Geometry of configuration spaces of tensegrities

GEOMETR Y OF CONFIGURA TION SP ACES OF TENSEGRIT I ES FRANCK DORA Y 1 , OLEG KARPENKO V 2 AND JAN SCHEPERS 1 Abstract. Consider a graph G with n v ertices. In this paper we study g eometric con- ditions for an n -tuple of points in R d to admit a tensegrit y with underlying gra ph G . W e int ro duce and inv estigate a natural stratifica tion, depe nding on G , of the configur ation space of all n -tuples in R d . In par ticula r w e find sur geries on g r aphs that give relations betw een different strata. B ased on numerous examples w e give a description of geometric conditions defining the str ata for plane tenseg rities, we co njecture that the list of such conditions is sufficient to describ e any stratum. W e conclude the pap er with particular examples of strata for tensegrities in the plane with a small num ber of vertices. Contents 1. In tro duction 2 2. General definitions 3 2.1. Configuration spaces o f t ensegrities 3 2.2. Stratification of the base of a configuratio n space of tensegrities 5 3. On t he tense grit y d -c haracteristic o f g raphs 7 3.1. Definition and basic properties o f t he tense grit y d - c haracteristic 8 3.2. A toms and atom decomp osition 9 3.3. Calculation of tensegrit y d - c haracteristic in the simplest case s 9 4. Surgeries on graphs that preserv e the dimension of the fib ers 11 4.1. General surgeries in arbitra ry dimension 12 4.2. Additional surgeries in dimension t w o 13 5. Geometric relations fo r strata and complexit y of tensegrities in tw o-dimensional case 16 5.1. A simple example 16 5.2. Systems of geometric conditions 17 5.3. Conjecture on geometric structure of the strata 18 5.4. Complexit y of the strata 18 6. Plane tensegrities with a small n um b er o f vertices 19 6.1. On the tensegrit y 2-characteristic of graphs 19 6.2. Geometric conditions for realizabilit y of plane tensegrities for graphs with zero tensegrit y 2-characteristic 22 References 25 Key wor ds and phr ases. T ensegrity , self-tensional equilibrium frameworks, str atification. 1 Suppo rted by VICI-gr ant 639.03 3.402 of NWO. 2 Partially supp or ted b y RFBR grant SS-70 9.2008 .1 and b y NW O -DIAMANT gra n t 6 13.009 .001. 1 2 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS 1. Introduction In his pa p er [9] J. C. Maxw ell made one of the first approa c hes to the study of equilib- rium states f or frames under the action of static forces. He noted that the frames together with the forces giv e rise to recipro cal figures . In the second half of the tw en tieth cen- tury the artist K. Snelson built man y surprising sculptures consisting of cables and ba r s that are actually suc h frames in equilibrium, see [14]. R. Buc kminster F uller introduced the name “tensegrity ” for these constructions, comb ining the w ords “tension” and “in- tegrit y”. A nice o v erview of the history of tensegrit y constructions is made by R. Motro in his b o ok [10]. In mat hematics, tensegrities w ere in v estigated in sev eral pap ers. In [12] B. Roth and W. Whiteley and in [3] R. Connelly and W. Whiteley studied rigidit y and flexibility o f tensegrities, see also the surv ey ab out rig idity in [18]. N. L. White and W. Whiteley started in [17] the inv estigation of geometric realiz- abilit y conditions for a t ensegrity with pres crib ed bars and cables. In the preprint [6] M. de Guzm´ an describ es other ex amples of geometric conditions for t ensegrities. T ensegrities hav e a wide range of applications in differen t branche s of s cience and in arc hitecture. F or instance they are used in the study of viruses [2], cells [5], for construction of deplo yable mec hanisms [13, 16], etc. W e f o cus on the following imp orta n t question. Suppose a graph G is give n. Is the gr aph G r e alizable as a tense grity for a gener al c onfigur ation of its vertic es? W e dev elop a new tec hnique to study this question. W e in tro duce sp ecial op eratio ns ( sur gerie s ) that change the gr a ph in a certain w ay but preserv e the pro p ert y to b e (not to be) realizable as a tensegrit y . Let n b e the n um b er o f v ertices of G . Consider the configuration space o f all n -tuples of p oints in R d . In this pap er w e define a stratification o f the configuration space suc h, that each stratum corresp onds to a certain set of admissible tensegrities asso ciat ed to G . Supp ose t ha t one wan ts to obta in a construction with some edges of G replaced b y struts and the others b y cables, then he/she should ta k e a configuration in a sp ecific stratum of the stratification. In this pap er w e prov e that all t he strata are semialgebraic sets, and therefore a notion o f dimension is w ell-defined for them. This allows to generalize the previous question: wha t is the minimal c o dime nsion of the str ata i n the c onfigur ation sp ac e that c ontains n -tuples of p oin ts a dmitting a tense grity with unde rlying gr aph G ? Our tec hnique of surgeries on graphs also giv es the first answ ers in this case. In particular w e obtain the list of a ll 6, 7, and 8 vertex tensegrities in the plane that a re realizable fo r co dimension 1 strata. W e note tha t the complete a nsw ers t o t he abov e que stions are not kno wn to the authors. N. L. White and W. Whiteley [17] and M. de G uzm´ an and D. O rden [7, 8] hav e found the geometric conditions o f realizabilit y of plane tensegrities with 6 ve rtices and of some ot her particular cases. W e con tin ue the inv estigation for other gra phs (see Subsection 6 .2). In all the observ ed examples the strata are defined by certain systems of geometric conditions. It turns out t ha t a ll these geometric conditions are obtained from elemen tary ones: — two p oints c oincide ; GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 3 — thr e e p oints ar e on a line ; — fiv e p oints a , b , c , d , e satisfy: e is the interse c tion p oint of the line s p a ssing thr ough p oints a and b and p oints c and d r esp e ctively . W e conjecture that fo r plane tensegrities any stratum can b e defined by certain geometric conditions (see Section 5). This pap er is org anized as follows. W e start in Section 2 with general definitions. In Subsection 2.1 w e describ e the configuration space of tensegrities ass o ciated to a given graph as a fibration ov er the affine space of all framew o r ks. W e introduce a natural stratification on the space of all framew orks in Subsection 2.2. W e pro ve that all strata are semialgebraic sets and therefore the strata ha v e we ll-defined dimensions. In Section 3 w e study the dimension of solutions for g r aphs on general configura t ions of p oints in R d . Later in this section w e calculate t he dimensions in the simplest cases, and form ulate general op en questions. In Section 4 w e study surgeries on gra phs and f r a mew orks that induce isomor phisms of the spaces of self-stress es for the framew orks. W e giv e general definitions related to sys tems of geometric conditions for plane tensegrities in Section 5. W e conjecture that a n y stratum is a dense subset of the solution of one o f such systems. Finally in Section 6 we giv e particular examples of graphs and their strata for tensegrities in the plane. W e study the dimension of the space of self-stress es in Subsection 6.1 and giv e ta bles of geometric conditions for co dimension 1 strata for graphs with 8 ve rtices and less in Subsection 6.2 . Ac kno wledgemen ts. Th e authors are grateful to B. Edixho v en fo r rousing our intere st to the s ub ject, to A. Sossinski, V. Goryunov , and A. Peruc ca for helpful remarks and discussions , to S. Sp eed for useful information on gr aph classification, and Mathematisc h Instituut of Univ ersiteit Leiden for hospitalit y and excellen t w orking conditions. The second author is grat eful to Liv erp o ol Univ ersit y fo r t he organization o f a f r uitful visit. 