aa r X i v : . [ m a t h . M G ] S e p NON-EUCLIDEAN BRACED GRIDS
STEPHEN POWER
Abstract.
Necessary and sufficient conditions are obtained for the infinitesimal rigidity ofbraced grids in the plane with respect to non-Euclidean norms. Component rectangles of thegrid may carry 0, 1 or 2 diagonal braces, and the combinatorial part of the conditions is givenin terms of a matroid for the bicoloured bipartite multigraph defined by the braces. Introduction
In this note we consider how to rigidly brace an m × n grid of flexible squares when distancesare measured with respect to a general norm. The characterisation for the Euclidean norm,due to Bolker and Crapo [4] in 1977, is well-known: bracing some of the squares, by adding adiagonal bar, gives an infinitesimally rigid bar-joint framework if and only if the subgraph ofthe complete bipartite graph K m,n determined by the braced squares is connected and spanning.In the anisotropic non-Euclidean setting novel phenomena appear. A singly braced square isinfinitesimally flexible and when it is doubly braced its infinitesimal rigidity may depend on itsinclination relative to the principal axes. The natural braces graph is therefore a subgraph ofthe bicoloured bipartite multigraph K m,n , in which each edge of K m,n is doubled and carries adistinct colour, blue or red. The colour of the edge corresponds to the translation class of therepresented brace.Let B max be the set of 2 mn possible diagonal braces that could be added to an m × n gridof squares framework G , and let G ( B ) be the braced grid framework determined by a braces set B ⊆ B max . The fully doubly braced grid G ( B max ) may be infinitesimally flexible in ( R , k · k ) forcertain values of angular inclination relative to the x -axis. When the norm is differentiable andstrictly convex we show how these exceptional values are determined by the geometry of the unitsphere { ( x, y ) : k ( x, y ) k = 1 } and its set of tangents. See Lemma 3.1. Additionally, we obtainthe following complementary result, an analogue of the Bolker-Crapo theorem.A cycle of edges e , . . . , e k in K m,n is said to have a dependent colouring , or to be depen-dent , if the number of blue edges in { e , e , . . . , e k − } is equal to the number of blue edges in { e , e , . . . , e k } , otherwise the cycle is said to be independent , with a independent colouring . Theorem 1.1.
Let G ( B ) be an m × n braced grid bar-joint framework in R and let k · k be adifferentiable, strictly convex non-Euclidean norm. Then the following are equivalent.(i) G ( B ) is infinitesimally rigid in ( R , k · k ) .(ii) G ( B max ) is infinitesimally rigid in ( R , k · k ) and the bicoloured braces graph of G ( B ) is aspanning subgraph of K m,n with an independent cycle in each path-connected component. A simple graph is a cycle-rooted tree if there is an edge whose removal gives a tree, and is a cycle-rooted forest if each connected components is a cycle-rooted tree. We extend this tree andforest terminology to bicoloured multigraphs G whose monochrome subgraphs are simple. Thus G is a cycle-rooted tree if there is an edge whose removal gives a tree. The graph condition in (ii)means that the braces graph contains a spanning subgraph which is an independent cycle-rooted forest in the sense that it is a cycle-rooted forest and the unique cycle in each component isindependent. Figure 1 shows a bracing pattern for which the braces graph is actually equal tosuch a spanning subgraph. In such cases if the maximally braced graph is infinitesimally rigidthen G ( B ) is minimally infinitesimally rigid, or isostatic. Figure 1.
A bracing pattern for which the braces graph is a spanning cycle-rooted forest in K , , where each cycle has an independent colouring.We also show that Theorem 1.1 may be generalised to braced grid frameworks where theunderlying grid has an irregular spacing, and where independence for cycles and forests is definedin terms of a gain graph formalism.In the case of norms with 4-fold rotational symmetry, such as the classical norms k · k p , with1 < p < ∞ , p = 2, we find that the maximally braced square grids, G ( B max ), of any inclination,are infinitesimally flexible. On the other hand this is not so in the case of properly rectangulargrids and the combinatorial condition (ii) applies.In Kitson and Power [11] we began the analysis of the rigidity of bar-joint frameworks ( G, p ) innon-Euclidean spaces ( R d , k · k ). In particular for the non-Euclidean norms k · k p , ≤ q ≤ ∞ , weobtained analogues of the Laman/Pollaczek-Geiringer combinatorial characterisation of genericrigidity for the Euclidean plane. This has recently been generalised to arbitrary norms by Dewar[7]. See also Remark 3.7. For the proof of Theorem 1.1 however, nongeneric methods are requiredand we follow a similar path to Bolker and Crapo, relating infinitesimal rigidity to the maximalindependent sets in the matroid of an appropriate stress-sheer matrix. The independent sets inthis matroid are the independent cycle-rooted forests of K m,n .In the final section we note that there are similar characterisations for infinite braced grids.2. Euclidean braced grids
A bar-joint framework G = ( G, p ) in ( R d , k·k ) is a finite or countable simple graph G = ( V, E )together with a placement p : V → R d of its vertices. A (real) infinitesimal flex of G is a vectorfield u : p ( V ) → R d which satisfies the first order flex condition for every bar. In terms of thestandard inner product for R d this means that h u ( p ( v )) − u ( p ( w )) , p ( v ) − p ( w ) i = 0 , for vw ∈ E. A framework is infinitesimally rigid if every infinitesimal flex is a rigid motion infinitesimalflex [1]. For an m × n braced grid framework the vector space of rigid motion flexes coincide withthe space of infinitesimal flexes of the fully braced grid, and this is 3-dimensional being spannedby two infinitesimal translations and an infinitesimal rotation.We first recall that the sufficiency of the braces graph condition for infinitesimal rigidity in theBolker-Crapo characterisation is straightforward and follows quickly from the following lemma.Define a bar 4-cycle for a braced grid to be a 4-cycle of bars in the associated unbraced gridframework. ON-EUCLIDEAN BRACED GRIDS 3
Lemma 2.1.
