsix carbons) is an example where 3-fold symmetry (the chair) keeps the genericfirst-order rigidity and independence, and the 2-fold symmetry (the boat) is amodel of the flexible octahedron.Theorem 7.5 showing the flexibility of generically isostatic graphs in 3-spacerealized with 2-fold symmetry, extends from this example to general moleculesin 3-space with 2-fold symmetry and no atoms or bonds intersecting the axis.This is a common occurrence among dimers of proteins, so it has potentialapplications to the study of proteins [46].
Owen and Power have investigated other examples of geometric constraints inCAD under symmetry [25]. In general, constraint systems with matrix repre-sentations are open to analysis using group representations and symmetric blockdecompositions of their matrices. However, there are some surprises which con-firm that the analysis of corresponding orbit matrices may not be a simpletranslation of the results given here.It is well known that in the plane, infinitesimal motions correspond to paralleldrawings of the same geometric graph and configuration. The correspondenceinvolves turning all the velocities by 90 ◦ , which takes a trivial rotation to atrivial dilation. For symmetry, this turn takes an infinitesimal motion which isfully symmetric for a rotation to a parallel drawing which is fully symmetric forthe same rotation. However, this operation takes an infinitesimal motion whichis fully symmetric for a mirror to a parallel drawing which is anti-symmetric forthe same mirror (and vice versa). Clearly, there are changes in the developmentof the orbit matrix, even for this special example. There are also additionalfully symmetric trivial motions (dilations about the center of the point group aretrivial, for the mirror). More surprisingly, some edge orbits seem to disappear inthe obit matrix (edge AA (cid:48) in Figure 12(b)). Figure 12 illustrates two examples.This may be enough to confirm that the extensions to other constraint systemsare non-trivial, and worth carrying out!38 a) (b) Figure 12: For symmetric frameworks, the space of fully symmetric trivial paral-lel drawings may be larger than the space of fully symmetric trivial infinitesimalmotions (a), and frameworks without fully symmetric infinitesimal flexes mayhave fully symmetric parallel drawings (b).
References [1] V. Alexandrov,
Flexible polyhedra in the Minkowski 3-space , manuscriptamathematica (2003), no. 3, 341–356.[2] L. Asimov and B. Roth, The Rigidity Of Graphs , AMS (1978), 279–289.[3] E. Baker,
An Analysis of the Bricard Linkages , Mech. Mach. Theory (1980), 267–286.[4] D.M. Bishop, Group Theory and Chemistry , Clarendon Press, Oxford,1973.[5] E.D. Bolker and B. Roth,
When is a bipartite graph a rigid framework? ,Pacific J. Math (1980), 27–44.[6] O. Bottema, Die Bahnkurven eines merkw¨urdigen Zw¨olfstabgetriebes ,¨Osterr. Ing.-Arch. (1960), 218–222.[7] R. Bricard, M´emoire sur la th´eorie de l’octa`edre articul´e , J. Math. PuresAppl. (1897), no. 3, 113–148.[8] A. L. Cauchy, Sur les polygons et les poly`edres , Oevres Compl`etesd’Augustin Cauchy 2`e S´erie Tom 1 (1905), 26–38.[9] R. Connelly,
A counterexample to the rigidity conjecture for polyhedra , Inst.Haut. Etud. Sci. Publ. Math. (1978), 333–335.[10] , Highly symmetric tensegrity structures ∼ tens/, 2008[11] , The rigidity of suspensions , J. Differential Geom. (1978), no. 3,399–408.[12] , Rigidity and energy , invent. math. (1982), 11–33.3913] R. Connelly, P.W. Fowler, S.D. Guest, B. Schulze, and W. Whiteley, Whenis a symmetric pin-jointed framework isostatic? , International Journal ofSolids and Structures (2009), 762–773.[14] R. Connelly, T. Jord´an, and W. Whiteley, Generic Global Rigidity of Body-Bar Frameworks , Egerv´ary Research Group on Combinatorial Optimiza-tion, Technical Report TR-2009-13, 2009[15] H. Crapo and W. Whiteley,