2. G eneral definitions 2.1. Configuration spaces of t ensegrities. Recall a slightly mo dified definition of a framew ork from [8]. Definition 2.1. Fix a p ositiv e in t eger d . L et G = ( V , E ) be an arbitrary graph without lo ops and mu ltiple edges. Let it hav e n vertice s. • A fr amework G ( P ) in R d is a map of the graph G with v ertices v 1 , . . . , v n on a finite p oin t configuration P = ( p 1 , . . . , p n ) in R d with straigh t edges, suc h that G ( P )( v i ) = p i for i = 1 , . . . , n . • A str e s s w on a framew ork is an assignmen t o f real scalars w i,j (called tensions ) t o its edges p i p j . W e also put w i,j = 0 if there is no edge b et w een the corresp onding v ertices. Observ e that w i,j = w j,i , since they refer to the same edge. • A stress w is called a self-s tr ess if, in addition, the following equilibrium condition is fulfilled at eve ry verte x p i : X { j | j 6 = i } w i,j p i p j = 0 . 4 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS By p i p j w e denote the v ector f r o m the point p i to the p oint p j . • A couple ( G ( P ) , w ) is called a tense grity if w is a self-stress for the framew ork G ( P ). R emark 2.2 . T ensegrities are self-tensional equilibrium f r amew orks. F or instance, an y framew ork for the tw o v ertex graph without edges is alwa ys a tensegrit y , although it is not rigid. F or more information ab out rigidity of tens egrities w e refer to a pap er o f B. R o th and W. Whiteley [12]. Denote b y W ( n ) the linear space o f dimension n 2 of all tensions w i,j . Consider a framew ork G ( P ) and denote b y W ( G, P ) the subset of W ( n ) of all p ossible self-stress ed tensions for G ( P ). By definition of self-stressed tensions, the set W ( G, P ) is a linear subspace of W ( n ). The c onfigur ation sp ac e of tense grities corresp onding to the graph G is the set  ( G ( P ) , w ) | P ∈ ( R d ) n , w ∈ W ( G, P )  , w e denote it b y Ω( G ). The s et { G ( P ) | P ∈ ( R d ) n } is said to b e the b ase of the c onfigur ation sp ac e , we denote it by B d ( G ). If w e forget ab out the edges betw een the p oints in all the framew orks, then w e get natural bijections b etw een Ω( G ) and a subset of ( R d ) n × W ( n ) and b etw een B d ( G ) and ( R d ) n . Later on w e actually iden t if y the last t wo pairs of sets . The bij ections induce natural top olo g ies on Ω( G ) and B d ( G ). Let π b e the natural pro j ection of Ω( G ) to the base B d ( G ). This define s the structure of a fibration. F or a giv en framew o r k G ( P ) of the base w e call the set W ( G, P ) the line ar fib er at the p oint P (or at the framew ork G ( P )) o f the configuration space. Consider a se lf-stress w for the framew ork G ( P ). W e say that the edge p i p j is a c ab le if w i,j < 0 and a strut if w i,j > 0. R emark 2.3 . The definitions of struts and cables come from the follow ing ph ysical in ter- pretation. Supp ose w e w ould lik e to construct the lig h test p o ssible tensegrit y structure on a giv en framew ork and with a given self-stress using hea vy struts and relat ively ligh t cables. Then w e should replace the edges with p ositive w i,j with struts, and the edges with negativ e w i,j with cables. Suc h constructions w ould b e t he ligh test possible. Denote b y “sgn” the sign function ov er R . Definition 2.4. Consider a framew o rk G ( P ) and one of its self-stresses w . The n × n matrix (sgn( w i,j )) is called the s trut-c able matrix of the stress w and denoted by sgn( w ). Let us giv e one example of a strut- cable matrix. Example 2.5. Consider a configuration of four p o ints in the plane: p 1 (0 , 0), p 2 (1 , 0), p 3 (2 , 2), p 4 (0 , 1) and a self-stress w as on the picture: w 1 , 2 = 6, w 1 , 3 = − 3, w 1 , 4 = 6, GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 5 w 2 , 3 = 2, w 2 , 4 = − 4, w 3 , 4 = 2. Then we ha v e: p 1 (0 , 0) p 2 (1 , 0) p 3 (2 , 2) p 4 (0 , 1) 6 − 3 6 2 − 4 2 sgn( w )=     0 1 − 1 1 1 0 1 − 1 − 1 1 0 1 1 − 1 1 0     . 2.2. Stratification of the base of a configuration space of tensegrities. Supp ose w e ha v e some fra mew ork G ( P ) and w e w ant to find the ligh test cable-strut construction on it, as explained in R emark 2.3. Then the follo wing questions arise. Which e dges c an b e r eplac e d by c abl e s, a nd which by struts? What is the ge ome tric p osition of p oints in the c onfigur ations for which given e dges may b e r eplac e d by c ables and the others by struts? The questions lead to the following definition. Definition 2.6. A linear fib er W ( G, P 1 ) is said to b e e q uivalent to a linear fib er W ( G, P 2 ) if there exists a homeomorphism ξ b etw een W ( G, P 1 ) and W ( G, P 2 ), s uc h that for a n y self-stress w in W ( G, P 1 ) the self-stress ξ ( w ) satisfies sgn  ξ ( w )  = sgn  w  . The describ ed equiv alence relation g iv es us a stratification of the base B d ( G ) = ( R d ) n . A str atum is by definition a maximal connected set of p oin ts with equiv alen t linear fib ers. Once we ha ve prov en Th eorem 2.8, b y general theory of semialgebraic se ts (see for in- stance [1]) it follows that a ll strata are path- connected. Example 2.7. W e describ e the stratification of B 1 ( K 3 ) = R 3 for the complete g r a ph K 3 on three v ertices. The p oint ( x 1 , x 2 , x 3 ) in R 3 corresp onds to the fra mework with v ertices p 1 = ( x 1 ), p 2 = ( x 2 ), and p 3 = ( x 3 ). The stratification consis ts of 13 strata . There is 1 one-dimensional stratum, and there a r e 6 tw o-dimensional and 6 three-dimens ional strata. The one-dimensional stratum consists of frameworks with all v ertices coinciding. It is defined by the equations x 1 = x 2 = x 3 . The dimension of the fib er at a po int of this stratum is 3. An y of the tw o-dimensional strat a consists of framew o rks with exactly tw o v ertices coinciding. The strata are the connected comp onents of the complemen t to the line x 1 = x 2 = x 3 in the union of the three planes x 1 = x 2 , x 1 = x 3 , a nd x 2 = x 3 . The dimension of the fib er at a p oint o f a n y of t hese strata is 2. An y of the three-dimensional strata consists of framew orks with distinct ve rtices. The strata are the connecte d comp o nents of the compleme n t in R 3 to the union of the three planes x 1 = x 2 , x 1 = x 3 , and x 2 = x 3 . The dimension of the fib er at a p o in t of an y o f these strata is 1 . In general w e hav e the follo wing theorem. Theorem 2.8. Any s tr atum is a semialge b r a i c set. 6 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS F or the definition and basic pro p erties of semialgebraic sets w e refer t he reader to [1]. W e need t wo preliminary lemmas for the pro of of the theorem, but first w e introduce the follow ing notatio n. Let M b e an a r bit r a ry symmetric n × n - matrix with zero es on the diagonal and all the other en tries b elonging to {− 1 , 0 , 1 } . Let i b e an integer with 0 ≤ i ≤ n 2 . W e sa y that a couple ( M , i ) is a str atum symb ol . F or an a rbitrary framew ork G ( P ) we denote b y W M ( G, P ) the set of all self-stresses with strut-cable matrix sgn( w ) equal to M . The closure of W M ( G, P ) is a p oin ted p o ly- hedral cone with v ertex at the orig in. The set W M ( G, P ) is ho meomorphic to an op en k -dimensional disc, we call k the dimension of W M ( G, P ) and denote it b y dim ( W M ( G, P )). F or an y stratum sym b ol ( M , i ) w e denote by Ξ( M , i ) the set { ( G ( P ) , w ) | w ∈ W ( G, P ) , sgn( w ) = M , dim( W M ( G, P )) = i } ⊂ Ω( G ) . Lemma 2.9. F or any str atum symb ol ( M , i ) , the subset π (Ξ( M , i )) of the b a se B d ( G ) is either e m pty or it is a se m ialgebr aic s e t. Pr o of. Th e set Ξ( M , i ) is a semialgebraic set since it is defined b y a system of equations and inequalities in the co ordinates of the v ertices and the tensions of the fo llo wing three t yp es: a ) quadratic equilibrium condition equations; b ) linear equations or inequalities sp ecifying if the co ordinate v alues w i,j are zero es, p osi- tiv e, o r negativ e reals; c ) algebraic equations and inequalities defining resp ectiv ely dim( W M ( G, P )) ≤ i and dim( W M ( G, P )) ≥ i . Note that dim( W M ( G, P )) is equal to the dimension of the linear space spanned b y W M ( G, P ). Let us make a small remark ab out item ( c ). A t eac h framew ork we ta k e the system o f equilibrium conditions a nd equations of ty p e w i,j = 0 in the v a r ia bles w i,j . This sys tem consists of the e qualities of items ( a ) and ( b ). It is linear in the v aria bles w i,j . The coef- ficien ts of the corresp onding matrix dep end linearly on the co ordinates of the framew ork v ertices. The equations and inequalities of item ( c ) are define d b y some determinan ts of submatrices of this matrix b eing equal or not equal to zero. Therefore, they ar e algebraic. Since b y the T arski-Seiden b erg theorem any pro jection of a semialgebraic set is semi- algebraic, the set π (Ξ( M , i )) is sem ialgebraic.  Denote b y S ( G, P ) the set of all stratum sym b ols ( M , i ) that a re realized by the p o in t G ( P ), in other w ords S ( G, P ) = { ( M , i ) | G ( P ) ∈ π (Ξ( M , i )) } . Lemma 2.10. L et G ( P 1 ) and G ( P 2 ) b e two fr ame works. Then S ( G, P 1 ) = S ( G , P 2 ) if and only if the line ar fib e r W ( G, P 1 ) is e q uiva lent to the line ar fib er W ( G, P 2 ) . Pr o of. Let the linear fib er at the p oint G ( P 1 ) b e equiv alen t to t he linear fib er at the p oint G ( P 2 ) t hen b y definition w e ha v e S ( G, P 1 ) = S ( G, P 2 ) . GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 7 Supp ose now that S ( G, P 1 ) = S ( G, P 2 ). Let us denote by W ( G, P i ) the one p oint compactification of the fib er W ( G, P i ) for i = 1 , 2. So W ( G, P i ) is homeomorphic to a sphere of dimension dim W ( G, P i ). F or an y p oint P and any M the set W M ( G, P ) is a con v ex cone homeomorphic to an op en disc of dimension dim ( W M ( G, P )). So, for any point P w e ha v e a natural CW- decomp osition of W ( G, P ) with cells W M ( G, P ) and the new one p oin t cell. A cell W M ′ ( G, P 1 ) is adjacen t to a cell W M ′′ ( G, P 1 ) iff the ce ll W M ′ ( G, P 2 ) is adjacen t to the cell W M ′′ ( G, P 2 ). This is true, since the couples of cells corresponding to M ′ and to M ′′ are defined by the same sets of equations and inequalities of t yp e “ > ” , and the closures of W M ′ ( G, P i ) for i = 1 , 2 are defined b y the system defining W M ′ ( G, P i ) with all “ > ” in the inequalities replaced b y “ ≥ ”. Therefore, there exists a homeomorphism of W ( G, P 1 ) and W ( G, P 2 ), sending all the cells W M ( G, P 1 ) to the corresp onding cells W M ( G, P 2 ). W e lea v e the pro of of this state- men t a s an exercise for the reader, this can b e done b y inductive ly constructing the homeomorphism on the k -sk eletons of the CW-complex es. Hence, the linear fib ers W ( G , P 1 ) and W ( G, P 2 ) a re equiv alen t.  Pr o of of The or em 2 .8. Let us pro ve the theorem for a stratum con taining some p oint P . Consider any p oint P ′ in the stra t um. By definition, W ( G, P ) is equiv alent to the space W ( G, P ′ ), a nd hence by Lemma 2.10, we hav e S ( G, P ) = S ( G, P ′ ). Consider the following set \ ( M ,i ) ∈ S ( G,P ) π (Ξ( M , i )) \  [ ( M ,i ) / ∈ S ( G,P ) π (Ξ( M , i ))  , w e denote it Σ( P ). So Σ( P ) is the set of framew orks G ( P ′ ) for whic h S ( G, P ′ ) = S ( G, P ) . By Lemma 2.9 all the sets π (Ξ( M , i )) are semialgebraic. Therefore, the set Σ( P ) is semialgebraic. Denote by Σ ′ ( P ) the connected comp onen t o f Σ( P ) that contains the p oin t P . Since the set Σ( P ) is sem ialgebraic, the set Σ ′ ( P ) is also semialgebraic, see [1]. Let us show that Σ ′ ( P ) is the stratum of B d ( G ) containing the p oint P . First, the set Σ ′ ( P ) is contained in the stratum. This holds s ince Σ ′ ( P ) is connected and consists of p oin ts with equiv alen t sets S ( G, P ). And hence by Lemma 2 .1 0 all the p oin ts of Σ ′ ( P ) ha v e equiv alen t linear fibers W ( G, P ). Secondly , the stratum is contained in the space Σ ′ ( P ) . This holds since t he strat um is connected and consists of po in ts with equiv a len t linear fib ers W ( G, P ). Th us by Lemma 2.10 all the p oin ts of the stratum hav e equiv a len t sets S ( G, P ). As w e hav e shown , the stratum containing P coincides with Σ ′ ( P ) and hence it is semialgebraic.  F rom the ab ov e pro of it follo ws that the tota l num b er of strata is finite. 3. On the tensegrity d -c hara cteristic of graphs In this section we study the dimension o f the linear fib er for graphs on a g eneral p oint configuration in R d . W e giv e a natural definition of the tensegrit y d -c haracteristic o f a 8 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS v 1 v 2 v 3 v 4 v 5 v 6 v 1 v 2 v 3 v 4 v 5 Figure 1. A graph with zero tensegrit y 2-c haracteristic (the left one) and a graph whose 2-characteristic equals 2 (the rig h t one). graph and calculate it for t he simplest graphs. In addition we form ulate general open questions for further inv estigation. 3.1. Definition and basic prop erties of the tensegrit y d -cha racteristic. Note that for a ny t w o p oints P 1 and P 2 of the same stratum S of the space B d ( G ) for a graph G w e ha v e dim( W ( G, P 1 )) = dim( W ( G, P 2 )) . Denote this n um b er b y dim( G, S ). Denote also by co dim( S ) the integer dim( B d ( G )) − dim( S ) . Consider a graph G with at least one edge. W e call the integer min { co dim S | S is a stra t um of B d ( G ), dim( G, S ) > 0 } the c o dimensio n of G and denote it by co dim d ( G ). Definition 3.1. W e call the in teger  1 − co dim d ( G ), if co dim d ( G ) > 0 max  dim( W ( G, P ))   G ( P ) contained in a co dimension zero stratum  , otherwise the tense gri ty d -char acteristic of t he graph G (or the d -TC of G for short), and denote it b y τ d ( G ). Example 3.2. Consider the tw o graphs sho wn on Figure 1. The left one is a graph of co dimension 1 in the plane, it can b e realized as a tensegrit y iff either the t w o triangles are in p erspectiv e po sition or the p oin ts of one of the tw o triples ( v 1 , v 4 , v 5 ) o r ( v 2 , v 3 , v 6 ) lie on a line ( f or more details see [8 ]), so its 2-TC is zero. The graph on the righ t has a t wodimensional s pace of self-stresses for a general p osition plane framew o rk, and hence its 2-TC equals t w o (w e sho w this la ter in Prop osition 6 .1 ). Prop osition 3.3. L et S 1 and S 2 b e two str a ta of c o dimension 0 . L et G ( P 1 ) and G ( P 2 ) b e two p oints of the s tr ata S 1 and S 2 r esp e ctively. The n the fol lowing holds: dim( W ( G, P 1 )) = dim( W ( G, P 2 )) . Pr o of. Th e equilibrium conditions giv e a linear system of equations in the v ariables w i,j , at eac h framew ork linearly dep ending on the co ordinates of the ve rtices. The dimension of the solution space is determined b y the rank of the matr ix of this system. The subset of B d ( G ) where the rank is not maximal is an algebraic subset of p ositiv e co dimension. GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 9 By definition, this set does not ha v e elemen ts in the strata S 1 and S 2 . This yields the statemen t of the propo sition.  