Equilibrium stressability of multidimensional frameworks
Oleg Karpenkov, Christian Müller, Gaiane Panina, Brigitte Servatius, Herman Servatius, Dirk Siersma
EEQUILIBRIUM STRESSABILITY OFMULTIDIMENSIONAL FRAMEWORKS
OLEG KARPENKOV, CHRISTIAN M ¨ULLER, GAIANE PANINA, BRIGITTESERVATIUS, HERMAN SERVATIUS, DIRK SIERSMA
Abstract.
We prove an equilibrium stressability criterium fortrivalent multidimensional tensegrities. The criterium appears indifferent languages: (1) in terms of stress monodromies, (2) interms of surgeries, (3) in terms of exact discrete 1-forms, and (4)in Cayley algebra terms.
Contents
1. Introduction 22. Main definitions and constructions 33. Self-stressability of frameworks. 63.1. Self-stressability of face-paths and face-cycles 63.2. HΦ-surgeries 113.3. Stress transition and stress monodromy 143.4. Face-path equivalence 143.5. Face-path d -frameworks in d -frameworks 164. Geometric Characterizations of self-stressability forTrivalent d -Frameworks 164.1. Ratio condition for self-stressable multidimensionaltrivalent frameworks 184.2. Cayley algebra conditions 205. R-frameworks and their self-stressability. Examples. 225.1. R-frameworks 225.2. Self-stresses and liftings 235.3. Some examples 26Acknowledgement. 29 Key words and phrases. framework, tensegrity, equilibrium stress, self-stress,discrete multiplicative 1-form, Cayley algebra, Maxwell-Cremona correspondence,lifting, Cayley algebra.The collaborative research on this article was supported by the “Research-In-Groups” program of ICMS Edinburgh, UK. O. Karpenkov is partially supportedby EPSRC grant EP/N014499/1 (LCMH) and C. M¨uller by the Austrian ScienceFund (FWF) through project P 29981. a r X i v : . [ m a t h . M G ] S e p KMPSSS
References 291.
Introduction
In the previous century, Fuller [9] coined the term tensegrity , a com-bination of ‘tension’ and ‘integrity’, to describe networks of rods andcables, such as those created by artist Kenneth Snelson, in which thetension of the cables and the compression in the rods combine to yieldstructural integrity to the whole. More generally, the word tensegrityis used to describe a variety of practical and abstract structures, e.g.bicycle tires and tents, whose rigidity follows from the balance of ten-sion and compression , in the mathematical literature the stress , on themembers.Practically, structures exhibiting tensegrity may be generated andanalyzed using conventional techniques of structural engineering [17].Theoretically, tensegrity is often considered as part of the study ofgeometric constraint systems, [2, 15]. The classical tensegrity modelconsists of a set of vertices V , and two graphs, ( V, S ) and (
V, C ), thegraph of struts and cables , and a placement function p : V → R d .One looks for a motion of the placed vertices such that the distancesbetween pairs of vertices connected by struts do not fall below theirinitial values, and such that the distances between pairs of verticesconnected by cables do not expand beyond their initial value. If nosuch motion exists, apart from the rigid motions of the space itself, thesystem is said to be rigid . A stress is a function s : S ∪ C → R , with s ( t ) ≥ s ( c ) ≤ c ∈ C , and struts, t ∈ S . A stressis an equilibrium stress if for each vertex v (cid:88) ( v,w ) ∈ C ∪ S s (( v, w ))( p ( v ) − p ( w )) = . There are two avenues in which the existence of a proper, i.e. nowherezero, equilibrium stress may allow one to establish structural integrity.A result of Roth and Whitely [12] states that if a tensegrity has a properequilibrium stress, and if the placement for (
V, C ∪ S ) is statically rigidas a bar and joint framework, then that tensegrity must be first orderrigid and hence rigid. More delicately, if the proper equilibrium stresspasses the second-order stress test of Connelley and Whiteley [3], thenthe tensegrity structure is second order rigid, hence rigid.Another important aspect of proper equilibrium stresses is the con-nection between the existence of an equilibrium stress and the liftingof embedded graphs into higher dimensions, as provided by the theory QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 3 of Maxwell-Cremona. It was understood by Lee, Ryshkov, Rybnikov,and some others that the connection between equilibrium stresses andlifts extends in certain cases to arbitrary CW-complexes M realized inany dimension d , also not necessarily embedded. Although the subjectcan be traced earlier (see [8, 19]), the first systematic study of multi-dimensional stresses, liftings and reciprocal diagrams was undertakenby Rybnikov in [13, 14].In the present paper we introduce d -frameworks and their equilib-rium stresses, i.e. self-stresses, in a slightly broader way than it wasdone by Rybnikov (Section 2). We introduce face paths and stresstransition along such face paths in d -frameworks in Section 3. We alsogive (Section 4) necessary and sufficient conditions for the equilibriumstressability of trivalent frameworks. This result appears as a general-ization of equilibrium stressability criteria for classical tensegrities [7].The conditions are equivalently expressed in terms of exact discretemultiplicative 1-forms, Cayley algebra, or, in terms of some surgeriesintroduced in Section 4.In Section 5, we explain how Rybnikov’s frameworks (R-frameworks,for short) arise in the proposed context. We give an equilibrium stress-ability criterion and derive some examples that demonstrate similaritiesand differences between planar and multidimensional tensegrities.2. Main definitions and constructions
Our model for tensegrity in this paper will be based on the followingstructure. Let
D > d ≥ plane means an affine subspace in R D . Definition 2.1. A d -framework F = ( E, F, I, n ) consists of E , acollection of ( d − R D ; F , a collection of d -dimensional planes in R D ; a subset I ⊂ { ( p, q ) ∈ ( E × F ) | p ⊂ q } ;a function n assigning to each pair ( e, f ) with e ∈ E and ( e, f ) ∈ I ,a unit vector n ( e, f ) which is contained in f and which is normal to e . We call planes from F faces and planes from E edges . The set I iscalled the set of incidences .A d -framework is called generic if, for every e ∈ E , all the planes f with ( e, f ) ∈ I are distinct.Let F = ( E, F, I, n ) be a d -framework. A stress s on F is anyfunction s : F → R . A framework F together with a stress s is said tobe in equilibrium if for every e ∈ E we have(1) (cid:88) ( e,f ) ∈ I s ( f ) n ( e, f ) = 0 . KMPSSS
Such a stress is called an equilibrium stress , a self-stress or sometimesa prestress for F . Definition 2.2. A d -framework is said to be self-stressable or a tenseg-rity if there exists a non-zero self-stress on it. Example 2.3.
The simplest non-trivial example here is the classicalcase of graphs in the plane ( D = 2 , d = 1).We say that a d -framework is trivalent if each element of E is incident(i.e., contained in a pair in I ) to precisely 3 elements of F .Surface based tensegrities of this type are models for minimal sur-faces (or, more generally, harmonic surfaces) which meet at edges, suchas, soap bubbles or tents. In this model, in the D = 3 case, the sur-faces are flat, and we can think of them as rigid plates, each having anexpansion or contraction coefficient, say caused by heat or cooling, forwhich the equilibrium condition indicates that the forces cancel on theedges, so that the framework does not deform. Example 2.4.
