From graphs to tensegrity structures : geometric and symbolic approaches

Publ. Mat. 50 (2006), 279–299 FROM GRAPHS TO TENSEGRITY STRUCTURES: GEOMETRIC AND SYMBOLIC APPROACHES Miguel de Guzm´ an†and David Orden∗ Abstract A form-finding problem for tensegrity structures is studied; given an abstract graph, we show an algorithm to provide a necessary condition for it to be the underlying graph of a tensegrity in Rd (typically d= 2,3) with vertices in general position. Furthermore, for a certain class of graphs our algorithm allows to obtain necessary and sufficient conditions on the relative position of the vertices in order to underlie a tensegrity, for what we propose both a geometric and a symbolic approach. 1. Introduction In this paper we study an instance of the so-called form-finding problems for tensegrity structures. These have brought special attention both among mathematicians and engineers since the seminal works of Kenneth Snelson around 1948 (see [13]). Roughly speaking, a form-finding problem for a tensegrity structure asks to determine a geometric configuration of points and straight edges in Rd(typically d= 2,3) such that the whole structure is in a self-tensional equilibrium. The word tensegrity was coined from tension and integrity by Buckminster Fuller, deeply impressed by Snelson’s work. Apart from a purely mathematical interest [3], [11], understanding these structures has applications to architecture and structural engineering [14] and has led to interesting models for viruses and cellular structures [1], [7]. It is also considered a useful tool for the study of deployable structures [9], [12], [15]. Previous works have proposed a 2000 Mathematics Subject Classification. 05C85. Key words. Form-finding problems, tensegrity, graphs, polynomial elimination. †In memoriam. ∗Research partially supported by grants MEC MTM2005-08618-C02-02 and CAM S-0505/DPI/000235. 280 M. de Guzm´ an, D. Orden number of different approaches to solve form-finding problems, which can be found in the recent review [16]. In particular, the present paper deals with the form-finding problem of building tensegrity structures with a given underlying graph G, in a given Rd. The graph has to be understood as an abstract graph, i.e., a set of vertices and pairs of vertices (edges). We aim to solve the following two problems: •First, to decide whether Gcan be the underlying graph of a tensegrity structure in Rd. •In case such a tensegrity with underlying graph Gis possible, to characterize the relative position of its vertices. In order to solve these problems, we first look for decompositions of tensegrities into basic instances, called atoms. This motivates a combinatorial method that allows to decompose a graph Ginto the smallest graphs that can underlie a tensegrity. In order to build up a tensegrity with graph G, we propose to reverse its decomposition: We show that this solves the above problems for a certain class of graphs and we present two different approaches. The first one looks at the geometric structure of the tensegrity; it is quite visual and provides intuition of the intrinsic properties of tensegrity structures. However, it becomes difficult to use for complicated structures. The second approach condenses in a matrix the information about the tensegrity; this allows to use tools from Symbolic Computation, despite being less intuitive. The paper is organized as follows: The basic notions and results are introduced in Section 2. Then, Section 3 introduces a method to decompose a tensegrity into atoms, which motivates a decomposition of the abstract graph G, reversed then by geometric means. Finally, in Section 4 a rigidity matrix is used for a symbolic resolution. 2. Preliminaries In this section we introduce the basic notions and results used in the paper. Despite it aims to be self-contained, an interested reader can look at [18] for further examples and a more detailed overview of the mathematical concepts. Let us introduce first the rigorous definition of “self-tensional equilibrium”: Definition 2.1. Let G= (V, E) be an abstract graph: •Aframework G(P) in Rdis an embedding of Gon a finite point configuration P:= {p1, . . . , pn}in Rd, with straight edges. In the From Graphs to Tensegrities 281 sequel we will focus on general position point configurations (no d+ 1 points lie on the same hyperplane). •Astress won a framework is an assignment of scalars wij (called tensions) to its edges. Observe that wij =wji, since they refer to the same edge. •Such a wis called a self-stress if, in addition, the following equilibrium condition is fulfilled at every vertex: (1) ∀i, X ij edge wij (pi−pj) = 0. That