Corollary 3.4. L et G b e a gr aph. If τ d ( G ) ≥ 0 then for e v ery fr amework G ( P ) in a c o dimension 0 str atum we have dim W ( G, P ) = τ d ( G ) .  3.2. Atoms and atom decomp osition. In this subsection w e recall a definition and some results of M. de Guzm´ an and D. Orden [8] that we use la ter. Consider a p oin t configuration P of d +2 p oin ts in general position in R d . Throughout this subsec tion ‘general p o sition’ means that no d +1 of them are con tained in a hy p erplane. An a tom in R d is a tensegrit y ( K d +2 ( P ) , w ), where K d +2 is the complete graph on d +2 v ertices a nd where w is a nonzero self-stress. According to [8, Section 2] the linear fib er W ( K d +2 , P ) is o ne- dimensional f or P in general p osition, in particular this implies τ d ( K d +2 ) = 1. In a ddition the tension on ev ery edge in the atom is nonzero. A more general stateme n t holds. Lemma 3.5. [8, Lemma 2.2] L et G ( P ) b e a fr am e work on a p oin t c onfig ur ation P in R d in gener al p osition. L et p ∈ P . Given a non z er o self-str ess on G ( P ) , then either at le ast d +1 of the e dges incident to p r e c eive nonzer o tensio n , or al l of them h a ve z e r o tension. M. de Guzm´ an and D. Orden show ed tha t one can consider atoms as the building blo c ks of tensegrit y struc tures. First, w e explain ho w to add tensegrities. Let T = ( G ( P ) , w ) and T ′ = ( G ′ ( P ′ ) , w ′ ) b e t wo tensegrities. W e define T + T ′ as follows. The framework of T + T ′ is G ( P ) ∪ G ′ ( P ′ ), w e tak e the union of v ertices and edges. The tension on a common edge p i p j = p ′ k p ′ l is defined as w i,j + w ′ k ,l and on an edge app earing exactly in one of the original framew orks w e put the original tension. It is easy to se e that the defined stress is a self-stress, so T + T ′ is a tensegrit y . Theorem 3.6. [8, Theorem 3.2] Every tense gri ty ( G ( P ) , w ) with a gene r a l p osition p oint c onfigur ation P and w i,j 6 = 0 on al l e dges of G is a finite sum of atoms. This de c om p osition is n ot unique in gener al. 3.3. Calculation of tensegrity d -chara cteristic in the simplest cases. W e start this subsection with the fo rm ulatio n of a problem, we do not kno w the complete solution of it. Problem 1. Giv e a general form ula fo r τ d ( G ) in terms o f the com binato r ics of the gra ph. Let us calculate the d -TC for a complete graph, this will give us the maximal v alue of the d -TC for fixed num b er o f vertice s n a nd dimension d . Prop osition 3.7. F or any p ositive inte gers n and d satisfying n ≥ d +2 , we have τ d ( K n ) = ( n − d − 1 )( n − d ) 2 . Pr o of. W e w ork b y induction on n . F o r n = d + 2 the d -TC equals 1, as men tio ned ab ov e. F or n > d +2 w e c ho ose any p oin t configuration P on n p oints suc h that no d +1 of them lie in a h yp erplane. T ak e p ∈ P . An y tensegrit y ( K n ( P ) , w ) can be decomposed as a 10 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS e f g Figure 2. The tension on the edge e is alw a ys zero. sum of n − d − 1 atoms with p as ve rtex and a tensegrit y on P \ { p } with underlying graph K n − 1 . Indeed, we can use suc h atoms to cancel the giv en tensions o n n − d − 1 edges at p . Then there ar e only d edges left, so b y Lemma 3 .5 the t ensions on these edges equal zero. W e conclude b y induction t hat τ d ( K n ) = τ d ( K n − 1 ) + n − d − 1 = ( n − d − 1)( n − d ) 2 .  No w w e sho w ho w the d - TC b eha v es when we remo ve a n edge of the graph. Prop osition 3.8. L et G b e some gr aph satisfying τ d ( G ) > 1 . L et a gr ap h G ′ b e obtaine d fr om the gr aph G by e r asing one e dge. Then τ d ( G ) − τ d ( G ′ ) ∈ { 0 , 1 } . Pr o of. Erasin g one edge is equiv alent to adding a ne w linear equation w i,j = 0 to the linear system defining the space W ( G, P ) for the graph G (for any p oin t P ). This implies that the space of solutions coincides with W ( G, P ) o r it is a hyperplane in W ( G, P ). So, first, τ d ( G ′ ) ≤ τ d ( G ). Secondly , since τ d ( G ) = dim( W ( G, P 0 )) for some fra mew ork P 0 of a co dimension 0 stratum for G (and therefore it b elongs to a co dimension 0 stratum for G ′ ), t hen τ d ( G ′ ) = dim( W ( G ′ , P 0 )) ≥ dim( W ( G, P 0 )) − 1 = τ d ( G ) − 1 . This completes the pro of.  As we sho w in the example below, erasing an edge do es not alw ays reduce the tensegrit y c hara cteristic. Example 3.9. Consider the gra ph show n in Fig ure 2. Assume that this graph underlies a tensegrit y . Then w e can add an atom on the four leftmost v ertices, to cancel the tension on edge f for instance. This automatically cancels the tensions on the edges connecting the four leftmost v ertices b y Lemma 3.5. W e can do the same on the righ t. So the tension on e is zero a s w ell. Therefore the tension on e w as zero from the b eginning and hence deleting e do es not change the 2 - TC. In Example 6.3 we giv e a less trivial example of this phenomenon. Let us formulate tw o general corolla r ies of Prop osition 3.8. GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 11 Corollary 3.10. L et G b e a gr aph on n vertic es and m ∈ Z > 0 . If G has m + n ( n − 1) 2 − τ d ( K n ) = m + dn − d 2 + d 2 e dges, then τ d ( G ) ≥ m . Pr o of. Com bine Prop osition 3.7 and Propo sition 3.8.  The following coro llary is useful fo r the calculatio n of the tensegrit y d -c hara cteristic. In Subsection 6.1 w e use it to calculate all the tensegrity 2- c hara cteristics for sufficien tly connected graphs with less than 8 vertic es. Corollary 3.11. L et G b e a gr aph on n vertic es with τ d ( G ) ≥ 0 . Assume that G has dn − d 2 + d 2 + τ d ( G ) e dges. T h en f o r any gr aph H that c an b e obtaine d fr o m G by adding N e dges we have τ d ( H ) = τ d ( G ) + N . Pr o of. W e delete τ d ( K n ) − τ d ( G ) − N edges from K n to reac h H . If the d -TC do es no t drop by 1 at o ne of t hese steps, then w e apply Prop osition 3.8 an additional N times to H to reach G . This leads to a wrong v alue of τ d ( G ). So the d -TC drops by one in eac h of the first τ d ( K n ) − τ d ( G ) − N steps and the fo rm ula for τ d ( H ) follows .  Example 3.12. A pseudo- trian g le is a planar p olygon with exactly three v ertices at whic h the a ng les a r e less than π . Let G b e a planar graph with n v ertices a nd k edges that admits a pseudo-triangular em b edding G ( P ) in t he plane, i.e. a non-crossing embedding suc h that the outer f ace is con v ex and all in terior faces are pseudo-triangles. It is ob vious that a pseudo-triangular em b edding G ( P ) b elongs to a co dimension 0 stratum of B 2 ( G ). By Lemma 2 of [11] w e find tha t — τ 2 ( G ) = k − (2 n − 3) if k − (2 n − 3) ≥ 1, — τ 2 ( G ) ≤ 0 if k − (2 n − 3) = 0. (Note t ha t for pseudo-triangular em b eddings we alw ays hav e k ≥ 2 n − 3.) 