Let G ( B ) be a braced m × n grid in ( R , k · k ) with a triple of braces correspondingto the squares with labels ( m , n ) , ( m , n ) , ( m , n ) . Then the restriction of an infinitesimalflex z of G ( B ) to the bar 4-cycle for the square with label ( m , n ) is a rigid motion flex.Proof. Since k · k is isotropic we may assume that the boundary of the grid is parallel to thecoordinate axes. By adding a rigid motion flex we may assume that the restriction of z to thebraced bar 4-cycle for ( m , n ) is zero. In view of the linear geometry of the grid the restrictionof z to the joints of the braced bar 4-cycle for ( m , n ) is an infinitesimal vertical translation.Similarly, the restriction of z to the braced bar 4-cycle for ( m , n ) is a horizontal infinitesimaltranslation. By the linear geometry it follows now that at the 4 joints of the bar 4-cycle for( m , n ) the vectors of z have horizontal and vertical components equal to these horizontaland vertical velocities. In particular the restriction of z to the bar 4-cycle for ( m , n ) is aninfinitesimal translation, as required. (cid:3) The lemma implies that if the braces graph H contains the path ( v , w ) , ( v , w ) , ( v , w )in K m,n then, in determining the infinitesimal flex space of the braced grid G , we may assumethat H also contains ( v , w ). Suppose then that H is connected and spanning. Repeating thisedge addition principle we can assume that the connected spanning graph H is equal to K m,n ,the graph associated with the fully braced grid. Since this grid is infinitesimally rigid for theEuclidean norm the sufficiency direction follows.The necessity of the condition, that the braced grid is infinitesimally flexible if H is notconnected and spanning, is more subtle. It follows from the fact that the property of rigidity orflexibility of a braced grid is unchanged if one performs row and column permutations to changebrace positions. This invariance becomes clear on expressing infinitesimal flexes of the unbracedgrid in terms of sheering flexes. We now discuss this shift of viewpoint, which is one of maindevices in the analysis of braced grids in 2 and 3 dimensions in Bolker-Crapo[4] and in Bolker[3].2.1. The stress-sheer matrix.
We adopt the following terminology and notation associatedwith the geometry and linear algebra of a finite grid framework. This will also be useful fornon-Euclidean grids.Let C ( m, n ), the m × n grid , be the closed subset of the plane which is the union of theboundaries of the squares [ j, j + 1] × [ k, k + 1] , ≤ j ≤ m, ≤ k ≤ n .Let L x be the set of n + 1 subsets l = [0 , m ] × k , the lines of the grid in the x -direction. Define L y similarly and let L = L x ∪ L y , the set of all lines for the grid.Let R = R x ∪ R y be the set of n + m ribbons of C ( m, n ). These are rectangles labelled by anadjacent pair of parallel boundary lines l , l ∈ L , there being no repetitions if m + n > l in L = L x ∪ L y determine linear subframeworks, or line frameworks , of a braced gridframework G ( B ) associated with C ( m, n ). For each line l in L let u l be the unique vector fieldwhich vanishes everywhere except on the joints of the line, where it has unit norm and positivedirection parallel to the line. It is elementary to show that the set of these vector fields is a basisfor the infinitesimal flex space of the unbraced m × n grid, G say, associated with C ( m, n ). Inparticular this vector space of flexes has dimension m + n + 2.Let M (resp. M ( B )) be the infinitesimal flex space of the unbraced grid G (resp. G ( B ). Then M is identifiable with the vector space R L of functions from L to R .Let S be the vector space R R which we refer to as the space of ribbon sheers . A ribbon sheer isthus a scalar field on the set of ribbons. The terminology comes from the mechanical viewpointthat the difference of applied velocities along the bounding lines of a ribbon is a sheer. STEPHEN POWER
Let σ be the vector space homomorphism from M to S given by a choice of signs, sgn( l ; ρ ),for the boundary lines the ribbons ρ , with σ ( u )( ρ ) = sgn( l ; ρ ) d l + sgn( l ; ρ ) d l , for ρ ∈ R, u = X l ∈ L d l u l . We assume that for a vertical (resp. horizontal) ribbon the boundary line l with sgn( l ; ρ )positive is the right hand (resp. lower) line. The other boundary line has negative sign.Let M trans (resp. M rig ) be the subspace of M , and of M ( B ), consisting of translation flexes(resp. rigid motion flexes). Note in particular that σ : M → S is onto and ker σ = M trans .Let L x (resp. L y ) consist of the consecutive lines l , l , . . . , l m (resp. l ′ , l ′ , . . . , l ′ n ) and define u rot = m X k =0 ku l k + n X k =0 − ku l ′ k Then u rot ∈ M ( B ) is an infinitesimal rotation velocity field which fixes the south-west cornerjoint of the braced grid. Moreover we have σ ( u rot ) = 1, the ribbon sheer field (1 , , . . . ,
1) in R R .This leads to the following lemma. Lemma 2.2.