is, for each vertex pithe scaled sum of incident vectors −−→ pipj is zero. Observe that the null stress is always a self-stress, of no interest for us. Note also that all scalar multiples of a self-stress (in particular its opposite) are self-stresses as well. We will see later that, indeed, the space of self-stresses on a given graph is a vector space. Lemma 2.2. Let p∈Pbe a vertex of a d-dimensional framework G(P) such that Pis in general position. Given a non-null self-stress on G(P), either at least d+ 1 of the edges incident to preceive non-null tension, or all of them have null tension. Proof: The result is true for any d, but the reader may consider d= 2,3 here. Let kbe the number of edges incident to pthat have non-null tension. The equilibrium condition on pimplies having kvectors in Rd, with common tail, which add up to the zero vector. For k < d + 1, this is only possible if their k+ 1 endpoints do not span a k-space. But either k= 0 or this contradicts the general position assumption. As a consequence, the next property makes particularly interesting the study of a certain family of general position frameworks, the socalled (d+ 1)-regular ones, for which a null tension on a single edge propagates to the rest of them: Corollary 2.3. Given a d-dimensional framework all of whose points are in general position and have exactly d+1 incident edges, a self-stress is non-null if, and only if, it is non-null on every edge. The following definition introduces our final object of study, which is a physical model of the mathematical objects defined above: 282 M. de Guzm´ an, D. Orden Definition 2.4. We define a tensegrity structure T(P) to be a selfstressed framework in which: •Edges ij such that wij >0 have been replaced by inextensible cables (its endpoints constrained not to get further apart), •Edges with wij <0 have been replaced by unshrinkable struts (endpoints constrained not to get closer together), and •Edges with wij = 0 have been removed. If no confusion is possible, a tensegrity structure T(P) will be denoted by just T. For another physical interpretation, one can think of cables and struts as springs endowed with a certain tension, respectively inwards and outwards. That is; cables and struts incident to point pihave respectively tensions in the direction of +−−→ pipj(inwards) and −−−→ pipj(outwards), see Figure 1. pi + −pi + Figure 1. Left: two cables (+) and a strut (−) incident to pi. Right: Their representation as springs with inwards and outwards tensions. Observe that, given a tensegrity structure, it might be possible to replace the struts by bars which react to the surrounding tensions. For example, if we replace the strut in Figure 1 by a bar, this will receive an outwards tension at pi, as a reaction to the sum of cable tensions. Such a replacement is usual when constructing tensegrity sculptures, like those in [13]. The most emblematic tensegrity structure is shown in Figure 2. Named oblique triangular prism with rotational symmetry, it is composed of nine cables, six of which form two copies of an equilateral triangle, the top one rotated 30 degrees, joined by three struts alternating the rest of cables. Thick edges denote the struts, which could be replaced by bars as before. From Graphs to Tensegrities 283 p1 p2 p3 p4 p5 p6 Figure 2. The oblique triangular prism. The last definition in this section introduces the smallest tensegrities possible, which we will show in Section 3 to be the fundamental bricks for building up any tensegrity: Definition 2.5. We define a self-stressed atom in Rd(d= 2,3) to be a general position realization of the complete graph Kd+2, together with its unique (up to constant multiplication) non-null self-stress. The tensegrity atoms are then obtained replacing edges by cables and struts. When no confusion is possible, we will just refer to atoms. Figure 3 shows half of the possible tensegrity atoms in Rdfor d= 2,3, where thick edges denote struts. The other half is obtained by interchanging cables and struts or, equivalently, by considering the opposite tensions. It is not difficult to check, using Lemma 2.2, that configurations with fewer points or edges do not admit self-stresses apart from the null one. The non-trivial fact that the above frameworks do admit a unique (up to constants) non-null self-stress appears in [10], where existence is proved by the following result. 