Statics of Frameworks and Motions ofPanel Structures, a Projective Geometric Introduction , Structural Topol-ogy (1982), no. 6, 43–82.[16] ,
Spaces of stresses, projections, and parallel drawings for sphericalpolyhedra , Beitraege zur Algebra und Geometrie / Contributions to Algebraand Geometry 35 (1994), 259-281.[17] P.W. Fowler and S.D. Guest,
A symmetry extension of Maxwell’s rule forrigidity of frames , International Journal of Solids and Structures (2000),1793–1804.[18] J.E. Graver, B. Servatius, and H. Servatius, Combinatorial Rigidity , Grad-uate Studies in Mathematics, AMS, 1993.[19] S.D. Guest and P.W. Fowler,
Symmetry conditions and finite mechanisms ,Mechanics of Materials and Structures (2007), no. 6.[20] S.D. Guest, B. Schulze, and W Whiteley, When is a symmetric body-barstructure isostatic? , to appear in International Journal of Solids and Struc-tures.[21] L.H. Hall,
Group Theory and Symmetry in Chemistry , McGraw-Hill, Inc.,1969.[22] G. Laman,
On graphs and rigidity of plane skeletal structures , J. Engrg.Math. (1970), 331340[23] R.D. Kangwai and S.D. Guest, Detection of finite mechanisms in symmetricstructures , International Journal of Solids and Structures (1999), 5507–5527.[24] , Symmetry-adapted equilibrium matrices , International Journal ofSolids and Structures (2000), 1525–1548.[25] J.C. Owen and S.C. Power, Frameworks, symmetry and rigidity , preprint,2009.[26] B. Roth and W. Whiteley,
Tensegrity Frameworks , AMS (1981), no. 2,419–446.[27] F. V. Saliola and W. Whiteley,
Some notes on the equivalence of first-orderrigidity in various geometries , arXiv:0709.3354, 2007.[28] B. Schulze,
Combinatorial and Geometric Rigidity with Symmetry Con-straints , Ph.D. thesis, York University, Toronto, ON, Canada, 2009.4029] ,
Symmetry as a sufficient condition for a finite flex , submitted toSIAM Journal on Discrete Mathematics, arXiv:0911.2424, 2009.[30] ,
Block-diagonalized rigidity matrices of symmetric frameworks andapplications , to appear in Beitr. Algebra und Geometrie, arXiv:0906.3377,2010.[31] ,
Injective and non-injective realizations with symmetry , Contribu-tions to Discrete Mathematics (2010), 59–89.[32] , Symmetric versions of Laman’s Theorem , to appear in Discreteand Computational Geometry, 2010.[33] B. Schulze and W. Whiteley,
Coning, symmetry, and spherical frameworks ,in preparation, 2010.[34] B. Servatius and W. Whiteley,
Constraining plane configurations in CAD:combinatorics of directions and lengths , SIAM J. Discrete Methods (1999), 136–153.[35] H. Stachel, Zur Einzigkeit der Bricardschen Oktaeder , J. Geom. (1987),41–56.[36] , Flexible Cross-Polytopes in the Euclidean 4-Space , Journal for Ge-ometry and Graphics (2000), no. 2, 159–167.[37] , Flexible Octahedra in the Hyperbolic Space , Mathematics and itsapplications (J´anos Bolyai memorial volume) (2006), 209–225.[38] T. Tarnai,
Finite mechanisms and the timber octagon of Ely Cathedral ,Structural Topology (1988), 9–20.[39] N. White and W. Whiteley, The algebraic geometry of stresses in frame-works , SIAM J. Algebraic Discrete Methods (1983), 481–511.[40] , The algebraic geometry of bar and body frameworks , SIAM J. Al-gebraic Discrete Methods (1987), 1–32.[41] W. Whiteley, Cones, infinity and one-story buildings , Structural Topology(1983), no. 8, 53–70.[42] ,
Infinitesimally Rigid Polyhedra I. Statics of Frameworks , Trans.AMS (1984), no. 2, 431–465.[43] ,
Infinitesimal motions of a bipartite framework , Pac. J. Math. (1984), 233-255.[44] ,
A Matroid on Hypergraphs, with Applications in Scene Analysisand Geometry , Discrete and Computational Geometry (1989), 75–95.[45] , Some Matroids from Discrete Applied Geometry , ContemporaryMathematics, AMS (1996), 171–311.[46] ,
Counting out to the flexibility of molecules , Physical Biology2