Let G ( B ) be a finite braced grid in ( R , k · k ) . Then the space of rigid motioninfinitesimal flexes, M rig , is equal to { u : σ ( u ) ∈ R } .Proof. Let z be an infinitesimal flex with σ ( z ) = λ
1. Subtract λu rot from z to obtain z ′ with σ ( z ′ ) = 0. For some set of coefficients d l , z ′ = X l ∈ L x d l u l + X l ∈ L y d l u l . The condition σ ( z ′ )( ρ ) = 0 for every ribbon ρ implies that the coefficients are equal for l ∈ L x and are also equal for l ∈ L y . Thus z ′ is an infinitesimal translation and the lemma follows. (cid:3) The stress-sheer matrix of G ( B ), denoted SS ( B ), is the |B| × | R | matrix where the row labelledby b ∈ B has zero entries except for entries of +1 and -1 for the columns for the ribbons for b in the x - and y -directions, respectively. The row for the brace b defines a brace functional f b : S = R R → R , and we can regard the value f b ( σ ( u )) as a stress on the brace b induced bythe sheer σ ( u ). The raison d’etre for SS ( B ) is the following lemma. Lemma 2.3. (i) The velocity field u ∈ M restricts to an infinitesimal flex of the braced 4-cyclefor b if and only if f b ( σ ( u )) = 0 .(ii) G ( B ) is infinitesimally rigid if and only if rank SS ( B ) = | R | − .Proof. (i) f b ( σ ( u )) = 0 if and only if the 2 ribbons for b have the same sheer values, that is ifand only if the 2 sheers of the bar 4-cycle defined by u are equal. For the Euclidean norm thismeans that u restricts to a rigid infinitesimal motion of the bar 4-cycle for b .(ii) By (i) u is an infinitesimal flex of G ( B ) if and only if σ ( u ) ∈ ker SS ( B ). Since σ is onto,by Lemma 2.2 infinitesimal rigidity holds if and only if rank SS ( B ) = | R | − (cid:3) Proof of the Bolker-Crapo theorem.
Let us show once again that the graph condition issufficient, using Lemma 2.2 in place of Lemma 2.1. If u is an infinitesimal flex in M with sheerfield σ ( u ), and if σ ( u ) belongs to ker SS ( B ) then the values of σ ( u ) on the 2 ribbons for eachbrace b are equal. Thus if the brace-ribbon graph H is connected and spanning we deduce thatif σ ( u ) ∈ ker SS ( B ) then σ ( u ) is constant. Thus σ ( u ) = λ λ ∈ R and Lemma 2.2, G isinfinitesimally rigid.Suppose that H is not connected. Then there is a permutation of rows and columns of ribbonsresulting in a braced grid with the braces in blocked form, by which we mean the following: there ON-EUCLIDEAN BRACED GRIDS 5 exist 1 ≤ m < m, ≤ n < n such that if the bar 4-cycle with label ( k, l ) is braced then either k ≤ m and l ≤ n , or k > m and l > n . Also there is at least 1 braced square in each block.Such braced grids are not infinitesimally rigid since there exists an infinitesimal flex which fixesone block and gives an infinitesimal rotation of the other block. It follows from Lemma 2.3 thatthe original unpermuted braced grid also fails to be infinitesimally rigid. Finally, note that if H is not spanning then there is a ribbon which is free of braces and so the braced grid is notinfinitesimally rigid. Remark 2.4.