4. Surgeries on graphs tha t preser ve the dimension of the fibers In this section w e describ e op eratio ns that one can p erfo rm o n a gr aph without changing the dimensions of the corresp onding fib ers for the framew orks. W e refer to suc h op erations as sur geries . The first type of surgeries is for general dimension, while the other t w o are restricted to dime nsion d = 2. W e do not kno w other similar op erations that are not comp ositions of the surgeries described below. The idea of surgeries is analog ous to the idea of Reidemeister mo v es in knot theory . If t wo graphs are connected b y a sequence of surgeries, then one obtains tensegrities for the first graph from tensegrities for the second graph and vic e vers a. W e essen tially use surgeries to calculate the list of g eometric conditions for the strata for (sufficien tly connected) graphs with less than 9 v ertices a nd with zero 2- TC in Sub- section 6.2. 12 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS 4.1. General surgeries in arbitrary dimension. F or an edge e of a graph G w e denote b y G e the graph obtained f r o m G b y r emoving e . Recall that a subgraph G ′ of a graph G is said to be induc e d if, for an y pair of v ertices v i and v j of G ′ , v i v j is an edge of G ′ if and o nly if v i v j is an edge of G . Denote by Σ d ( G ) the union of co dimension zero strata in B d ( G ). Let G b e a graph and H a subgraph. Consider the map that takes a framew ork for G to the framework fo r H b y forgetting all the ve rtices and edges of G that are not in H . D enote b y Σ d ( G, H ) the preimage of Σ d ( H ) for this map. Prop osition 4.1. L et G b e a gr aph and H an induc e d sub gr aph with τ d ( H ) = 1 . C on- sider a c onfigur ation P 0 lying in Σ d ( G, H ) . S upp ose that ther e exists a self-str ess on the fr amework G ( P 0 ) that ha s nonze r o t ensions for al l e d ges of H and zer o tensions on the other e d g es. L et e 1 , e 2 b e e dges o f H . Then for any P ∈ Σ d ( G, H ) we h a ve W ( G e 1 , P ) ∼ = W ( G e 2 , P ) . The corresp onding surgery tak es the graph G e 1 to G e 2 , o r vice v ersa. R emark 4.2 . W e alw a ys ha ve the inclusion Σ d ( G ) ⊂ Σ d ( G, H ), this follo ws directly from the definition of the strata. Nev ertheless the set Σ d ( G, H ) usually contains many strata of B d ( G ) of p ositive codimension. So Prop osition 4.1 is applicable t o all strata of co dimension zero as w ell as to some strata of p ositiv e codimension. F or the pro o f o f Pro p osition 4.1 w e need the fo llo wing lemma. Lemma 4.3. L et G b e a gr aph w ith τ d ( G ) = 1 and e one of its e dg es. Supp ose that ther e exists a c onfigur ation P 0 ∈ Σ d ( G ) and a nonze r o se l f -str ess w 0 such that w 0 ( e ) = 0 . Then for a n y tense grity ( G ( P ) , w ) with P ∈ Σ d ( G ) we get w ( e ) = 0 . Pr o of. Sinc e τ d ( G ) = 1 and P 0 ∈ Σ d ( G ), any tensegrity ( G ( P 0 ) , w ) satisfies the condition w ( e ) = 0. The refore, an y tensegrit y with P in the same strat um as P 0 has zero tension at e . So the condition always to have zer o tension at e defines a somewhere dense subset S in B d ( G ). Since the condition is defined by a solution of a certain linear system , S is dense in B d ( G ). It follows that Σ d ( G ) is a subset of S .  Pr o of of Pr op osition 4.1. F rom Lemma 4.3 w e ha v e that for an y configuration of Σ d ( H ) there ex ist a unique up t o a scalar se lf-stress that is nonzero at eac h edge of H . The uniqueness follows from the fact that τ d ( H ) = 1. Hence for any configuration of Σ d ( G, H ) there exists a unique up to a scalar self-stress that is nonzero at each edge of H a nd zero at all other edges of G . F or any P ∈ Σ d ( G, H ) w e obtain an isomorphism betw een W ( G e 1 , P ) and W ( G e 2 , P ) b y adding the unique tensegrit y o n the underlying subgraph H of G that cancels the tension on e 2 , considered as edge o f G e 1 .  In particular one can use atoms ( i.e. H = K d +2 ) in the a b o v e propo sition. GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 13 v 1 v 2 v 3 v 4 v 5 v 1 v 2 v 3 v 4 v 5 v 1 v 2 v 3 v 4 v 2 v 3 v 4 v 5 G 0 = G e 1 H G e 2 e 2 e 2 e 1 e 1 Figure 3. This sho ws that τ 2 ( G 0 ) = 1. Corollary 4.4. In the notation and wi th the c o n ditions of Pr op osition 4.1 we have: if either τ d ( G e 1 ) > 0 or τ d ( G e 2 ) > 0 then τ d ( G e 1 ) = τ d ( G e 2 ) . Pr o of. Th e statemen t follow s directly from Prop osition 4.1 and Corollary 3.4.  Let us sho w how to use t he ab ov e corollary to compute the tensegrity characteristic . Example 4.5. W e calculate the 2 - TC of the graph G 0 sho wn in Figure 3. Consider the ato m H on the v ertices v 1 , v 2 , v 3 , and v 4 and let e 1 , e 2 b e the edges v 2 v 4 , v 1 v 3 resp ectiv ely . D enote by G the graph obtained from G 0 b y adding the edge e 2 . So the graph G 0 is actually G e 1 . By Corollary 3 .10 w e ha ve τ 2 ( G 0 ) ≥ 1, and hence it is p ossible to apply Corollary 4.4. Consider the gra ph G e 2 , it is sho wn in Fig ure 3 in the middle. The degree of the v ertex v 1 in this graph equals 2, so b y Lemma 3.5 the tensions on its incoming edges equal zero if the p oin ts v 1 , v 2 , and v 4 are not on a line. After removin g these tw o edges and the v ertex v 1 w e g et the gr a ph of a n atom. Therefore, τ 2 ( G 0 ) = 1 . 4.2. Additional surgeries in dimen sion t wo. In this subsection we study t wo surgeries on edges of plane fra meworks that do not change the dimension of the fib ers o f the framew orks. Surgery I. Consider a graph G a nd a framew o rk G ( P ). Let G con ta in the complete graph K 4 with vertice s v 1 , v 2 , v 3 , and v 4 as a n induced subgraph. Supp ose that the edges b et w een v 1 , v 2 , v 3 , v 4 and other vertic es of G are a s follows: — pv 2 and q v 3 for unique v ertices p and q ; — the edges pv 1 and q v 1 ; — an y set of edges from v 4 . 14 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS v 1 v 2 v 3 v 4 p q G ( P ) v 1 v 4 p q G I 1 ( P I 1 ) v 2 v 3 v 4 p q G I 2 ( P I 2 ) Figure 4. Surgery I. In addition w e require that t he fra mew ork G ( P ) has the tr iples of p oin ts ( p, v 1 , v 2 ) and ( q , v 1 , v 3 ) o n one line. See Fig ure 4 in the middle. Let us dele te f rom the gr aph G t he v ertices v 2 and v 3 (the v ertex v 1 ) with all edges adjacen t to them. W e denote the resulting graph by G I 1 (b y G I 2 resp ectiv ely). The corresp onding framew o r k is denoted by G I 1 ( P I 1 ) (b y G I 2 ( P I 2 ) resp ectiv ely). See Fig ure 4 on the left (on the right). Surgery I take s G I 1 to G I 2 or vice v ersa. Prop osition 4.6. Consider the fr ameworks G ( P ) , G I 1 ( P I 1 ) , and G I 2 ( P I 2 ) as ab ove. If the triples of p oints ( p, v 2 , v 3 ) , ( q , v 2 , v 3 ) , ( p, v 2 , v 4 ) , ( q , v 3 , v 4 ) and ( v 2 , v 3 , v 4 ) ar e not on a line then w e have W ( G I 1 , P I 1 ) ∼ = W ( G I 2 , P I 2 ) . Pr o of. W e explain how to go from W ( G I 2 , P I 2 ) to W ( G I 1 , P I 1 ). The in v erse map is simply giv en b y the reve rse construction. By the conditions the in tersection p oin t v 1 of pv 2 and q v 3 is uniquely defined and not on the lines through v 2 and v 4 or v 3 and v 4 . W e add the uniquely defined atom on v 1 , v 2 , v 3 , v 4 to G I 2 ( P I 2 ) that cancels the tension on v 2 v 3 . Since p, v 2 , v 1 lie on one line, this surgery also cancels the tension on v 2 v 4 and similarly for v 3 v 4 . Due to t he equilibrium condition at v 2 , we can replace the edges pv 2 and v 2 v 1 with their tensions w p, 2 and w 2 , 1 b y an edge pv 1 with tension w p, 1 defined b y one of the fo llowing v ector eq uations: w p, 2 pv 2 = w p, 1 pv 1 = w 2 , 1 v 2 v 1 . This uniquely defines a self-stress on G I 1 ( P I 1 ).  