Let d = 2 and D = 3. Consider a 2-framework whoseedges, E , and faces, F , correspond to the edges and triangles of thegraph K , embedded in R . If four of the vertices, { , , , } , of K are placed as vertices of a regular tetrahedron and the fifth one astheir centroid, then the resulting 2-framework is generic in our sense.For each edge-face pair choose the normal to be a unit vector pointinginto the interior of the face-triangle. Note that the chosen normalssum to the zero vector around the lines corresponding to edges { i, } ,so choosing equal stress on these interior triangles leaves those edgesequilibrated. Then, it is easy to see that choosing stresses on theexterior and interior triangles in the ratio −√ / Example 2.5.
Again, let d = 2 and D = 3. We may create a different2-framework based on the graph K in R by keeping E as before,and associating the faces F to the K subgraphs of K , with the usualincidence relation. Since any two K ’s intersect in 3 edges, their faceplanes must be identical, and the 5 vertices of the embedded K mustbe coplanar.Since the normal vectors all lie in the plane of the K , it is no loss ofgenerality to assume that the vertices lie on a regular pentagon, and itis quickly checked that only the zero stress satisfies Equation (1). QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 5
Example 2.6.
Let d = 2 and D = 3. Consider the vertices of aregular cube and set E to be the set of all lines joining a pair of non-antipodal vertices (see Figure 1). The face planes F consist of all sixplanes containing the faces of the cube, together with the six planescontaining antipodal pairs of cube edges, as well as the eight planes ofthe dual tetrahedra. Let incidences be induced by containment. Since Figure 1.
A 3-framework based on the cube with three types of faces. each plane contains a polygon of edges supported by incident lines ofthe structure, we may take the normals to be inwardly pointing unitvectors. It is easy to check that the self-stresses on the three types offaces are in the ratio 1 : −√ √ / Example 2.7.
This example has two versions, both with d = 2 and D = 3. Consider the vertices of an octahedron, regularly embedded in R . The set of 12 edge lines E lie along the edges of the octahedron, Figure 2.
Three face types: triangles, squares, or faces with just two edges. and the set F of 11 face planes will consist of those eight supportingtriangles of the octahedron, together with the three planes which passthrough four coplanar vertices. Let the incidences be all those inducedby containment and, as before, let all normals be chosen inwardly point-ing with respect to the triangle or square to which they belong. Thenthis is easily computed to be stressable, and hence a tensegrity. In fact,the self-stress is unique, since each each line is incident to three distinctplanes. KMPSSS f e e e e e f f f f e e e f f f f f f Figure 3.
A face path contained in the framework described in Example 2.4 with f , = { , , } , f , = { , , } , f , = { , , } , f , = { , , } . As an alternative, we can take each of the three planes containingfour vertices to have multiplicity 2, with each one incident to a differ-ent pair of opposite lines, and with the same choice of normals. Thisstructure consisting of 12 lines and 14 planes is also a tensegrity.3.
Self-stressability of frameworks.
In this section we study self-stressability of trivalent d -frameworksin R d +1 , so starting from now on, we assume that D = d + 1.3.1. Self-stressability of face-paths and face-cycles.
Let us startwith the following general definition.3.1.1.
Face-path and face-cycle.
Let E = ( e , . . . , e k ) be a sequenceof distinct ( d − R D ; F = ( f , , f , , . . . , f k,k +1 )be a sequence of d -dimensional planes in R D ; ˆ F = ( ˆ f , . . . , ˆ f k ) be asequence of d -dimensional planes in R D . Then the collection ( E, F, ˆ F )is said to be a face-path if for i = 1 , . . . , k we have e i ⊂ f i,i +1 , e i ⊂ f i − ,i , and e i ⊂ ˆ f i . For example, removal of two antipodal triangles of an octahedron leavesa face-cycle with six faces.Denote the set of pairs defined by these inclusions by I . For a partic-ular choice of normals n we obtain a d -framework F = ( E, F ∪ ˆ F , I, n )which is called a face-path d -framework , see Figure 3. It is called a face-cycle d -framework if f , = f k,k +1 . In this case it is denoted by C ( E, F, ˆ F , n ), see Figure 4.3.1.2. Self-stressability of face-path d -frameworks. Proposition 3.1.
Any generic face-path d -framework which containsno face-cycle has a one-dimensional space of self-stresses. All thestresses for all the planes of F and ˆ F are either simultaneously zero,or simultaneously non-zero. QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 7 f e e e e e f f f f f e e e e e e f f f f f f f f4 f Figure 4.
A face path contained in the framework described in Example 2.4, withlabeling as before and f , = { , , } . Proof.
Setting s ( f , ) = 1 we inductively define all stresses for all otherplanes using Equation (1). Therefore a non-zero self-stress exists. Byconstruction the obtained stress is non-zero at all planes of F and ˆ F .Once we know any of the stresses at one of the planes of F and ˆ F , wereconstruct the remaining stresses uniquely using Equation (1). Hencethe space of stresses is at most one-dimensional. Therefore, all self-stresses are proportional to a self-stress that is non-zero at all planesof F and ˆ F . (cid:3) Edge-orientation transition.
Suppose we have a face path whoseedges are e , e , . . . , and we are given an orientation of e i by declaringa frame spanning e i as positive. We pass over this positive orientationon e i to a positive orientation on e i +1 by requiring that the given frameof e i together with n ( e i , f i,i +1 ) and a new (chosen to be positive) frameof e i +1 together with n ( e i +1 , f i,i +1 ) differ by an orientation reversingautomorphism on f i,i +1 . We call this the edge-orientation transition .A face-cycle d -framework C ( E, F, ˆ F , n ) is said to be edge-orientable ifthe edge-orientation transition around the cycle returns to the startingedge in its initial orientation.Non-orientable face-cycles are a usual phenomenon in frameworks.Indeed, we see them even in small examples like Example 2.4. Figure 5depicts such a face-cycle. Here the first and the last edges coincide,but are oppositely oriented.Note that the edge-orientability of a cycle depends neither on thechoice of the first element e ∈ E , nor the choice of direction in thecycle. We observe the following proposition. Proposition 3.2.
A face-cycle d -framework C = C ( E, F, ˆ F , n ) hasthe following properties. KMPSSS f e e e e e e f f f f
403 34 15 22
Figure 5.
A face-cycle contained in K which is not edge-orientable. For thisparticular example faces correspond to triangles, and we choose all the normals topoint inward. (i) Reversing simultaneously all the normals at a single e i ∈ E (namely n ( e i , f i − ,i ), n ( e i , f i,i +1 ), and n ( e i , ˆ f i )) does not changethe self-stressability or orientability of C .(ii) Reversing simultaneously the normals at f i,i +1 ∈ F (namely n ( e i , f i,i +1 ) and n ( e i +1 , f i,i +1 )) does not change the self-stressabilityor orientability of C .(iii) Reversing the normal n ( e i , ˆ f i ) does not change the self-stressabilityor orientability of C . Proof.
In all the items we change altogether an even number of normalsfor all the faces in F . Therefore, orientability is preserved.The change in (i) does not change the equations of self-stressability,so it preserves self-stressability. For (ii) and (iii) the change of the signsof stresses s ( f i,i +1 ) and s ( ˆ f i ) respectively delivers the equivalence of theconditions of self-stressability. (cid:3) Self-stressability of face-cycles of length 3.
Let us consider atrivalent cycle C ( E, F, ˆ F , n ) of length 3 with F = { f , f , f } , ˆ F = { ˆ f , ˆ f , ˆ f } , E = { e , e , e } (for a schematic sketch see Figure 8). Create a new plane ˆ f (cid:48) by thefollowing Cayley algebra algorithm (see Figure 6):(i) g = ( e ∨ e ) ∧ ˆ f ,(ii) g = f , ∧ ˆ f ,(iii) g = ( g ∨ g ) ∧ f , ,(iv) g = ( g ∨ e ) ∧ ( g ∨ e ),(v) ˆ f (cid:48) = e ∨ g .In this notation, the stressability conditions are given by the follow-ing. Proposition 3.3.