Every real finite matrix A defines a matroid, M ( A ) say, on a finite set X withcardinality equal to the number of rows of the matrix. The independent sets of the matroid arethe subsets of X corresponding to sets of rows which are linearly independent. Bolker and Crapouse the term vector geometry for the matroid defined by a set of row vectors. The Bolker-Crapotheorem for a braced grid G may be expressed in an alternative form as a matroid isomorphism:the stress-sheer matroid for stress-sheer matrix for the completely braced grid is isomorphic tothe graphic matroid of K m,n , the braces graph for the completely braced m × n grid. Indeed,the mn × ( m + n ) stress-sheer matrix provides a linear representation of this graphic matroid.For discussions of the k · k -rigidity of braced grids in R see Bolker [3] and Recksi [16]. Inparticular Recksi obtains a matroidal condition which is necessary for rigidity. As far as theauthor is aware there is no combinatorial characterisation of infinitesimal rigidity known for 3Dbraced grids. (The articles [6], [5] are minor variations of [4], [3].) Remark 2.5.
The infinitesimal rigidity of a general bar-joint framework, with nonzero edgelengths, is determined by its | E | × d | V | rigidity matrix R ( G, p ) [1]. The combinatorial charac-terisation of Euclidean plane bar-joint frameworks (
G, p ) whose joints are generically positioned is due to Pollaczek-Geiringer [15] and Laman [13]. (See also Bernstein [2] for a recent matroidalproof.) The condition is that G should contain a spanning graph H which is (2 , -tight in thesense that 2 | V |−| E | = 3 and for each subgraph H ′ , with at least 2 vertices, 2 | V ′ |−| E ′ | ≥ . Sincethe joints of a grid of squares are not generically placed (the set of 2( n + 1)( m + 1) coordinates isnot an algebraically independent set) special position considerations are required for a proof ofthe Bolker-Crapo theorem. In particular if G ( B ) is the 3 by 3 braced grid with the corner squaresand the central square braced then this is a bar-joint framework ( G, p ) which is infinitesimallyflexible. On the other hand G is (2 , G, p ′ ) is infinitesimallyrigid. 3. Non-Euclidean braced grids
Let G = ( G, p ) be a bar-joint framework in R , with n vertices, and let k · k be a general normfor R . An infinitesimal flex with respect k · k is vector (or velocity) field u = ( u , . . . , u n ), with u i ∈ R for all i , such that k ( p i + tu i ) − ( p j + tu j ) k − k p i − p j k = o ( t ) , as t → , for each edge v i v j of G . Assume that the unit sphere of the normed space ( R , k·k ) is differentiableat the point p / k p k . In this case if p = (0 ,
0) then the velocity field (0 , u ) for the bar p p is a k · k -infinitesimal flex if and only if u is zero or is tangential to this unit sphere at the point p .An infinitesimal rigid motion or rigid motion flex of a bar-joint framework G in ( R , k · k ) maybe defined as an infinitesimal flex which extends to an infinitesimal flex of any framework G ′ in( R , k · k ) which contains G . As in the Euclidean setting, a bar-joint framework is infinitesimallyrigid in ( R , k · k ) if all its infinitesimal flexes are rigid motion flexes.Let us say that an m × n braced or unbraced grid has inclination α ∈ [0 , π/
2) if it is associatedwith the image of the closed set C ( m, n ) under a translation and positive rotation by α . Such a STEPHEN POWER framework is equivalent to its uninclined variant in the normed space ( R , k · k ′ ), where the unitsphere for k · k ′ is the counterclockwise rotation by α of the k · k -sphere.Assume for the remainder of this section that the norm k · k is differentiable and strictlyconvex, so that that the unit sphere is a curve which has well-defined tangents at all points.For a general non-Euclidean norm the space of rigid motion flexes is the 2-dimensional space oftranslation flexes [11]. Under our assumption on the norm this follows on showing that thereexists a framework ( K , p ) whose only infinitesimal flexes are infinitesimal translations.3.1. Tangent vectors and 4-fold symmetry.
As in the Euclidean case, there is a natural basisfor the space of all k · k -infinitesimal flexes of an unbraced grid G in ( R , k · k ) with inclination α .This is the set { u l,α : l ∈ L } where we label the (now possibly inclined) lines of the grid as before,where u l,α is supported by the joints of the line. The individual velocities at these joints areequal to the unique unit vector which is tangential to a k · k -sphere centred on an orthogonal linethrough such a joint, with this centre to the left of the line with respect to its positive direction.This positive direction corresponds to increasing coordinates when the grid is not inclined. With α fixed we denote this tangent vector as u x or u y , according to whether l belongs to L x or L y .The vector u x and an associated basis vector field u l,α are illustrated in the diagrams of Figure2. The first diagram of Figure 2 shows tangent vectors to the k · k -sphere and a k · k -spherefor the particular radial angle, θ = α , measured from a downward radius. For a general radialangle θ these tangents have angles θ and τ ( θ ) respectively, for some strictly increasing function τ : θ → τ ( θ ) , ≤ θ < π . Figure 2. (i) Tangent vectors to the unit spheres for k · k and k · k , and (ii) acorresponding basis element u l,α for the inclined grid framework.The normed space ( R , k · k ) is said to be , or have 4-fold symmetry, ifrotation by π/ τ ( θ ) = τ ( θ + π/ ≤ θ < π/
4. In particular the usual p -norms k · k p , < p < ∞ , are differentiable strictlyconvex norms with 4-fold symmetry.3.2. Brace parameters and stress-sheer matrices.