Corollary 4.7. Assume that one of the fol low ing c onditions holds: (1) τ 2 ( G I 1 ) > 0 or τ 2 ( G I 2 ) > 0 . (2) τ 2 ( G I 1 ) = 0 and ther e is a c o d i m ension 1 str atum S of B 2 ( G I 1 ) such that — dim W ( G I 1 , P ) > 0 for a G I 1 ( P ) in the str atum S , — the str atum S is not c on taine d i n the subset of B 2 ( G I 1 ) of fr amew orks having one of the triples of p oints ( p, v 1 , q ) , ( p, v 1 , v 4 ) , or ( q , v 1 , v 4 ) on one line. (3) τ 2 ( G I 2 ) = 0 and ther e is a c o d i m ension 1 str atum S ′ of B 2 ( G I 2 ) such that — dim W ( G I 2 , P ′ ) > 0 for a G I 2 ( P ′ ) in the str atum S ′ , — the str atum S ′ is n o t c ontaine d in the subset of B 2 ( G I 2 ) of fr ameworks ha v i ng ( p, v 2 , v 3 ) , ( q , v 2 , v 3 ) , ( p, v 2 , v 4 ) , ( q , v 3 , v 4 ) , or ( v 2 , v 3 , v 4 ) on one line. GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 15 v 1 v 2 v 3 v 4 p q r s G ( P ) v 2 v 3 p q r s G I I 1 ( P I I 1 ) v 1 v 4 p q r s G I I 2 ( P I I 2 ) Figure 5. Surgery I I. Then τ 2 ( G I 1 ) = τ 2 ( G I 2 ) . Pr o of. Let A b e the subset of B 2 ( G I 2 ) of framew orks ha ving ( p, v 2 , v 3 ), ( q , v 2 , v 3 ), ( p, v 2 , v 4 ), ( q , v 3 , v 4 ) or ( v 2 , v 3 , v 4 ) on one line. Let B b e t he subset of B 2 ( G I 1 ) of framew orks ha ving ( p, v 1 , q ), ( p, v 1 , v 4 ) or ( q , v 1 , v 4 ) on one line. Note that A and B are of co dimension 1. The pro of of Prop osition 4.6 give s a surjectiv e map ϕ : B 2 ( G I 2 ) \ A → B 2 ( G I 1 ) \ B inducing an isomorphism b etw een the linear fib ers ab o v e G ( P ) ∈ B 2 ( G I 2 ) \ A and ϕ ( G ( P )). No w in a ll the cases (1)— (3) the statemen t of the corollary follows directly f r o m the definition of the tensegrit y c haracteristic.  Surgery I I. Consider a graph G and a f ramew ork G ( P ). Let G con tain the complete graph K 4 with vertice s v 1 , v 2 , v 3 , and v 4 as a n induced subgraph. Supp ose that the set of edges b et we en v 1 , v 2 , v 3 , v 4 and other v ertices of G is { pv 1 , pv 2 , q v 1 , q v 3 , r v 2 , r v 4 , sv 3 , sv 4 } , for unique p oin ts p, q , r , s . In additio n w e require that the framew ork G ( P ) has the triples of p oints ( p, v 1 , v 2 ) , ( q , v 1 , v 3 ) , ( r , v 2 , v 4 ) , and ( s, v 3 , v 4 ) on one line. See F igure 5 in the middle. Let us delete from the gra ph G the v ertices v 1 and v 4 ( v 2 and v 3 ) with all edges adj a cen t to them. W e denote the resulting graph by G I I 1 (b y G I I 2 resp ectiv ely). The corresp onding framew ork is de noted b y G I I 1 ( P I I 1 ) (b y G I I 2 ( P I I 2 ) resp ectiv ely). See Figure 5 on the left (on the righ t). Surgery I I tak es G I I 1 to G I I 2 or vice v ersa. The pro o f s of the pro p osition and corollary b elow are similar to the pro ofs o f Prop osi- tion 4.6 and Corollary 4.7. Prop osition 4 .8. Co nsider the fr a m eworks G ( P ) , G I I 1 ( P I I 1 ) , and G I I 2 ( P I I 2 ) as ab ove. If non of the triples of p oints ( p, q , v 1 ) , ( p, v 1 , v 4 ) , ( r , v 1 , v 4 ) , ( q , v 1 , v 4 ) , ( s, v 1 , v 4 ) , or ( r , s, v 4 ) lie on a line then we have W ( G I I 1 , P I I 1 ) ∼ = W ( G I I 2 , P I I 2 ) .  16 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS Corollary 4.9. Assume that one of the fol low ing c onditions holds: (1) τ 2 ( G I I 1 ) > 0 or τ 2 ( G I I 2 ) > 0 . (2) τ 2 ( G I I 1 ) = 0 and ther e is a c o d i m ension 1 str atum S of B 2 ( G I I 1 ) such that — dim W ( G I I 1 , P ) > 0 for a G I I 1 ( P ) in the str atum S , — the str atum S is no t c ontaine d in the subset of B 2 ( G I I 1 ) of fr ameworks ha ving ( p, v 2 , v 3 ) , ( q , v 2 , v 3 ) , ( p, v 2 , r ) , ( q , v 3 , s ) , ( r, v 2 , v 3 ) , or ( s, v 2 , v 3 ) on one line. (3) τ 2 ( G I I 2 ) = 0 and ther e is a c o d i m ension 1 str atum S ′ of B 2 ( G I I 2 ) such that — dim W ( G I I 2 , P ′ ) > 0 for a G I I 2 ( P ′ ) in the str atum S ′ , — the str atum S ′ is not c ontaine d in the subset of B 2 ( G I I 2 ) of fr ameworks havi n g ( p, q , v 1 ) , ( p, v 1 , v 4 ) , ( r , v 1 , v 4 ) , ( q , v 1 , v 4 ) , ( s , v 1 , v 4 ) , or ( r, s, v 4 ) on one line. Then τ 2 ( G I I 1 ) = τ 2 ( G I I 2 ) .  5. G eometric re la tions f or s tra t a and complexity of tense grities in two -dimensional case In all t he observ ed examples o f plane tensegrities with a give n graph the strata for whic h a tensegrit y is realizable are defined by certain geometric conditio ns o n the p oints of the corresp onding framew orks. In this sec tion w e study suc h geometric conditions. In Subsection 5.1 w e des crib e an example of a geometric c ondition for a par t icular graph. F urther, in Subsection 5 .2 we giv e general definitions related to systems of geometric conditions. Finally , in Subsections 5.3 a nd 5.4 we form ulate t w o o p en questions related to the geometric nature of tensegrity strata. T o av oid problems with describing anno ying cases of parallel/nonpara llel lines we exte nd the plane R 2 to the pro jectiv e space. It is conv enien t for us to consider the fo llowing mo del of the pro jectiv e space: R P 2 = R 2 ∪ l ∞ . The set of p oints l ∞ is the set of all “directions” in the plane. The set of lines of R P 2 is the set o f all plane lines (eac h plane line con tains no w a new p oin t of l ∞ that is the direction of l ) together with the line l ∞ . Now any t w o lines intersec t at exactly one p oin t. 5.1. A simple example. F irst, we study the graph sho wn in Figure 1 on the left, we denote it b y G 0 . In [17] N. L. White and W. Whiteley pro v ed that the 2- TC of this gr a ph is zero. They sho w ed that there exists a no nzero tensegrit y with graph G 0 and framework P iff the p oints of P satisfy one o f t he follo wing t hr ee conditio ns: i ) the lines v 1 v 2 , v 3 v 4 , a nd v 5 v 6 ha v e a common nonempty intersec tion (in R P 2 ); ii ) the v ertices v 1 , v 4 , and v 5 are in one line; iii ) the v ertices v 2 , v 3 , and v 6 are in one line. W e remind that the base B ( G 0 ) of the configuratio n space is R 12 with co ordinates ( x 1 , y 1 , . . . , x 6 , y 6 ), where ( x i , y i ) are the co ordinates of v i . Condition ( i ) defines a degree GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 17 4 h yp ersurface with equation det   y 1 − y 2 y 3 − y 4 y 5 − y 6 x 2 − x 1 x 4 − x 3 x 6 − x 5 x 1 y 2 − x 2 y 1 x 3 y 4 − x 4 y 3 x 5 y 6 − x 6 y 5   = 0 . and Conditions ( ii ) and ( iii ) define the conics x 1 y 4 + x 4 y 5 + x 5 y 1 − x 1 y 5 − x 4 y 1 − x 5 y 4 = 0 , and x 2 y 3 + x 3 y 6 + x 6 y 2 − x 2 y 6 − x 3 y 2 − x 6 y 3 = 0 resp ectiv ely . 5.2. Systems of geometric conditions. Let us define t hr ee elemen tary geometric con- ditions. Consider an ordered subse t P = { p 1 , . . . , p n } of the pro jectiv e plane. 2-p oint c ondition . W e say that the subset P satisfie s the c ondition p i = p j if p i coincides with p j . 