A face-cycle d -framework C (cid:0) ( e , e , e ) , ( f , , f , , f , ) , ( ˆ f , ˆ f , ˆ f ) , n (cid:1) QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 9 is self-stressable if and only ifdim( ˆ f ∩ ˆ f ∩ ˆ f ) = d − C is edge-orientable, anddim( ˆ f ∩ ˆ f ∩ ˆ f (cid:48) ) = d − C is non-edge-orientable, where ˆ f (cid:48) is constructed as above in step (v)(see also Figure 6). e e e g g g g ˆ f ˆ f ˆ f (cid:48) Figure 6.
The plane ˆ f (cid:48) can be constructed using Cayley algebra since the twolines f , , f , separate the two lines ˆ f , ˆ f (cid:48) harmonically. Proof.
Let us first examine the case where d = 1 and D = 2, withstresses at all edges of the triangles equal to 1. The triangle in Figure 7(left) corresponds to an edge-oriented d -framework and therefore ˆ f ∩ ˆ f ∩ ˆ f is not empty. Note that this intersection is empty for the Figure 7.
Orientable ( left ) and non-orientable ( right ) self-stressed d -frameworks. triangle in Figure 7 (right), which corresponds to a non-edge-orientedchoice of normals. The condition for non-edge-oriented tensegrities ismore complicated as we will see below. Let us consider a self-stressed trivalent cycle C ( E, F, ˆ F , n ) of length3 (for a schematic sketch see Figure 8). Then let us change the direc-tion of the normal n ( e , f , ). Consequently, its orientability changesand the old stress for this new cycle is not a self-stress. However, bychanging the d -plane ˆ f to a new d -plane ˆ f (cid:48) we can resolve the stressesaround e again to reobtain a self-stressed framework. n ( e , f , ) e e e f , f , ˆ f ˆ f ˆ f (cid:48) n ( e , f , ) Figure 8.
Self-stressed orientable and non-orientable face-cycles of length 3.
Left :An orientable face-cycle.
Right : Reversing the normal n ( e , f , ) yields a non-orientable face-cycle which is still self-stressable after replacing ˆ f by ˆ f (cid:48) . Let us consider the following two cases in the planar situation asdepicted by Figure 8. The classical tensegrity (Figure 8), which cor-responds to the orientable case, has the property that the lines ˆ f , ˆ f , ˆ f meet in a point (see, e.g., [6]). Now changing the orientation of n ( e , f , )yields ˆ f (cid:48) as the new d -plane (see Figure 8 right). From the parallelo-gram in Figure 8 (right) we derive the condition for the non-orientable case. Standard projective geometry implies [11] that the two lines f , , f , separate the two lines ˆ f , ˆ f (cid:48) harmonically. This property ischaracterized by incidence relations of points and lines and thereforeexpressible in terms of Cayley algebra.Next we describe how to reduce any dimension d to the above one-dimensional case. Denote by Π the intersection Π = f , ∩ f , ∩ f , .Observe that dim Π = d −
2. It it well known that self-stressability isa projective invariant, see [18], so we can consider the plane Π to be atinfinity.Fix a two-dimensional plane π orthogonal to e , e , and e (this ispossible since Π is at infinity).Now the face-cycle d -framework C is a Cartesian product of R d − with the two-dimensional tensegrity F = C ∩ π in the plane π (seeFigure 9). Stresses of F are in a bijection with the stresses of the QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 11 initial tensegrity. So the problem is reduced to the planar situation,i.e., to a 1-framework in the two-dimensional plane.Let us now consider the edge-orientable case. According to Propo-sition 3.2 the problem has been reduced to the case of normals in Fig-ure 9. The necessary and sufficient condition in the plane is that thethree lines ˆ f ∩ π, ˆ f ∩ π, ˆ f ∩ π, intersect in one point, say a (see, e.g., in [6]). Therefore C is self-stressable if and only if the three planes ˆ f , ˆ f , and ˆ f intersect in acommon ( d − a and Π). π Figure 9.
A face-cycle d -framework C and the corresponding tensegrity C ∩ π . The non-edge-orientable case is reduced to the edge-orientable in thefollowing way. Let us make our three-cycle orientable by changing thelast normal n ( e , f ) (denote the resulting set of normals by n (cid:48) ). Inthis case, in order to preserve the property of self-stressability condi-tion for at the edge e , we should also change the sign of one of thecoordinates for the plane ˆ f . The resulting plane is the plane ˆ f (cid:48) , whoseCayley algebra expression is described above (see step (v)). Now thestressability of the original non-edge-orientable cycle is equivalent tothe stressability of an edge-oriented cycle C (cid:0) ( e , e , e ) , ( f , , f , , f , ) , ( ˆ f , ˆ f , ˆ f (cid:48) ) , n (cid:48) (cid:1) . This concludes the proof. (cid:3) -surgeries.
In this section we discuss HΦ-surgeries and el-ementary surgery-flips on face-paths d -frameworks and face-cycle d -frameworks which preserve self-stressability in R d +1 . Definition 3.4.
Let 1 ≤ i ≤ n be positive integers ( n ≥ C = (cid:0) ( e , . . . , e n ) , ( f , , f , , . . . , f n, ) , ( ˆ f , . . . , ˆ f n ) , n ) (cid:1) be a face-path (or a face-cycle if f n, = f , ) d -framework. Denoteˆ f (cid:48) i = (cid:104) f i − ,i ∩ f i,i +1 , ˆ f i ∩ ˆ f i +1 (cid:105) . We say that the HΦ i -surgery of C is the following face-path (face-cycle) d -framework (see Figure 10)HΦ i ( C ) = (cid:0) ( e , . . . , e i − , f i − ,i ∩ f i +1 ,i +2 , e i +2 , . . . , e n )( f , , . . . , f i − ,i , f i +1 ,i +2 , . . . , f n, );( ˆ f , . . . , ˆ f i − , ˆ f (cid:48) i , ˆ f i +2 , . . . , ˆ f n ) , n (cid:48) (cid:1) . The normals n (cid:48) coincide with the normals of n for the same adjacent a) b)c) d) Figure 10.
An HΦ i -surgery. pairs. We have three extra normals in n (cid:48) to the new element in E : n (cid:48) ( f i − ,i ∩ f i,i +1 , f i − ,i ) , n (cid:48) ( f i − ,i ∩ f i,i +1 , f i,i +1 ) , and n (cid:48) ( f i − ,i ∩ f i,i +1 , ˆ f i ) . The first two are defined by the fact that the cycle: (cid:16) ( e i , f i − ,i ∩ f i,i +1 , e i +1 ) , ( f i − ,i , f i +1 ,i +2 , f i,i +1 ) , ( ˆ f i , ˆ f (cid:48) i , ˆ f i +1 ) , n (cid:48) (cid:17) of length 3 is edge-orientable. The orientation of ˆ f i does not play anyrole here (and hence can be chosen arbitrarily).Alternatively, in terms of Cayley algebra ˆ f (cid:48) i readsˆ f (cid:48) i = ( f i − ,i ∧ f i,i +1 ) ∨ ( ˆ f i ∧ ˆ f i +1 ) . In the planar case we have precisely HΦ-surgeries on framed cycles(i.e., face-cycle 1-frameworks) that were used for the conditions of pla-nar tensegrities (for further details see [6]).
QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 13
In order to have a well-defined HΦ-surgery, one should consider sev-eral simple conditions on the elements of F and ˆ F . Definition 3.5.
We say that an HΦ i ( C )-surgery is admissible if(i) the planes f i − ,i and f i,i +1 do not coincide;(ii) the planes ˆ f i and ˆ f i +1 do not coincide;(iii) the planes f i − ,i ∧ f i,i +1 and ˆ f i ∧ ˆ f i +1 do not coincide.For an admissible HΦ i ( C )-surgery we havedim f i − ,i ∩ f i +1 ,i +2 = d − f (cid:48) i = d. dim ˆ f i ∩ ˆ f i +1 = d − . where the last follows from Items (i) and (ii) above. Then by Item (iii)we have dim ˆ f (cid:48) i ≥ d . Since the d -planes f i − ,i , f i +1 ,i +2 , ˆ f i , and ˆ f i +1 byconstruction share a ( d − f (cid:48) i ≤ d .Let us distinguish the following elementary surgery-flips. Definition 3.6. An elementary surgery-flip is one of the followingsurgeries. • An admissible HΦ i -surgery or its inverse. • Removing or adding consecutive duplicates at position i . Herewe say that we have a duplicate at position i if e i = e i +1 , f i,i +1 = f i +1 ,i +2 , and ˆ f i = ˆ f i +1 . • Removing or adding a loop of length 2. Here we say that wehave a simple loop of length at position i if e i = e i +2 , f i,i +1 = f i +2 ,i +3 , and ˆ f i = ˆ f i +2 . Proposition 3.7.
Assuming that a surgery is admissible, a face-cycle d -framework C is self-stressable if and only if the face-cycle d -frameworkHΦ i ( C ) is self-stressable. Proof.
Assume that C has a non-zero self-stress s . Let us show thatHΦ i ( C ) has a self-stress.Consider the face-cycle d -framework C i = (cid:0) ( e i , f i − ,i ∩ f i,i +1 ) , ( f i − ,i , f i +1 ,i +2 , f i,i +1 ) , ( ˆ f i , ˆ f (cid:48) i , ˆ f i +1 ) , n (cid:1) , where ˆ f (cid:48) i = (cid:10) f i − ,i ∩ f i,i +1 , ˆ f i ∩ ˆ f i +1 (cid:11) . and n is constructed according Definition 3.4. This face-cycle d -frame-work admits a self-stress by Proposition 3.3 since three d -planes of( f i , f (cid:48) i , f i +1 ) intersect in a plane of dimension d −
2. Now let us add C i to C taking the self-stress s i which negates the stress at e i,i +1 . Thenthe stresses at ˆ f i for C and C i negate each other; and the stresses at f i − ,i for C and C i coincide. For the same reason the stresses at f i +1 ,i +2 for C and C i coincide. Therefore, the constructed self-stress is in facta non-zero self-stress on HΦ i ( C ).The same reasoning works for the converse statement. In fact addingthe C i to C provides an isomorphism between the space of self-stresseson C and the space of self-stresses on HΦ i ( C ). (cid:3) Remark 3.8.
It is possible to describe one HΦ-surgery in terms ofCayley algebra. Consider a face-cycle d -framework C with admissibleHΦ i ( C )-surgery. We have only one new plane ˆ f (cid:48) i in this case, and itsCayley expression isˆ f (cid:48) i = ( f i − ,i ∧ f i,i +1 ) ∨ ( e i ∧ e i +1 ) . Stress transition and stress monodromy.
We will now adaptto our setting the notion of “quality transfer” due to Rybnikov [13].
Definition 3.9.
Let Γ be a generic face-path d -framework with startingplane f a ∈ F and ending plane f z ∈ F . Assign some stress s to the firstplane. Due to genericity, it uniquely defines the stress on the secondface. The stress on the second face uniquely defines the stress on thethird face, and so on. So the stress on f a uniquely defines the stress on f z . This is called the stress transition along the face path.If f a = f z , that is, we have a face-cycle, we arrive eventually atsome stress s (cid:48) assigned to f a again. The ratio s ( f a ) /s ( f z ) is called the stress-monodromy along C . A stress monodromy of 1 is trivial .It is clear that: Lemma 3.10. (1) A generic face-cycle is self-stressable if and onlyif the stress monodromy is trivial.(2) The monodromy does not depend on the choice of the first face.(3) Reversal of the direction of the cycle takes monodromy m to1 /m .(4) Monodromy behaves multiplicatively with respect to homolog-ical addition: the monodromy of the homological sum is theproduct of monodromies. (cid:3) Face-path equivalence.
Let us now introduce the notion of equiv-alent face-path d -frameworks. Definition 3.11.
Two face-path (face-cycle) d -frameworks Γ andΓ starting from the plane f a and ending at the plane f z are equivalent if there exists a sequence of elementary surgery-flips taking Γ to Γ . QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 15
It turns out that equivalent face-path d -frameworks have equivalentstress-transitions. Proposition 3.12.
The stress-transition of two equivalent face-path d -frameworks coincide. Proof.
It is enough to prove this statement for any elementary surgery-flip. In case of HΦ-surgeries we must show that the face-path d -frameworks C = (cid:0) ( e , e ) , ( f , , f , , f , ) , ( ˆ f , ˆ f ) , N (cid:1) , and HΦ ( C ) = (cid:0) ( e ) , ( f , , f , ) , ( ˆ f ) , N (cid:48) (cid:1) have the same stress-transition (see Figure 11). This is equivalent to e e e f , f , f , ˆ f ˆ f ˆ f Figure 11.
An elementary flip. the fact that the face-cycle d -framework (cid:0) ( e , e , e ) , ( f , , f , , f , ) , ( ˆ f , ˆ f , ˆ f ) , n (cid:48)(cid:48) (cid:1) (where n (cid:48)(cid:48) is as in the cycle of Definition 3.4) has a unit stress-transition(i.e. trivial monodromy or, equivalently, is self-stressable).By the construction of Definition 3.4 we get that this cycle is edge-orientable, and that the intersectionˆ f ∩ ˆ f ∩ ˆ f (cid:54) = ∅ . Therefore, by Proposition 3.3 it is self-stressable.The cases of removing duplicates or loops of length 2 are straight-forward. (cid:3)
Face-path d -frameworks in d -frameworks. In this subsectionwe briefly discuss face-path d -frameworks and face-cycle d -frameworksthat are parts of a larger d -framework. Definition 3.13.
Let F be a trivalent d -framework. Then for every(cyclic) sequence of adjacent d -planes γ we naturally associate a face-path d -framework (face-cycle d -framework) Γ( T, γ ) with • F is the sequence of the planes spanned by the corresponding d -planes of γ ; • E is the sequence of the intersections of the d -planes of theabove F ; • ˆ F is the sequence of planes of F that are adjacent to the planesof E and distinct to the faces already considered in F ; • n is the corresponding sequence of normals defined by the nor-mals of F .We say that a face-path d -framework (face-cycle d -framework) Γ( T, γ )is induced by γ on F .Induced face-path and face-cycle d -frameworks have a natural homo-topy relation, which is defined as follows. Definition 3.14. • Two induced face-path d -frameworks Γ and Γ for F startingfrom the plane f a and ending at the plane f z are face-homotopic if there exists a sequence of elementary surgery-flips taking Γ to Γ and such that after each surgery-flip we have an inducedface-path d -framework for F . • Two face-cycle d -frameworks Γ and Γ are face-homotopic ifthere exists a sequence of elementary surgery-flips taking Γ toΓ and such that after each surgery-flip we have an inducedface-path d -framework for G ( M ).Finally we formulate the following important property of face-homo-topic face-path and face-cycle d -frameworks. Proposition 3.15.