We now determine an appropriate stress-sheer matrix for a possibly inclined braced grid G ( B ) in the normed space ( R , k · k ). This isgiven in terms of positive real numbers λ, λ ′ which we refer to as the brace parameters for G ( B ).As in the Euclidean case write M = R L for the vector space of coefficients representinginfinitesimal flexes of the unbraced grid G with respect to the line-labelled basis { u l,α : l ∈ L } .The space S = R R , of ribbon sheers, and the surjection σ : M → S are defined in terms of this
ON-EUCLIDEAN BRACED GRIDS 7 basis as before. Thus σ ( u )( ρ ) = sgn( l ; ρ ) d l + sgn( l ; ρ ) d l , for ρ ∈ R, u = X l ∈ L d l u l,α . Once again ker σ is the 2-dimensional space of translation rigid motion flexes. For a given ribbon ρ let us write u ρ for the specific infinitesimal flex u , of the unbraced grid, with σ ( u )( ρ ) = 1and support on the boundary lines of the ribbon. This means that d l = sgn( l ; ρ )1 / d l = 0 otherwise.The braces of B are now of 2 types, indicated in Figure 2(ii) as solid (blue) and dashed (red)line segments, and we write b, b ′ for a pair of these two types for a particular bar 4-cycle. Considerthe x -ribbon sheer, ρ x say, and the y -ribbon sheer, ρ y say, which are the two basis elements of S = R R associated with a pair of braces b, b ′ . By the strict convexity of the norm the vectors u x , u y are linearly independent. Also, we claim that there are unique linear combinations of theform(1) u b = λu x + u y , u b ′ = − λ ′ u x + u y , λ, λ ′ > , such that u b (resp. u b ′ ) is in the tangential direction for b (resp b ′ ). That is, u b (resp. u b ′ )is parallel to τ ( α + π/
4) (resp. τ ( α + π/ b, b ′ do not coincide with the direction of u y or − u y . However, by thestrict convexity of the norm the direction of u b (resp. u b ′ ) is strictly intermediate between thedirections of u x and u y (resp. u x and − u y ) and so the claim follows. See also Figure 3. u b b u x u y ++ -- b ′ λ p p u b ′ λ ′ Figure 3.
The brace parameters λ, λ ′ are determined by the tangent vectors for b, b ′ and the tangent vectors u x , u y .The non-Euclidean stress-sheer matrix of G ( B ) for the strictly convex differentiable norm k · k ,denoted SS λ,λ ′ ( B ), is the |B| × | R | matrix where the row labelled by a solid (blue) brace (resp.dashed (red) brace) has zero entries except for entries of 1 and − λ (resp. 1 and − λ ′ ) for thecolumns for the ribbons ρ x , ρ y associated with the brace. We view the row for each brace b ∈ B as determining a brace functional f b : S = R R → R . The rationale for the definition of SS λ,λ ′ ( B )is that the velocity field u ∈ M restricts to an infinitesimal flex of the bar 4-cycle with addedbrace b (resp. b ′ ) if and only if f b ( σ ( u )) = 0 (resp. f b ′ ( σ ( u )) = 0). To see this in the case of f b let u be a velocity field with restriction velocity field (0 , u ) for the bar p p indicated in Figure3. This is a flex of the bar if and only if u = cu b = c ( λu x + u y ) for some c ∈ R . On the other STEPHEN POWER hand note that from the definition of σ this is equivalent to σ ( u )( ρ x ) = cλ, σ ( u )( ρ y ) = c, for some c , which is the same as f b ( σ ( u )) = 0. The argument for f b ′ is similar.Let 1 x (resp. 1 y ) be the sum of the basis elements for the x -ribbons (resp. y -ribbons). Ifthe brace parameters agree then the ribbon sheer field s = λ x + 1 y lies in the nullspace ofthe stress-sheer matrix and so u is a non rigid motion infinitesimal flex if σ ( u ) = s , assumingthe norm is not Euclidean. It follows that the fully braced grid (of the same inclination) is notinfinitesimally rigid. Thus we have obtained the next lemma. Lemma 3.1.
The following are equivalent for a non-Euclidean, differentiable, strictly convexnorm k · k and an angle α ∈ [0 , π/ .(i) The doubly braced × square grid of inclination α is infinitesimally flexible in ( R , k · k ) .(ii) The brace parameters λ, λ ′ for k · k and α are equal.(iii) A maximally braced m × n square grid framework of inclination α is infinitesimally flexiblein ( R , k · k ) . It is straightforward to see that a monochrome cycle of edges in the braces graph gives a circuitin the matroid M ( SS λ,λ ′ ( B )). More generally we have the following. Lemma 3.2.