284 M. de Guzm´ an, D. Orden Figure 3. Types of tensegrity atoms in R2(left) and R3(right). Proposition 2.6. Let Pn i=1 λipi= 0,Pn i=1 λi= 0 be an affine dependence on a point set P={p1,...,pn}. Then, wij := λiλjdefines a self-stress on the complete graph K(P). Proof: For any pi∈P, we have: X ij∈K wij (pi−pj) = n X j=1 λiλj(pi−pj) = λipi n X j=1 λj−λi n X j=1 λjpj, which equals zero. In order to prove the uniqueness up to constants, consider two different self-stresses, one not a scalar multiple of the other. Then some linear combination of them would cancel the tension at a particular edge but not at all of them, in contradiction with Corollary 2.3. Note that we are using the claimed fact that self-stresses form a vector space, as will be shown at the beginning of the next subsection. From Graphs to Tensegrities 285 3. Geometric approach In this section we present a geometric algorithm to decompose a tensegrity into atoms. This decomposition motivates a combinatorial one, which opens the way towards the resolution of the two problems posed in Section 1. 3.1. Decomposing into atoms. Let G(P) and G′(P′) be two self-stressed frameworks such that P∪P′ is a point configuration in general position. Let wand w′be their selfstresses. For the framework G(P)∪G′(P′) obtained by union of vertices and edges, one can define the sum of self-stresses w+w′in the natural way: Assign tension wij +w′ ij to common edges ij and maintain the initial tension at the others. It is easy to observe that equations (1) are fulfilled and hence w+w′is indeed a self-stress. Furthermore, the space of self-stresses on a given graph G= (V, E) together with this sum and the product by a scalar form a vector subspace of RE, when the latter is identified with the space of all self-stresses. Abusing notation, we denote by G+G′the self-stressed framework obtained. Observe that, after this addition is performed, one can appropriately replace edges by cables and struts in order to obtain a tensegrity structure T+T′. Hence, the sum of tensegrities yields another tensegrity. Observation 3.1. We will only consider this kind of addition when P and P′have at least dpoints in common; otherwise we obtain either two separate tensegrity structures or one of them hanging from the other. The main result in this section states that, reciprocally, under our conditions every tensegrity can be decomposed into a sum of tensegrity atoms: Theorem 3.2 (Atomic decomposition of tensegrities).Every non-null tensegrity structure T(P),Pin general position, is a finite sum of tensegrity atoms. This decomposition is not unique in general. Proof: Let G(P) and wbe the framework and non-null self-stress associated to T(P). We show how to obtain, by addition of atoms, a chain of non-null self-stresses w′on G(P) in which the number of vertices with only null incident tensions (null vertices) is increased at each step. At the end we come up with a self-stressed framework with only null vertices, so that the original tensegrity Twill be the sum of the opposites of those atoms that have appeared in the process. 286 M. de Guzm´ an, D. Orden Let us focus on the two-dimensional case, since the d-dimensional one is carried out analogously: At each step, an arbitrary non-null vertex a∈Pis chosen to be converted in a null one. By Lemma 2.2, only the following two cases are possible (see Figures 4 and 5): •Type 1: If exactly three incident edges ab,ac,ad have non-null tension, we consider the atom Kof vertices a,b,c,d. Since this atom has a non-null self-stress wKwhich is unique up to constants, we can choose wK ab to be the opposite of the tension assigned to edge ab at the current stress w′, i.e. wK ab := −w′ ab. Because of the equilibrium at a, it turns out that also wK ac =−w′ ac and wK ad =−w′ ad. Therefore, adding wKto the current self-stress makes vertex ahave only null tensions at incident edges (i.e. makes it disappear from the induced tensegrity). See Figure 4, where dashed interior edges in the second picture are opposite to those in the first one. Note that at b,c,dthe edges bc,bd and cd may have appeared with non-null tension, but these extra edges do not affect a. However, we will be concerned about them later. a b cd + bb aa cdcd Figure 4. Type 1 step, exactly three incident edges with non-null tension. •Type 2: If a∈Phas incidence degree greater than 3, let b,c,dbe neighbors of a. Consider the atom ¯ Kof vertices a,b,c,d(and all the possible edges between them) and choose it to have tension w¯ K ab := −w′ ab at edge ab. Hence, obviously w′+w¯ Khas null tension at edge ab. Again, other edges bc,bd,cd may appear with non-null tension, but not incident to a. Hence, repeating this process if needed, we obtain a self-stress on G(P) in which ahas only three incident edges with non-null tension (i.e. in the induced tensegrity, ahas only three incident edges). Now we are in the previous case. See Figure 5, where now only dashed edge ab is