Face-homotopic face-path (face-cycle) d -frameworkshave the same stress-transition (stress-monodromy). Proof.
The proof directly follows from Proposition 3.12. (cid:3) Geometric Characterizations of self-stressability forTrivalent d -Frameworks In this section we discuss the practical question of writing geomet-ric conditions for cycles. We characterize self-stressable trivalent d -frameworks in terms of exact discrete multiplicative 1-forms and in QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 17 terms of resolvable cycles. Before that we show that a trivalent d -framework is self-stressable if and only if every path and every loop isself-stressable. Definition 4.1.
A face-cycle d -framework is called a face-loop d -frame-work if it contains no repeating planes.Let us formulate the following general theorem. Theorem 4.2.
Consider a generic face-connected trivalent d -framework.Then the following three statements are equivalent. (i) F has a non-zero self-stress ( which is in fact non-zero at any d -plane ) . (ii) For every two d -planes f a , f z in F the stress-transition does notdepend on the choice of an induced face-path d -framework on F . (iii) Every induced face-loop d -framework on F is self-stressable.Proof. (ii) ⇔ (i): Item (i) tautologically implies Item (ii). Let us showthat Item (ii) implies Item (i). Fix a starting face f a and put a stress s ( f a ) = 1 on it. Expand the stress to all the other faces. By assump-tion this can be done uniquely. Therefore, this stress is a self-stress.(Indeed, if we do not have the equilibrium condition at some plane e , then at the planes incident to e we have more than one possiblestress-transition.)(ii) ⇒ (iii): Indeed any simple face-cycle d -framework on F can beconsidered as one long face-path d -framework with f a = f z . By condi-tion of Items (ii) the stress-transition equals 1, and therefore this face-cycle d -framework is self-stressable. The last is equivalent to Item (iii).(iii) ⇒ (ii): Let us use reductio ad absurdum. Suppose Item (iii)is true while Item (ii) is false. If Item (ii) is false then there existat least two face-path d -frameworks with the same f a and f z wherethe stress-transitions fail to be the same. Now the union C of thefirst face-path d -framework and the inverse second is a induced face-cycle d -framework on G ( M ) with non-unit stress-transition. Let ussplit C into consecutive loops C , . . . , C k . At least one of them shouldhave a non-unit translation. Therefore, Item (iii) is false as well, acontradiction.For completeness of the last proof we should add the following twoobservations regarding cycles of small length. Firstly, the stress-tran-sition remains constant at planes that repeat successively two or moretimes. This happens due to genericity of F : there are zero contribu-tions from ˆ f i in case if f i − ,i = f i,i +1 . And secondly, if it happens that f i − ,i = f i +1 ,i +2 then we immediately have e i = e i + 1 and thereforeagain the stress-transitions at f i − ,i and at f i +1 ,i +2 coincide. (cid:3) Ratio condition for self-stressable multidimensional triva-lent frameworks.
In this section we characterize generic trivalent d -frameworks F with respect to their self-stressability in terms of specificproducts of ratios. More precisely, we equip each d -framework with aso called discrete multiplicative 1-form which turns out to be exact ifand only if the d -framework is self-stressable. Let us start with thedefinition of discrete multiplicative 1-forms (see, e.g., [1]). Definition 4.3.
A real valued function q : (cid:126)E ( G ) → R \ { } (where (cid:126)E ( G ) denotes the set of oriented edges of the graph G ) is called a discrete multiplicative -form , if q ( − a ) = 1 /q ( a ) for every a ∈ (cid:126)E ( G ).It is called exact if for every cycle a , . . . , a k of directed edges the valuesof the 1-form multiply to 1, i.e., q ( a ) · . . . · q ( a k ) = 1 . Now, as a next step we will equip any general trivalent d -frameworkwith a discrete multiplicative 1-form q . However, we will not define q directly on the d -framework but on what we call its dual graph. Definition 4.4.
The vertices of the dual graph of a d -framework arethe d -dimensional planes and the edges “connect d -dimensional planes”that are sharing a ( d − F can be identified withtriples of successive ( d − a i := ( e i − , e i , e i +1 ) (where e i ∈ E ).So let us now equip the dual graph of F with a discrete multiplicative1-form. For an illustration see Figure 12. e i e i − e i +1 f i − , i f i , i + ˆ f i n ( e i , f i − , i ) n ( e i , f i , i + ) r i Figure 12.
Illustration of some edges of the dual graph in a face loop of a d -framework. The values of the discrete multiplicative 1-form q ( a i ) (cf. Eqn. (3)) isthe affine ratio q ( a i ) = (cid:0) n ( e i , f i − ,i ) − r i (cid:1) : (cid:0) r i − n ( e i , f i,i +1 ) (cid:1) . QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 19
As our d -framework is trivalent the ( d − e i is contained inthree d -planes f i − ,i , f i,i +1 , ˆ f i and therefore the corresponding normals n ( e i , f i − ,i ), n ( e i , f i,i +1 ), n ( e i , ˆ f i ) are linearly dependent, i.e., lie in a2-plane. This together with the fact that the d -framework is genericimplies that there are λ i − ,i , λ i,i +1 , ˆ λ ∈ R \ { } such that(2) λ i − ,i n ( e i , f i − ,i ) + λ i,i +1 n ( e i , f i,i +1 ) + ˆ λ n ( e i , ˆ f ) = 0 . Now we are in position to define our discrete multiplicative 1-formon the oriented dual graph by(3) q ( a i ) = q ( e i − , e i , e i +1 ) := λ i,i +1 λ i − ,i , since clearly q ( − a i ) = 1 /q ( a i ) is fulfilled. The geometric meaning of q ( a i ) is the following (cf. Figure 12 right). Denote by r i the intersectionpoint of the straight line with direction n ( e i , ˆ f ) and intersect it withthe line through n ( e i , f i − ,i ) and n ( e i , f i,i +1 ). A simple computationshows r i = λ i − ,i λ i − ,i + λ i,i +1 n ( e i , f i − ,i ) + λ i,i +1 λ i − ,i + λ i,i +1 n ( e i , f i,i +1 ) . Thus q ( a i ) is the affine ratio of the three points n ( e i , f i − ,i ) , r i , n ( e i , f i,i +1 ),i.e., q ( a i ) = (cid:0) n ( e i , f i − ,i ) − r i (cid:1) : (cid:0) r i − n ( e i , f i,i +1 ) (cid:1) .With that definition of a discrete multiplicative 1-form we can nowcharacterize self-stressable d -frameworks. Theorem 4.5.