3-p oint c ondition . W e sa y that the subset P satisfie s the c ondition p i ▽ p j ▽ p k = 0 if the p oints p i , p j , and p k are o n a line. 5-p oint c ondition . W e sa y that the subset P satisfie s the c ondition p i = [ p j , p j ′ ; p k , p k ′ ] if the four p oints p j , p j ′ , p k , and p k ′ are on a line and p i also belongs to this line, or if p i = p j p j ′ ∩ p k p k ′ otherwise. W e sa y that [ p j , p j ′ ; p k , p k ′ ] is the interse ction symb ol of the lines p j p j ′ and p k p k ′ . Note that w e define the last condition in terms of closures, since [ p, q ; r, s ] is not defined for all 4-tuples, but fo r a dense subset. Definition 5.1. Consider a system of elemen tary geometric conditions for o r dered n -p o int subsets of R P 2 , and let m ≤ n . — W e say that the o rdered n -p oin t subset P o f pro jectiv e plane satisfies the system of elemen tary geometric conditions if P satisfies eac h of these conditions. — W e sa y that the ordered subset { p 1 , . . . , p m } satisfies c onditional ly the system of elemen tary geometric conditions if there exist p oin ts q 1 , . . . , q n − m suc h that the ordered set { p 1 , . . . , p m , q 1 , . . . , q n − m } satisfies the system. W e call the n um b er n − m the c onditional numb e r of the sys tem. Example 5.2. The condition that six p oints p 1 , . . . , p 6 lie on a c on i c is equiv alen t to the follo wing geometric conditional sy stem:        q 1 = [ p 1 , p 2 ; p 4 , p 5 ] q 2 = [ p 2 , p 3 ; p 5 , p 6 ] q 3 = [ p 3 , p 4 ; p 1 , p 6 ] q 1 ▽ q 2 ▽ q 3 = 0 . 18 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS This is a refo r m ulatio n of P ascal’s theorem. The conditional n umber is 3 here. W e can rewrite the sy stem as fo llo ws, fo r short: [ p 1 , p 2 ; p 4 , p 5 ] ▽ [ p 2 , p 3 ; p 5 , p 6 ] ▽ [ p 3 , p 4 ; p 1 , p 6 ] = 0 . Example 5.3. The condition for six points p 1 , . . . , p 6 that the lines p 1 p 2 , p 3 p 4 , and p 5 p 6 have a c ommon p oint is equiv alen t to the following geometric conditiona l sys tem:  q 1 = [ p 1 , p 2 ; p 3 , p 4 ] q 1 ▽ p 5 ▽ p 6 = 0 , or in a shorter form: [ p 1 , p 2 ; p 3 , p 4 ] ▽ p 5 ▽ p 6 = 0 . The conditio nal n um b er of the system is 1. 5.3. Conjecture on geometric str uct ure of the strata. F or a giv en p ositiv e in teger k and a graph G consider the set of all framew orks G ( P ) at whic h the dimension of the fib er W ( G , P ) is g r eat er tha n or equal to k . W e call this set the ( G, k )- str atum . Since an y ( G, k )-stratum is a finite union o f stra t a o f the base B 2 ( G ), it is semialgebraic. Definition 5.4. Let G b e a graph and k b e a p ositive in teger. The ( G , k )-stratum is said to b e ge ometric if it is a finite union o f the sets of conditional solutio ns of systems of geometric conditions ( in these systems p 1 , . . . , p m corresp ond to the v ertices of the gra ph). Conjecture 2. F or an y graph G a nd integer k the ( G, k )- stratum is g eometric. The conjecture is c heck ed for all the graphs with sev en and few er ve rtices, see Section 6 for the tec hniques. Problem 3. Find analogous elemen tary geometric conditions in the three- (higher-) di- mensional case. W e refer to [17] for examples of geometric conditions in dimens ion 3. 5.4. Complexit y of t he st rata. W e end t his section with a discussion of the complexit y of geometric ( G, k )-strata . A geometric ( G, k )-stratum is defined b y some union of the conditional solutions of sys- tems of geometric conditions. Eac h system in this union has its o wn conditional n um b er. T ak e the maximal among all the conditional num b ers in the union. W e call the minimal n umber among suc h maximal n um b ers for all the unions of s ystems defining the same ( G, k )-stratum the ge om e tric c omplexity of the ( G, k )- stratum. Example 5.5. The geometric complexit y of ( G 0 , 1) stratum fo r the graph G 0 described in Subsection 5.1 and sho wn in Figure 1 on the left equals 3. Problem 4. Find the asymptotics o f t he maximal complexit y of geometric ( G, k )-strata with b ounded num b er of vertice s k while k tends to infinit y . GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 19 Figure 6. A graph G with 9 vertice s, 15 edges and τ 2 ( G ) = 1. 6. Plane tensegritie s with a small number of ver tic es In this section w e w o rk in the t wo-dimens ional case (unless o therwise stated). In Subsec- tion 6.1 w e study the 2-TC o f graphs. In particular, w e calculate the 2 -TC fo r sufficien tly connected g raphs with sev en or less v ertices. In Subsection 6.2 w e giv e a list of geometric conditions for realizabilit y of tensegrities in the plane for gra phs with zero 2-TC. 6.1. On the tensegrity 2 -c haracteristic of graphs. Recall the f o llo wing definitions from graph theory . Let G b e a graph. The vertex c onne ctivity κ ( G ) is the minimal n um b er of v ertices whose deletion disconnects G . The e dge c onne c tivity λ ( G ) is the minimal n umber of edges whose deletion disconnects G . It is well kno wn tha t κ ( G ) ≤ λ ( G ). F or general dimension d , let G ( P ) b e a framew ork in R d with underlying graph G . If κ ( G ) < d or λ ( G ) < d + 1 then G ( P ) consists of t w o or more piec es that can rotate with resp ect to each other. So f or us the mo st interes ting graphs are tho se with κ ( G ) ≥ d and λ ( G ) ≥ d +1. Prop osition 6.1. L et G b e a 2-vertex and 3-e dge c onn e cte d g r aph with k e dges and n vertic es. I f n ≤ 7 , then τ 2 ( G ) = k − 2 n + 3 . R emark 6.2 . In particular w e hav e equalit y in Coro lla ry 3.10 under the conditions of Prop osition 6.1 . The form ula of Prop osition 6.1 holds for many graphs in g eneral, see for instance Example 3.12. It do es no t alw ays hold for gra phs with 9 v ertices as the example b elo w sho ws. Example 6.3. Let G b e the graph with 9 vertice s and 15 edges as in Figure 6. If w e use the form ula of Prop osition 6.1, then we hav e τ 2 ( G ) = 0. Nev ertheless, G con tains K 4 as an induced subgraph. Hence for an y framew o rk G ( P ) the dimension o f W ( G, P ) is at least 1 (w e put zero tensions on a ll edges not b elonging to K 4 and c ho ose a nonzero self-stress on K 4 ). So τ 2 ( G ) ≥ 1. In fact it is not har d t o pro v e that τ 2 ( G ) = 1. This in particular implie s that the t ensions on all edges not b elonging to K 4 are zero fo r a framew ork in a co dimension zero stratum. Notice that the g raph G of Example 6.3 is not a L aman gr ap h , i.e. a g r aph with 2 n − 3 edges, where n is the n umber of v ertices, for whic h each subset of m ≥ 2 v ertices spans at most 2 m − 3 edges. Theorem 1.1 of [4] sho ws that eve ry plana r Laman graph H can b e em b edded as a pseudo-triangulatio n and hence τ 2 ( H ) ≤ 0 b y Example 3.12. W e susp ect that equalit y holds here, and more generally for all Laman gra phs. 20 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS G 1 5 G 2 5 K 5 Figure 7. T he three p ossible graphs with fiv e ve rtices, κ ≥ 2 a nd λ ≥ 3. G 1 6 G 2 6 G 3 6 G 4 6 Figure 8. T he four graphs with six v ertices, κ ≥ 2, λ ≥ 3 and a minimal n umber o f edges. Pr o of of Pr op osi tion 6.1. W e use a classification argumen t. F our v ertic es. F or the complete graph K 4 w e ha v e τ 2 ( K 4 ) = 1 = 6 − 8+3. There are no other graphs satisfying the conditions of the prop osition. Fiv e v ertices. There ar e three possibilities, w e sho w them in Figure 7. F rom Prop o- sition 3.7 we kno w that τ 2 ( K 5 ) = 3 = 10 − 10 + 3 and in Example 4.5 w e ha v e seen that τ 2 ( G 1 5 ) = 1 = 8 − 10 + 3. T o see that τ 2 ( G 2 5 ) = 2 w e apply Corollary 3.1 1. Six v ertices. F rom t he classification of gra phs on six vertice s (see for instance [15]) w e know that an y suc h 2 - v ertex and 3-edge connected graph can b e obtained b y adding edges to one of the four graphs sho wn in Figure 8. By Coro llary 3.1 1 it suffice s to c hec k the form ula of the prop o sition for them. Note tha t G 1 6 and G 2 6 ha v e 9 edges. They both ha v e zero 2- TC (9 − 12+3 = 0). Indeed, in Subsection 5.1 w e men tioned that B 2 ( G 1 6 ) has co dimension 1 strata with nontrivial