guaranteed to be opposite to its filled counterpart. From Graphs to Tensegrities 287 a b cd + bb aa cdcd eee Figure 5. Type 2 step, more than three incident edges with non-null tension. Since a sum of self-stresses is another self-stress, after a finite number of these steps we get a self-stress with at least one more null vertex, for which we can iterate the process until all vertices become null. Note that different choices of vertices to make null may lead to different decompositions. Remark 3.3.The reader should notice that the addition of an atom only changes the value of the self-stress on edges of the atom. Hence, adding an atom might cancel other tensions than the intended ones, but only at edges contained in the atom. Motivated by the geometric process in Theorem 3.2, we define now the following combinatorial algorithm, that can be applied to any abstract graph G: Algorithm 3.4 (Combinatorial decomposition). input: abstract graph G= (V, E)and dimension d. output: list Lof “atoms”, where each atom is a subset of (d+ 2) elements of V. 1. Initialize L=∅. 2. While Eis not empty, choose a vertex a∈Vwith minimum degree and: 2.1 If ahas degree ≤d, remove its incident edges from E. 2.2 If ahas degree d+ 1, let a0,...,adbe its neighbors. Remove the edges aaifrom E. Add to Eall the edges aiajthat were not in E. Insert the atom {a, a0,...,ad}to the list L. 294 M. de Guzm´ an, D. Orden edges of one of the three cycles of length 4 in G. Equivalently, if and only if the planes containing four alternating triangles intersect. We start with the combinatorial decomposition of Gin which the vertex 6 is chosen at step 2 of Algorithm 3.4. This gives the atoms 2,3,4,5,6 and 1,2,3,4,5 (see Figure 2). In the sequel we detail the steps of the reconstruction process using Maple 7, and omitting some outputs of no interest: •According to the proof of Theorem 3.7, the first d+2 points p1,...,p5 can be arbitrarily chosen: > p1:=[0,0,0]: p2:=[1,1,1]: p3:=[0,1,0]: p4:=[1,0,0]: p5:=[0,0,1]: •Solving the corresponding equation (2) we get the tensions of the atom p1, p2, p3, p4, p5. In order to generate the equations, we use the command geneqns of the linalg package, which needs the equivalent transpose form Rt·wt= 0 of (2). Then we solve them and, since according to the proof of Theorem 3.7 the stress of the first atom can be considered a normalization constant, we take the value 1 for the parameter obtained. > d:=3: zeros:=[0,0,0]: with(linalg): > R:=matrix(10,5*d,[ op(p1-p2),op(p2-p1),op(zeros),op(zeros),op(zeros), op(p1-p3),op(zeros),op(p3-p1),op(zeros),op(zeros), op(p1-p4),op(zeros),op(zeros),op(p4-p1),op(zeros), op(p1-p5),op(zeros),op(zeros),op(zeros),op(p5-p1), op(zeros),op(p2-p3),op(p3-p2),op(zeros),op(zeros), op(zeros),op(p2-p4),op(zeros),op(p4-p2),op(zeros), op(zeros),op(p2-p5),op(zeros),op(zeros),op(p5-p2), op(zeros),op(zeros),op(p3-p4),op(p4-p3),op(zeros), op(zeros),op(zeros),op(p3-p5),op(zeros),op(p5-p3), op(zeros),op(zeros),op(zeros),op(p4-p5),op(p5-p4) ]): > Rt:=transpose(R): > eqs:=geneqns(Rt, [w12,w13,w14,w15,w23,w25,w26,w34,w36,w45,w46,w56], vector(n*d,0)): > tensions:=solve(eqs, w12,w13,w14,w15,w23,w24,w25,w34,w35,w45); From Graphs to Tensegrities 295 tensions := {w15 = −2w45, w35 = w45, w12 = 2 w45, w14 = −2w45, w23 = −w45, w34 = w45, w13 = −2w45, w45 = w45, w25 = −w45, w24 = −w45}. > tensions:=subs(w45=1,tensions); tensions := {1 = 1, w35 = 1, w12 = 2, w14 = −2, w23 = −1, w34 = 1, w13 = −2, w25 = −1, w24 = −1, w15 = −2}. •Then we have to add the atom p2, p3, p4, p5, p6, in which p6has unknown coordinates x,y,zand the stress of the atom is determined by the cancelation of tensions at edges p2p4and p3p5. The same operations as above lead to a second system of equations eqs2, in which we substitute w24 = 1, w35 = −1 to get eqs2subs := {w23+w25−w26 x+w26 = 0,−w23−w34−x w36 = 0, w34+ w45−w46 x+w46 = 0,−w25−w45−x w56 = 0,−w26+w26 x+x w36− w46+w46 x+x w56 = 0,−w26+w26 y−w36+w36 y+yw46+y w56 = 0,−w26+w26 z+z w36+z w46−w56+w56 z= 0,1+w25−w26 y+w26 = 0, w23+1−w26 z+w26 = 0, w34−1−w36 y+w36 = 0,−w23+1−z w36 = 0,−1−w34 −y w46 = 0,−1−w45 −z w46 = 0,−w25 + 1 −y w56 = 0,−1 + w45 −w56 z+w56 = 0}. •In order to look for non-null tension on every edge, Corollary 2.3 turns out to be crucial; if there is a non-null self-stress, then all its tensions are null. Therefore, the self-stress obtained when adding the two atoms is non-null since w12 = 2 6= 0 and this tension is not affected by the second atom. In particular, it is not needed to consider extra variables tij: In order to obtain the polynomial elimination, we compute a Groebner basis of the polynomials in eqs2subs, which we call polys2subs, for an elimination order in