A generic trivalent d -framework is self-stressable if andonly if the discrete multiplicative -form defined by (3) is exact.Proof. Suppose the d -framework has a self-stress s . Therefore Equa-tion (1) implies s ( f i − ,i ) n ( e i , f i − ,i ) + s ( f i,i +1 ) n ( e i , f i,i +1 ) + s ( ˆ f ) n ( e i , ˆ f ) = 0 . Comparison with Equation (2) implies that the coefficients in bothequations are just a multiple of each other, i.e.,( s ( f i − ,i ) , s ( f i,i +1 ) , s ( ˆ f )) = µ ( λ i − ,i , λ i,i +1 , ˆ λ ) . Consequently, the value of the discrete multiplicative 1-form is the ratioof neighboring stresses: q ( a i ) = λ i,i +1 λ i − ,i = s ( f i,i +1 ) s ( f i − ,i ) . Therefore it is easy to see that the product of values q ( a i ) along anyclosed loop in the dual graph multiplies to 1: q ( a ) · . . . · q ( a k ) = s ( f , ) s ( f k, ) s ( f , ) s ( f , ) · . . . · s ( f k − ,k ) s ( f k − ,k − ) s ( f k, ) s ( f k − ,k ) = 1 . Now conversely, let us assume that the discrete multiplicative 1-form q is exact. By Theorem 4.2 it is sufficient to show that each loopof the form e , . . . , e k of ( d − s ( f , ) ∈ R \ { } for the first d -plane. Equation (1) andthe d -framework being generic then uniquely determines the stressesof the two other d -planes incident to e , that is, s ( f k, ) and s ( ˆ f ).Continuing, determining stresses this way defines all stresses along theloop including the last stress that we now denote by ˜ s ( f k, ) because itwas defined before. However, the exactness of q gives1 = q ( a ) · . . . · q ( a k ) = λ , λ k, λ , λ , · . . . · λ k − ,k λ k − ,k − λ k, λ k − ,k = s ( f , ) s ( f k, ) s ( f , ) s ( f , ) · . . . · s ( f k − ,k ) s ( f k − ,k − ) ˜ s ( f k, ) s ( f k − ,k )= ˜ s ( f k, ) s ( f k, ) , so ˜ s ( f k, ) = s ( f k, ). Consequently, we can consistently define a non-zero stress. (cid:3) Cayley algebra conditions.
Let us start with the following im-portant definition.
Definition 4.6. • A face-cycle d -framework of length 3 is in general position if all6 planes in the sequences F and ˆ F are pairwise distinct. • A face-cycle d -framework is resolvable if there exists a sequenceof HΦ-surgeries transforming it to a face-cycle d -framework oflength 3 in general position. • A d -framework is resolvable if all its simple induced face-cycle d -frameworks are resolvable.We continue with the following definition. Definition 4.7 ( Cayley algebra condition for a single face-cycleresolvable d -framework). Consider a resolvable face-cycle d -frame-work C = ( E, F, ˆ F , n ) QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 21 and any sequence of HΦ-surgeries transforming it to a face-cycle d -framework C (cid:48) = ( E (cid:48) , F (cid:48) , ˆ F (cid:48) , n (cid:48) )of length 3 in general position. • Let us write all elements of F (cid:48) and ˆ F (cid:48) in C (cid:48) as Cayley algebraexpressions of the elements of F and ˆ F in C . The resultingexpressions are compositions of expressions of Remark 3.8. • Finally, we use the dimension condition of Proposition 3.3 for C (cid:48) to determine if C (cid:48) is stressable or not.The composition of the above two items gives an existence conditionfor nonzero self-stresses on C . We call this condition a Cayley algebrageometric condition for C to admit a non-zero self-stress.Note that one can write distinct Cayley algebra geometric conditionfor C using different sequences of HΦ-surgeries transforming C to aface-cycle d -framework of length 3 in general position.As we have already mentioned in the above definition the Cayleyalgebra geometric conditions detect self-stressability on cycles. So wecan write self-stressability conditions for a general trivalent resolvable d -framework. Namely we have the following theorem. Theorem 4.8.
Let F be a trivalent resolvable d -framework and let fur-ther C , . . . , C n be all pairwise non-face-homotopic face-loop d -frame-works on F . Then F has a self-stress if and only if it fulfills Cayleyalgebra geometric conditions for cycles C , . . . , C n as in Definition 4.7.Proof. First of all the self-stressability C , . . . , C n is equivalent to self-stressability of all face-loops on F . It follows directly from definitionof face-homotopic paths. Hence by Theorem 4.2 the self-stressabilityof all C , . . . , C n is equivalent to self-stressability of F itself.Finally the geometric conditions for face-loop d -frameworks C i ( i =1 , . . . , n ) are described in Definition 4.7. (cid:3) Corollary 4.9.
Each realization of K in R is self-stressable. Here wemean a realization of a 2-framework associated with K , in the spiritof Example 2.4. Namely, the edges are all the edges of K , and facesare all the associated triangles.Indeed, each face-loop in K is face-homotopic to a face-loop of lengththree. For them, the stressability condition is automatic. Anotherargument of stressability of K will appear later in Remark 5.11 as aconsequence of Theorem 5.10. R-frameworks and their self-stressability. Examples.
In this section we work in the settings of Rybnikov’s papers [13, 14].5.1.
R-frameworks.
Informally, R-frameworks are PL (piecewise lin-ear) realizations of CW-complexes in R d +1 . To make this precise, letus start with a reminder about CW complexes.A finite CW-complex is constructed inductively by defining its skeleta(for details see, e.g., [5]). The zero skeleton sk is a finite set of pointscalled vertices . Once the ( k − sk k − is constructed, a finitecollection of closed k -balls B i (called cells ) is attached by some contin-uous mappings φ i : ∂B i → sk k − . The images of B i in the complex arecalled closed cells . Definition 5.1. A regular CW-complex is a CW-complex such that(i) For each k -cell B i , the mapping φ i is a homeomorphism between ∂B i and a subcomplex of the skeleton sk k − .(ii) The intersection of two closed cells is either empty or somesingle closed cell of this CW-complex.Let M be a regular finite CW-complex with no cells of dimensiongreater than d .Its faces of dimension d will be called d -faces . The ( d − d -edges . The ( d − d -vertices . Example 5.2.
Let M be a regular finite CW-complex whose support | M | is a connected ( d +1)-manifold, either closed or with boundary. Let M be its d -skeleton.In this setting we also have cells of M of dimension d + 1. These willbe called chambers . Definition 5.3.
Assume that a mapping p : Vert( M ) → R d +1 is suchthat the image of the vertex set of each k -cell spans some affine k -plane.We say that p realizes M in R d +1 ; we also say that the pair ( M, p )is a realization of M , or a Rybnikov-framework , or
R-framework , forshort.Notation: given a cell f ∈ M , we abbreviate the image of the vertexset p (Vert( F )) as p ( f ) and denote by (cid:104) p ( f ) (cid:105) its affine span. Definition 5.4.
An R-framework is generic if, whenever two d -faces f and f share a d -edge, then (cid:104) p ( f ) (cid:105) (cid:54) = (cid:104) p ( f ) (cid:105) . The support of a CW-complex is the topological space represented by the com-plex. That is, one forgets the combinatorics and leaves the topology only.
QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 23
From now on we assume that all R-frameworks we deal with aregeneric.As before, we say that an R-framework is trivalent if each d -edge isincident to exactly three d -faces.An R-framework is (3 , -valent if each d -vertex is incident to exactlyfour d -edges (and therefore, to six d -faces). Informal remark: by construction, vertices are mapped to points.One may also imagine that the 1-cells of the complex are mapped to linesegments. Therefore 2-cells are mapped to some closed planar brokenlines (polygons). Here self-intersections may occur. As the dimensionof faces grows, the more complicated the associated geometrical objectis.However, there exist nice examples with convex polyhedra as imagesof the faces. In particular, if a face is a (combinatorial) simplex, onemay think that its image is a simplex lying in R d . Example 5.5. (i)
The Schlegel diagram [20] of a convex ( d + 2)-polytope K is a realisation of the boundary complex of K .(ii) More generally, the projection of a ( d +2)-dimensional polyhe-dral body K (that is, of a body with piecewise linear boundary)to R d +1 yields a realization ( M, p ) where | M | is homeomorphicto ∂K .5.2. Self-stresses and liftings.