Let λ = λ ′ . A cycle of edges in K m,n gives a circuit in the matroid M ( SS λ,λ ′ ( B max )) if and only if its colouring is a dependent colouring.Proof. For notational convenience consider a coloured 6-cycle e , . . . , e in K m,n . The nonzeroentries of the rows of SS λ,λ ′ ( B max ) determine a 6 × λ λ
00 1 0 0 λ
00 1 0 0 0 λ λ λ . with λ i ∈ {− λ, − λ ′ } , for 1 ≤ i ≤
6. Under row operations this matrix is equivalent to λ λ λ − λ λ
00 0 − λ λ λ − λ and this has zero determinant if and only if λ λ λ = λ λ λ . Similarly, a coloured 2 n -cycle gives a dependent set in the stress-sheer matroid if and only if theassociated odd and even products are equal. Since λ, λ ′ > λ in the odd and even products are the same, as required. (cid:3) A subgraph of the bicoloured graph K m,n is an independent cycle-rooted tree if it is a cycle-rooted tree whose unique cycle is independent . A independent cycle-rooted forest is a subgraphwhose components are independent cycle-rooted trees. In an extreme case each component couldbe a bicoloured cycle with 2 vertices. ON-EUCLIDEAN BRACED GRIDS 9
Lemma 3.3.
Let λ = λ ′ . Then the maximal independent sets of the matroid M ( SS λ,λ ′ ( B max )) are the independent cycle-rooted forests which are spanning subgraphs of K m,n .Proof. An independent cycle-rooted forest which is spanning has m + n edges and so, by theprevious lemma and the fact that SS λ,λ ′ ( B max ) has m + n columns, it follows that it is a maximalindependent set in the matroid. On the other hand if F is a maximal independent set of edgesin the matroid then it has m + n edges. Each component cannot be a tree for otherwise anappropriately coloured edge could be added to create an independent cycle-rooted tree. Thuseach component contains an independent cycle rooted-tree and by the previous lemma must beequal to it. (cid:3) Lemma 3.4.
Let G ( B ) be a braced grid with brace parameters λ = λ ′ . Then G ( B ) is infinitesi-mally rigid if and only if SS λ,λ ′ ( B ) has rank m + n .Proof. It follows from the definition of the brace functionals that a velocity field u ∈ M restrictsto an infinitesimal flex of the bar 4-cycle with added braces b, b ′ if and only if f b ( σ ( u )) = 0 and f b ′ ( σ ( u )) = 0. Also ker σ is the space of infinitesimal translations. Thus G ( B ) is infinitesimallyrigid if and only if ker SS λ,λ ′ ( B ) = { } . (cid:3) The proof of Theorem 1.1.
Suppose that G ( B ) is infinitesimally rigid. Then the braceparameters must be distinct, by Lemma 3.1, and so rank SS λ,λ ′ ( B ) = m + n by Lemma 3.4.Thus there exists an independent set of m + n rows. By Lemma 3.3 these rows correspond to theedges of a independent cycle-rooted forest, and so the bicoloured graph condition in (ii) follows.On the other hand if H ⊂ K m ,n is a spanning independent cycle-rooted tree then by Lemma3.3 the rows of SS λ,λ ′ ( B ) are a maximal linearly independent subset and rank SS λ,λ ′ ( B ) = m + n ,completing the proof.3.3. Irregularly spaced grids.
Consider the bar-joint frameworks G ( B ) arising from an irreg-ularly spaced m × n grid framework G , aligned with the coordinate axes, and a set B of diagonalbraces. Assuming that the underlying norm is differentiable and strictly convex there is a set ofpositive brace parameters, Λ say, together with a stress-sheer matrix SS Λ ( B ) defined as before.Lemma 3.2 suggests the following gain-graph formalism, with edge-labelling by elements of theabelian group R + .Denote the braces and their parameters as b e and λ e , where e is a coloured edge of the bracesgraph. This graph is bipartite with vertices v , . . . , v m , w , . . . , w n . Choose an associated edgedirectedness, so that an edge is positively directed from its v -labelled vertex to its w -labelledvertex. Also, define the directed edge gain map γ : E ( K m,n ) → R + where γ ( e ) = λ e when e is positively directed. Following Lemma 3.2 we see that a directed cycle c in the braces graphcorresponds to a dependent set in M ( SS Λ ( B )) if and only if γ ( c ) = 1, that is, if and only if thecycle has no gain. As before we say that the cycle is dependent in this case and independent otherwise. The statement and proof of Theorem 1.1 generalise, with no essential changes, toirregularly spaced grids. Remark 3.5.