which variables x,y,zare smaller than the wij ’s. The command gbasis of the Groebner package is used: with(Groebner): G:=gbasis(polys2subs, lexdeg([w23,w25,w26,w34,w36,w45,w46,w56],[x,y,z])); G:= [x2−x−z2−y2+y+z,... and 40 polynomials more, involving wij ’s. We conclude that the hyperboloid x2−y2−z2−x+y+z= 0 contains the set of points p6:= (x, y, z) such that the framework admits a self stress non-null on every edge. Therefore this is a necessary condition. In order to test its sufficiency, we have considered an extra edge in the framework and forced its tension to be null, obtaining the following stress 296 M. de Guzm´ an, D. Orden from the left-kernel of the corresponding rigidity matrix: w:=                                     y+z−1−x −y+x−z+ 1 −y+x−z+ 1 −y+x−z+ 1 −y+x x−z 1 −y(−y+x−z+ 1) y+z+x−1 −−y+z+x−1 y+z+x−1 −z(−y+x−z+ 1) y+z+x−1 −y+x−z+ 1 y+z+x−1 −y−z−1 + x y+z+x−1                                     . We omit the (easy) computations checking that: •Under the condition x2−y2−z2−x+y+z= 0 of the hyperboloid, this is indeed a self-stress. •The denominator is not null: We observe that points p3,p4and p5 already lie on the plane y+z+x−1 = 0. Therefore, if point p6lied on that plane, the configuration would not be in general position. •Similar arguments prove that all the components of ware non-null. In conclusion, points p6= (x, y, z) for which a tensegrity exists are precisely those lying on x2−y2−z2−x+y+z= 0. We now observe that this hyperboloid contains the edges of the quadrilateral p2p3p4p5. Indeed, it is the only one passing through the initial five points and containing those four edges, since the space of all hyperboloids has dimension 9. In order to conclude the first condition in the statement we just have to observe that the combinatorial decompositions which at step 2 of Algorithm 3.4 choose vertices 1 or 6, lead to the 4-cycle 2345, those choosing 3 or 5 lead to 1264, and those choosing 2 or 4 to 1365 (see Figure 2). From Graphs to Tensegrities 297 For the equivalent condition in the statement, which appears in [5], easy computations check that the four planes defined by p1p2p3,p1p4p5, p3p4p6and p2p5p6intersect precisely for x2−y2−z2−x+y+z= 0, and the same condition is obtained for the other four alternate planes. Let us finally note that an interested reader can find in [8] a different symbolic approach to the resolution of this problem, that uses comprehensive Groebner basis. Remark 4.4.We have to call the attention of the reader to the fact that, in spite of dealing with more complicated problems than the geometric approach, the usefulness of the algebraic one is also limited by the size and type of the problem. For instance, if Corollary 2.3 cannot be used and all the inequations wij 6= 0 have to be considered in order to look for non-null tension on every edge. This would introduce more auxiliary variables t12, t13,...,t56 and a polynomial (w12t12 −1)(w13t13 −1) ···(w56t56 −1), making the computations infeasible. Finally, let us recall that the above symbolic computations follow the decomposition-reconstruction method from Section 3. We already noted that following Observation 4.2 it is also possible to compute symbolically the kernel of the 12 ×18 rigidity matrix R(P) for the whole framework G(P). Instead, with the decomposition-reconstruction approach we compute the kernel of the two 10 ×15 submatrices corresponding to the two atoms. The main advantage of the decomposition reconstruction method is related to this fact: Since w24 and w35 were no longer variables and the system had fewer equations, the computation of the Groebner basis was easier. In addition, we did not need to introduce an extra variable tij. Acknowledgements Both authors would like to thank the initial help and later encouraging comments and revisions by professors Juan Llovet and Tom´as Recio, specially about the algebraic part. The second author also wants to thank an anonymous reviewer for several valuable comments that helped improving the results in the paper. Final note The present paper is dedicated to the memory of Miguel de Guzm´an, who left us before it was finished. This work started when he gave a conference about tensegrities at the Departamento de Matem´aticas of the Universidad de Alcal´a in November 2003. Five months later he died 298 M. de Guzm´ an, D. Orden after a seminal version had been finished. 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Departamento de Matem´aticas Facultad de Ciencias Universidad de Alcal´a E-28871 Alcal´a de Henares (Madrid) Spain E-mail address:[email protected] Primera versi´o rebuda el 10 de mar¸c de 2005, darrera versi´o rebuda el 13 de mar¸c de 2006.