Now we turn to a particular notion ofstresses, which is borrowed from Rybnikov’s paper [13] and representsa special case of Definition 2.1. The principal difference is that inRybnikov’s setting the choice of normal vectors n is dictated by theR-framework.As Examples 2.4 and 5.5 show, in certain cases, a realization ( M, p )represents all the faces as convex polytopes. Let us call such a real-ization non-crossing . Otherwise, we say that the realization is self-crossing .Let us start by introducing stresses for the non-crossing version : Definition 5.6 (cf. [14]) . Assume that a non-crossing realization (
M, p )is fixed. Let us assign to each pair ( f, e ) where e is a d -edge containedin a d -face f a unit normal n ( e, f ) to p ( e ) pointing inside the convexpolytope p ( f ).A real-valued function s defined on the set of d -faces is called a self-stress if at each d -edge e of the complex, (cid:88) f ⊃ e s ( f ) n ( e, f ) = 0 . ( ∗ ) To relax the non-crossing condition, let us make some preparationfollowing [14]. The informal idea is to triangulate the faces of thecomplex, since the representation of a simplex is never self-crossing.Pick a (combinatorial) orientation of each of the cells of M , and a(combinatorial) triangulation of M without adding new vertices. Soeach d -face now is replaced by a collection of (combinatorial) sim-plices. The realization ( M, p ) yields a realization (
M , p ) of the newCW-complex.
Definition 5.7 (cf. [14]) . Assume that a generic realization (
M, p ) isfixed. Choose a triangulation (
M , p ) as is described above.For a d -edge e ∈ M and a d -face f containing g , choose n ( e, f ) tobe the unit normal to the oriented cell f at its simplicial face g whoseorientation is induced by the orientation of f .A real-valued function s on the set of d -cells of M is called a self-stress if for every d -edge e of M , the condition ( ∗ ) is fulfilled.This definition is proven to be independent on the choice of thecombinatorial triangulation and also on the choice of the orientationsof the faces.The notion of stressed realizations has the following physical mean-ing. One imagines that the d -faces are realized by planar soap film.The faces are made of different types of soap, that is, with differentphysical property. Each of the faces creates a tension, which shouldbe equilibrium at the d -edges. The tension is always orthogonal tothe boundary of a face and lies in the affine hull of the face. A self-intersecting face produces both compression and tension as is depictedin Figure 13. Figure 13.
Tensions for a convex quadrilateral (left) and a self intersecting quadri-lateral (right).
We say that a R-framework (
M, p ) is self-stressable whenever thereexists a non-zero stress.
Proposition 5.8.
The following two statements hold.
QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 25 (i) Each R-framework yields a d -framework ( E, I, F, n ) which agreeswith the Definition 2.1. The incidences are dictated by com-binatorics of M . Its self-stressability agrees with the Defini-tion 2.2 (cid:3) (ii) For d = 1, an R-framework is a planar realization of somegraph. Its self-stressability agrees with the classical notion ofself-stresses of graphs in the plane. (cid:3) Assume now that M is the d -skeleton of some ( d +1)-dimensionalmanifold M , that is, the chambers are well-defined. Definition 5.9. A lift of ( M , p ) is an assignment of a linear function h C : R d → R to each chamber C . By definition, a lift satisfies thefollowing: whenever two chambers C and C (cid:48) share a d -face f , therestrictions of h C and h C (cid:48) on the affine span (cid:104) p ( f ) (cid:105) coincide.A lift is non-trivial if (at least some of) the functions h C are differentfor different chambers.Let us fix some chamber C . Lifts that are identically zero on C forma linear space Lift( M , p ). Theorem 5.10 (cf. [14]) . Let M be the d -skeleton of some ( d +1) -dimensional manifold M . (i) If the first homology group of M vanishes, that is, H ( M, Z ) = 0 , then the linear spaces Lift(
M , p ) and the space of self-stresses Stress(
M, p ) are canonically isomorphic. (ii) Liftability of ( M , p ) implies self-stressability of ( M, p ) . (cid:3) Remark 5.11.
The theorem gives another proof of Corollary 4.9 In-deed, each realization of K can be viewed as a projection of a four-dimensional simplex. In other words, it is liftable, and hence stressable.Each d -vertex v of a R-framework yields in a natural way a sphericalframework via the following algorithm:(1) We may assume that each face is a simplex, otherwise triangu-late the faces.(2) Take an affine h which is orthogonal to the affine span (cid:104) p ( v ) (cid:105) .Clearly, we have dimh = 3.(3) Take a small sphere S lying in the plane e and centered at theintersection point O = h ∩ (cid:104) p ( v ) (cid:105) .(4) For each d -face f incident to v , take the projection P r h ( f ) tothe plane h and the intersection P r h ( f ) ∪ S . Since f is asimplex, the intersection is a geodesic arch. This yields a framework S v placed in the sphere S . Self-stressability ofspherical graphs is well understood since it reduces to self-stressabilityof planar graphs (see [4, 10, 16]). A face loop is called local with respecta d -vertex v if all the d -faces and d -edges participating in the path areincident to v . Lemma 5.12.
The two statements are equivalent:(i) The stress monodromy of each local (writh respect to some d -vertex v ) face loop is trivial.(ii) The spherical framework S v is self-stressable. Proof.
Triviality of any local stress monodromy implies that stressescan be assigned to the faces incident to v in such a way that locallythe equilibrium condition holds.The same stress assignment gives a self-stress of S v , and vice versa. (cid:3) Example 5.13.
If a d -vertex v has exactly four incident d -edges, then S v is stressable. Indeed, in this case S v is a K placed on the sphere,which is always stressable. Theorem 5.14.
Assume that R is a trivalent R-framework. (i) R is self-stressable iff for each face loop the stress monodromyis trivial. (ii) R is self-stressable iff the two conditions hold: (a) For each vertex v , the induced spherical framework S v isself-stressable. (b) For some generators g , . . . , g k of the first homolog group H ( R ) , and some collection of representatives γ , . . . , γ k that are face-cycles, all the stress monodromies are trivial.(One representative for one generator). (iii) In particular, if R is one-connected, its self-stressability is equiv-alent to self-stressability of S v for all the d -vertices.Proof. (i) Take any face, assign to it any stress, and extend it to otherfaces. Triviality of the monodromy guarantees that no contradictionwill arise.(ii) Keeping in mind Lemma 3.10, observe that any face loop is a lin-ear combination of γ , . . . , γ k and some local face loops. By Lemma 5.12,the monodromies of local loops are trivial. It remains to apply (i). Now(iii) follows. (cid:3) Some examples.
Let us start by two elementary examples.
QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 27 (1) Take the Schlegel diagram of a 3-dimensional cube (that is theprojection of the edges of a cube). It is a trivalent graph in the plane.It is self-stressable since it is liftable by construction.However, one easily can redraw it keeping the combinatorics in sucha way that the realization is no longer self-stressable. For this, it issufficient to generically perturbe the positions of the vertices.(2) Now let us work out an analogous example in R . Take theSchlegel diagram of a 4-dimensional cube (that is, the projections ofall the 2-faces). It is a (3 , Are all trivalent R-frameworks self-stressable? Are all (3 , -valentR-frameworks self-stressable? The answer is negative, which is demonstrated through the belowexample, which is interesting for its own sake.
Example 5.15.