The matroid theory of signed graphs and gain graphs has been developed exten-sively by Zaslavsky. See for example [18], [19]. In fact Bolker [3] made use of signed graphs todetermine the circuits for the 3-dimensional braced cube grid rigidity matroid in graphic terms.For a (monochrome) simple graph the matroid whose independent sets are the cycle-rootedforests is known as the bicycle matroid [17]. Also cycle-rooted spanning forests appear in expan-sion formulae for the determinant of the combinatorial Laplacian on a graph [8].We now consider the special case of the classical p -norms, k · k p with k ( x, y ) k pp = | x | p + | y | p . Theorem 3.6.
Let < p < ∞ with p = 2 and let G ( B ) be a diagonally braced m × n grid ofcongruent orthogonal rectangles with inclination α ∈ [0 , π/ .(i) If the rectangles are squares then G ( B ) is infinitesimally flexible in ( R , k · k p ) .(ii) If the rectangles are not squares then G ( B ) is infinitesimally rigid in ( R , k · k p ) if and onlyif the braces graph contains an independent cycle-rooted forest which is spanning.Proof. (i) The classical p -norms for 1 < p < ∞ are differentiable, strictly convex and 4-foldsymmetric. By 4-fold symmetry the vectors u x and u y are orthogonal for any inclination value α . Also by 4-fold symmetry, the braces tangents u b and u b ′ for a grid of squares are orthogonalfor any value of α . In view of this double orthogonality, Equation 1 has solutions λ = λ ′ , and (i)follows.(ii) When the rectangles are not squares then their diagonals subtend an angle 0 < β < π with β = π/
2. Also, the tangent function τ ( θ ) for k · k p has the property that τ ( θ + β ) = τ ( θ ) + π/ β = π/
2, and so the braces tangents are not orthogonal for any value of α . By the braceparameter equations (1), λ = λ ′ (since u x , u y are orthogonal) and so Lemma 3.1 applies. (cid:3) Remark 3.7.
A graph G is (2 , | V | − | E | = 2 and for each subgraph G ′ we have2 | V ′ | − | E ′ | ≥ . The characterisations in Dewar [7], and Kitson and Power [11], show that theexistence of a (2 , G is necessary and sufficient for the infinitesimalrigidity of a bar-joint framework ( G, p ) which is “sufficiently generic”. Under our assumptionsfor the underlying norm “sufficiently generic” corresponds to the non-Euclidean variant of theEuclidean rigidity matrix R ( G, p ) having maximum rank, in which case the framework (
G, p )is said to be regular . It follows from our analysis that the braced square grid G ( B ) is regularif and only if the brace parameters are distinct and the cycles of the cycle-rooted forest areindependently coloured.In the case of the nondifferentiable norm k ( x, y ) k ∞ =max {| x | , | y |} the tangent function τ ( θ )is not defined for θ = π/ , π/
4. Let us say that a braced rectangle grid G ( B ) is well-positionedfor k · k ∞ if α = π/ , π/ G ( B ) is well-positioned then it isinfinitesimally rigid for k · k ∞ if and only if the braces graph is spanning and each componenthas an independent cycle. See also the general characterisation of infinitesimal k · k ∞ -rigidity,for well-positioned, regular frameworks, given in Kitson and Power [11].Let us also note the curiosity of the special rigidity requirements for a braced square grid G ( B ),with zero inclination, with respect to the non-differentiable norm k · k ∞ . By the square geometryof the unit sphere, a diagonal brace, say p p with p = (0 , , p = (1 , u = ((0 , , ( − , u = ((0 , , (0 , − Proposition 3.8.
An axis-aligned braced grid of squares is infinitesimally rigid with respect to k · k ∞ if and only if each vertex of the braces graph is incident to blue and red edges. The graph condition is equivalent to requiring that each ribbon contains at least one brace ofeach type. This is necessary and sufficient to rule out a sheering flex of G ( B ) associated with theribbon. Note, in particular that a ribbon bar-joint framework containing braces of only one type(colour) has a one-sided sheering flex. The proposition follows readily from this.For some further discussions of non-Euclidean frameworks see also Dewar [7], Kitson [9],Kitson, Nixon and Schulze [10], and Nixon and Power [14].4. Infinite braced grids
The definitions of infinitesimal flex and infinitesimal rigidity for a countably infinite bar-jointframework are the same as those for a finite bar-joint framework [11]. Let G ∞ be the infinite ON-EUCLIDEAN BRACED GRIDS 11 unbraced grid bar-joint framework in the usual Euclidean space R with joints located at pointswith integer coordinates. Once again there is a distinct set of infinitesimal flexes, { u l : l ∈ L } ,which is indexed by the lines of G ∞ . Also they form a generalised basis in the following sense. Lemma 4.1.
Every infinitesimal flex u : Z → R of the grid framework G ∞ has a uniquerepresentation u = X l ∈ L d l u l = X l ∈ L x a l u l + X l ∈ L y b l u l . The proof of the Bolker-Crapo Theorem carries over to give the following.
Theorem 4.2.