Take a prism P in R , a pyramid over P , and theprojection of the 2-skeleton of the pyramid back to R . We obtain aR-framework R which is defined with some freedom: firstly, one mayalter the position of the vertex of the pyramid, and secondly, one mayapply a projective transform to P . Lemma 5.16.
The R-framework R is self-stressable, and the space ofstresses is one-dimensional. Proof.
The R-framework R is a projection of the 2-skeleton of 4-dimensionaltetrahedron. In other words, R is liftable, and therefore, self-stressable.Since it is trivalent, the space of stresses is at most one-dimensional. (cid:3) The prism P has two disjoint triangular faces. They are also facesof R ; let us call them green . The edges of these faces are also called green . The faces of R that are not green are called white . The edgesthat are not green are called white . Fix also one of the white faces, letus call it the test face for R .It is easy to check that any two of white faces of R are connected bya face path which uses white edges only. Main construction, first step:
Take two copies of R , say, R and R such that one of the greenfaces of R coincides with a green face of for R . Patch R and R along these faces, and eliminate these green faces. Define the result by R . Lemma 5.17. R is self-stressable, and the space of the self-stresseshas dimension 1. Proof.
Self-stressability: Take a self-stress s of R and a stress s of R such that s − s on the patched green faces vanishes. Then s − s represents a non-trivial stress of R .Dimension one: Consider a stress s on R . Take a stress s of R which agrees with s on the test face of F and a stress s of R whichagrees with s on the test face of R . Take s − s − s . It is a stress on R plus the green face which is zero everywhere on R , except, maybe, the green face, which possible only if s − s − s vanishes everywhere.Therefore, each stress of R is a linear combination of two stresses of R and R that cancel each other on the green face. (cid:3) Main construction, second step:
We proceed in the same man-ner: we take one more copy of R , which is called R and patch it to R along green faces, and eliminate the green face which was used forthe patch. We get R . Analogously, we have: Lemma 5.18. R is self-stressable, and the space of the stresses hasdimension 1. (cid:3) Main construction, next steps:
Now we have a chain of threecopies of R patched together. Only two green faces survive. By ad-justing the shapes of the components, we may assume that these greenfaces coincide. Patch the last two green faces and remove them. Afterthat, we get a R-framework (cid:101) R . Generically we have: Lemma 5.19. (cid:101) S is not self-stressable, but it is locally self-stressable,that is S v is self-stressable for each vertex v . Proof.
Non-self-stressability: Before the last step, the space of stresseswas one-dimensional. After the last step (=after patching two lastgreen faces), the dimension can only drop. Let us patch back the twolast green faces and get a framework (cid:101) R (cid:48) .Assume R is self-stressable. This means that (cid:101) R (cid:48) has a stress whichsums up to zero on the last two green faces. By the above lemmata,the value of the stress on the test face of R uniquely defines the stresson the first green face and the stress on the test face of R (and all thefaces of R and R ), this uniquely defines the stress on the test face of R , and so on. We conclude that the stress on the second green face isalso uniquely defined, and generically, the green stresses do not canceleach other.Local self-stressability follows from the fact that putting back any ofthe green faces creates a self-stressable R-framework. (cid:3) Let us observe that (cid:101) R is trivalent, but not (3 , QUILIBRIUM STRESSABILITY OF MULTIDIMENSIONAL FRAMEWORKS 29 higher that 4. However one can prove that a number of local surgeriesturns it to a (3 , ,
4) R-framework which is locally self-stressable, but not globally self-stressable.
Acknowledgement.
The collaborative research on this article was initiated during “Research-In-Groups” programs of ICMS Edinburgh, UK. The authors are grate-ful to ICMS for hospitality and excellent working conditions.Christian M¨uller gratefully acknowledges the support of the AustrianScience Fund (FWF) through project P 29981.
References [1] Alexander I. Bobenko and Yuri B. Suris.
Discrete differential geometry. In-tegrable structure , volume 98 of
Graduate Studies in Mathematics . AmericanMathematical Society, 2008.[2] Robert Connelly.
Tensegrities and Global Rigidity , pages 267–278. SpringerNew York, New York, NY, 2013.[3] Robert Connelly and Walter Whiteley. Second-order rigidity and prestress sta-bility for tensegrity frameworks.
SIAM J. Discrete Math. , 9:453–491, 1990.[4] Henry Crapo and Walter Whiteley. Autocontraintes planes et poly`edres pro-jet´es. I. Le motif de base.
Structural Topology , (20):55–78, 1993. Dual French-English text.[5] Allen Hatcher.
Algebraic topology . Cambridge University Press, Cambridge,2002.[6] Oleg Karpenkov. Geometric conditions of rigidity in nongeneric settings. In
Handbook of geometric constraint systems principles , Discrete Math. Appl.(Boca Raton), pages 317–339. CRC Press, Boca Raton, FL, 2019.[7] Oleg Karpenkov and Christian M¨uller. Geometric criteria for realizability oftensegrities in higher dimensions.
ArXiv e-prints , 2019.[8] Carl W. Lee. P.L.-spheres, convex polytopes, and stress.
Discrete Comput.Geom. , 15(4):389–421, 1996.[9] Robert W. Marks.
The Dymaxion world of Buckminster Fuller . Reinhold Pub.Corp., 1960.[10] G. Yu. Panina. Pointed spherical tilings and hyperbolic virtual polytopes.
Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) , 372(Ge-ometriya i Topologiya. 11):157–171, 208, 2009.[11] J¨urgen Richter-Gebert.
Perspectives on projective geometry . Springer, Heidel-berg, 2011.[12] Ben Roth and Walter Whiteley. Tensegrity frameworks.
Transactions of TheAmerican Mathematical Society - TRANS AMER MATH SOC , 265, 02 1981.[13] Konstantin Rybnikov. Stresses and liftings of cell-complexes.
Discrete Comput.Geom. , 21(4):481–517, 1999.[14] Konstantin Rybnikov.
Polyhedral partitions and stresses . ProQuest LLC, AnnArbor, MI, 2000. Thesis (Ph.D.)–Queen’s University (Canada). [15] Meera Sitharam, Audrey St. John John, and Jessica Sidman.
Handbook ofGeometric Constraint Systems Principles . Discrete Mathematics and Its Ap-plications. CRC Press, Taylor & Francis Group, 2018.[16] Ileana Streinu and Walter Whiteley. Single-vertex origami and spherical expan-sive motions. In
Discrete and computational geometry , volume 3742 of
LectureNotes in Comput. Sci. , pages 161–173. Springer, Berlin, 2005.[17] Gunnar Tibert and Sergio Pellegrino. Review of form-finding methods fortensegrity structures.
International Journal of Space Structures , 18:209–223,12 2003.[18] Neil L. White and Walter Whiteley. The algebraic geometry of stresses inframeworks.
SIAM Journal on Algebraic Discrete Methods , 4(4):481–511, 1983.[19] Walter Whiteley. A matroid on hypergraphs with applications in scene analysisand geometry.
Discrete and Computational Geometry , 4:75–95, 12 1989.[20] G¨unter M. Ziegler.
Lectures on polytopes , volume 152 of
Graduate Texts inMathematics . Springer-Verlag, New York, 1995.
Oleg Karpenkov, University of Liverpool
E-mail address : [email protected] Christian M¨uller, TU Wien
E-mail address : [email protected] Gaiane Panina, PDMI RAS, St. Petersburg State University
E-mail address : [email protected] Dirk Siersma, University of Utrecht
E-mail address : [email protected] Brigitte Servatius, Worcester Polytechnic Institute
E-mail address : [email protected] Herman Servatius, Worcester Polytechnic Institute
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