An infinite braced grid G ∞ ( B ) in ( R , k · k ) is infinitesimally rigid if and onlyif the braces graph is a connected spanning subgraph of K ∞ , ∞ .Proof sketch. Once again there are brace functionals f b : R → R and G ∞ ( B ) is infinitesimallyrigid if and only if the intersection of the nullspaces ker f b , for b ∈ B , is equal to R
1. Here 1is the sheer field function, on the infinite set of ribbons, R , which is identically equal to 1. Itfollows that infinitesimal rigidity is preserved on permuting the (infinite) rows or columns of thebraced grid. If the braces graph is not connected, with at least 2 infinite components, then,by permuting, we may assume that the braces are on bar 4-cycles in either the first or thirdquadrant of Z . Thus G ∞ ( B ) fails to be infinitesimally rigid since there is a nonzero flex fixingthe bar 4-cycles in the first quadrant. A similar arguments applies whatever the cardinality ofthe components. The braces graph must be spanning, or else there is a ribbon that is free ofbraces and so there is a sheering infinitesimal flex. Thus the graph condition is necessary.By Lemma 2.1 the graph condition implies that the infinitesimal flex space of G ∞ ( B ) is equalto that of G ∞ ( B max ) and so the sufficiency direction follows. (cid:3) In a similar way the equivalence in Theorem 1.1 extends to infinite braced grids G ∞ ( B ). Example 4.3.
The subgraph H of K ∞ , ∞ indicated in Figure 4 is a braces graph for a Euclideaninfinite braced grid G ∞ ( B ) where the first (resp. second) row of vertices corresponds to a con-secutive ordering of the horizontal (resp. vertical) ribbons. By Theorem 4.2 this braced grid isinfinitesimally rigid. On the other hand it is straightforward to show that it contains no finitesubframework, with more than 2 brace bars, which is infinitesimally rigid. Figure 4.
A braces graph subgraph of K ∞ , ∞ .We note that this phenomenon is not possible for an infinite generic bar-joint framework ( G, p )in R . Such a framework is infinitesimally rigid if and only if ( G, p ) is sequentially infinitesimallyrigid (Kitson and Power [12]). This means that there exists an increasing chain of infinitesimallyrigid finite subframeworks ( G n , p ) with G equal to the union of the G n . References [1] L. Asimow and B. Roth, The rigidity of graphs, Trans. Amer. Math. Soc., 245 (1978), 279-289.[2] D. Bernstein, Generic symmetry-forced infinitesimal rigidity: translations and rotations, arXiv:2003.10529v3.[3] E. D. Bolker, Bracing grids of cubes, Environment and Planning B, vol. 4 (1977), 157-172.[4] E. D. Bolker and H. Crapo, How to brace a one-story building, Environment and Planning B, vol 4 (1977),125-152. [5] E. D. Bolker, Bracing Rectangular Frameworks. II, SIAM J. App. Math., 36, (1979), 491-508[6] E. D. Bolker and H. Crapo, Bracing Rectangular Frameworks. I, SIAM J. App. Math., 36 (1979), 473-490.[7] S. Dewar, Infinitesimal rigidity in normed planes, SIAM J. Discrete Math., 34 (2020), 1205-1231.[8] R. Kenyon, Spanning forests and the vector bundle Laplacian, Annals of Probability, 39 (2011), 1983-2017.[9] D. Kitson, Finite and infinitesimal rigidity with polyhedral norms, Discrete & Computational Geometry, 54(2015), 390-411.[10] D. Kitson, A. Nixon and B. Schulze, Rigidity of symmetric frameworks in normed spaces, Linear Algebra andits Applications, 607 (2020), 231-285.[11] D. Kitson and S.C. Power, Infinitesimal rigidity for non-Euclidean bar-joint frameworks. Bull. Lond. Math.Soc. 46 (2014), no. 4, 685–697.[12] D. Kitson and S.C. Power, The rigidity of infinite graphs, Discrete & Computational Geometry, 60 (2018),531-557.[13] G. Laman, On graphs and the rigidity of plane skeletal structures, J. Engineering Mathematics, 4 (1970),331-340.[14] A. Nixon and S.C. Power, Double-distance frameworks and mixed sparsity graphs, Discrete and Computa-tional Geometry, 63 (2020), 294-318.[15] H. Pollaczek-Geiringer, Ueber die Gliederung ebener Fachwerke, Ztschr. f. Angew. Math. und Mech. Band 7,Heft 1, February 1927, pp. 58–72.[16] A. Recksi, Bracing cubic grids - a necessary condition, Discrete Math., 73 (1988/89), 199-206.[17] W. Whiteley, The union of matroids and the rigidity of frameworks, SIAM J. Disc. Math. Vol. 1 (1988),237-255.[18] T. Zaslavsky, Signed graphs, Discrete Applied Mathematics, 4 (1982), 47-74.[19] T. Zaslavsky, Biased graphs. I. Bias, balance, and gains, J. Comb. Theory, Ser. B 47 (1989) 32-52.
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