linear fib er. As it is stated in [17] the graph G 2 6 underlies a tensegrit y if and only if the six p oin ts lie o n a c onic, whic h is a lso a co dimension 1 condition. Note that G 2 6 is the complete bipartite graph K 3 , 3 . F or G 3 6 w e pro ceed as follows. F ro m Corollary 3.10 it follo ws that τ 2 ( G 3 6 ) ≥ 1. Then w e use Prop osition 4.1 in the same w ay as in Example 4.5 to sho w that τ 2 ( G 3 6 ) = τ 2 ( G 1 5 ) = 1 , and again 10 − 12 + 3 = 1 , see F ig ure 9. It is easy to see that the same a r g umen t w orks to sho w that τ 2 ( G 4 6 ) = 1. Sev en vertices. F r o m t he classification of graphs with sev en ve rtices (see [15]) w e get that a ll 2- v ertex and 3-edge connected graphs on sev en v ertices can b e obtained by adding edges to one of the sev en graphs sho wn in Figure 10. By Corollary 3.11 it suffices again to c heck these graphs. GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 21 G 3 6 K 4 e 2 → e 1 G 1 5 e 2 e 1 Figure 9. Using Prop osition 4.1 w e get that τ 2 ( G 3 6 ) = τ 2 ( G 1 5 ) = 1. p 1 p 2 p 3 p 4 p 5 p 6 p 7 G 1 7 G 2 7 G 3 7 G 4 7 G 5 7 G 6 7 G 7 7 Figure 10. The se v en graphs with sev en v ertices, κ ≥ 2, λ ≥ 3 and a minimal n um b er of edges. T o pro v e that τ 2 ( G 1 7 ) = τ 2 ( G 2 7 ) = 0 we use Corollary 4.7 a pplied to G 1 6 . Note that the geometric conditions for G 1 6 to underlie a nonzero tensegrit y (see Subsection 5.1) a llo w to apply Corollary 4.7 . Similarly , w e apply Coro llary 4.7 to G 2 6 to conclude that τ 2 ( G 3 7 ) = 0. By computations analogous to [8, Section 4] w e find that τ 2 ( G 4 7 ) = 0. Indeed, one can sho w tha t t his graph underlies a nonzero tensegrit y if and only if at least one of the follo wing co dimension 1 conditions holds: p 1 ▽ p 2 ▽ p 3 = 0 , p 1 ▽ p 5 ▽ p 6 = 0 , p 2 ▽ p 4 ▽ p 7 = 0 , p 3 ▽ p 4 ▽ p 7 = 0 , p 3 ▽ p 5 ▽ p 6 = 0 . So the first four graphs with 11 edges hav e zero 2-TC. The other t hree hav e 12 edges. W e apply Corollary 4.7 to G 3 6 and G 4 6 to obtain that τ 2 ( G 5 7 ) = 1 and τ 2 ( G 7 7 ) = 1 . T o pro v e t ha t the 2-TC o f G 6 7 = K 3 , 4 is 1 w e pro ceed as follows. F ir st, τ 2 ( G 6 7 ) ≥ 1 b y Corollary 3 .10. Then we apply Prop osition 4.1 as sho wn in F igure 11 . The graph G has 6 v ertices and 10 edges and thus w e hav e τ 2 ( G ) = 1. It is easy to c hec k that fo r a general p osition framew ork G ( P ) with a no nzero self-stress, all edges of G ( P ) hav e nonzero stress. On the middle picture w e get a v ertex o f degree 2, so w e reduce to the graph H o n the righ t. Note that H is isomorphic to G , so τ 2 ( H ) = 1. Hence τ 2 ( G 6 7 ) = 1 as w ell.  22 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS G 6 7 G e 2 → e 1 H e 2 e 1 Figure 11. Using Prop osition 4.1 one sees that τ 2 ( G 6 7 ) = 1. 6.2. Geometric conditions for realizabilit y of plane tensegrities for graphs with zero tensegrit y 2 -c haracteristic. Like in in tersection theory of alg ebraic v arieties, it often happ ens that strata for a graph with negativ e 2-TC are obtained as in tersections of closures of some strata of graphs with zero 2-TC. So the conditions for realizabilit y of plane tensegrities for graphs with zero 2-TC are the most imp orta n t. In this subs ection w e give all the conditio ns for the zero 2-TC graphs with num b er of v ertices not exceeding 8. In practice one w ould lik e to construct a tense grit y without struts or cables with zero tension. So it is natur a l to give the follo wing definition. W e sa y that a graph G is visib l e at the framework P if there exists a self-stress that is nonzero at eac h edge of this framew or k. R emark 6.4 . Vis ibilit y restrictions remo ve many degenerate stra t a . F or instance if a zero 2-TC graph G has a complete subgraph on v ertices v 1 , v 2 , and v 3 , then the co dimension 1 stratum defined by the condition: the p oints v 1 , v 2 , and v 3 ar e on one line do es in many cases not con tain visible frameworks . Let us list the geometric conditions for the v ertices of all visible 2 -v ertex and 3-edge con- nected graphs with n v ertices a nd zero 2-TC for n ≤ 8. T o find t he geometric conditions w e essen tially use t he surgeries of Section 4, see Propo sitions 4.1, 4.6 and 4.8. In the next table we use b esides the elemen t a ry also the followin g t wo additio na l geo- metric conditions: — six p oints are on a conic; — for six p oints p 1 , . . . , p 6 the lines p 1 p 2 , p 3 p 4 , and p 5 p 6 ha v e a common nonempt y in tersection. As w e ha v e seen in Examples 5.2 and 5.3 these conditions are equiv alen t to geometric conditional systems. GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 23 Graph (6 v ert .) Sufficien t geometric conditions v 1 v 2 v 3 v 4 v 5 v 6 the lines v 1 v 2 , v 3 v 4 , and v 5 v 6 ha v e a common nonempt y in tersec- tion v 1 v 2 v 3 v 4 v 5 v 6 the six p o ints v 1 , v 2 , v 3 , v 4 , v 5 , a nd v 6 are on a conic Graph (7 v ert .) Sufficien t geometric conditions v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 ▽ v 2 ▽ v 3 = 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 the lines v 1 v 2 , v 3 v 4 , and v 5 v 6 ha v e a common nonempt y in tersec- tion v 1 v 2 v 3 v 4 v 5 v 6 v 7 the lines v 1 v 2 , v 3 v 4 , and v 5 p where p = [ v 2 ,v 6 ; v 3 ,v 7 ] hav e a com- mon nonempt y inte rsection v 1 v 2 v 3 v 4 v 5 v 6 v 7 the six p o ints v 1 , v 2 , v 3 , v 4 , v 5 , and p , where p = [ v 1 ,v 6 ; v 3 ,v 7 ] are on a conic Graph (8 v ert .) Geometric conditions v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the lines v 1 v 2 , v 3 v 4 , and v 5 v 6 ha v e a c ommon nonempt y in- tersection v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 ▽ v 2 ▽ v 3 = 0 24 FRANCK DORA Y, OLEG KARPENKO V , AND JAN SCHEPERS Graph (8 v ert .) Geometric conditions v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oin ts v 1 , v 2 , v 3 , v 4 , v 5 , and v 6 are o n a conic v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the lines v 1 v 2 , v 3 v 4 , and v 5 p , where p = [ v 2 , v 6 ; v 3 , v 7 ] ha v e a common nonempt y in t ersec- tion v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the lines v 1 v 2 , v 3 v 4 , and v 5 p , where p = [ v 2 , v 6 ; v 7 , v 8 ] ha v e a common nonempt y in t ersec- tion v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oin ts v 1 , v 2 , v 3 , v 4 , v 5 , and p , where p = [ v 1 , v 6 ; v 3 , v 7 ], are o n a conic v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the lines v 1 v 2 , v 3 p , and v 5 q , where p = [ v 1 , v 4 ; v 5 , v 8 ] a nd q = [ v 2 , v 6 ; v 3 , v 7 ] hav e a com- mon no nempty in tersection v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the lines v 5 v 6 , v 1 p , and v 4 q , where p = [ v 2 , v 3 ; v 6 , v 7 ] a nd q = [ v 2 , v 3 ; v 6 , v 8 ] hav e a com- mon no nempty in tersection v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oints v 1 , v 2 , v 4 , v 6 , p , and q , whe re p = [ v 2 , v 3 ; v 6 , v 7 ] and q = [ v 2 , v 5 ; v 6 , v 8 ], a re o n a conic v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oints v 1 , v 3 , v 4 , v 6 , p , and q , whe re p = [ v 2 , v 3 ; v 5 , v 7 ] and q = [ v 5 , v 7 ; v 6 , v 8 ], a re o n a conic v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oints v 1 , v 2 , v 3 , v 5 , p , and q , whe re p = [ v 1 , v 6 ; v 3 , v 7 ] and q = [ v 3 , v 4 ; v 5 , v 8 ], a re o n a conic v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oin ts v 1 , v 2 , v 3 , v 5 , v 6 , and q , where p = [ v 1 , q ; v 3 , v 4 ] and q = [ v 5 , v 7 ; v 4 , v 8 ], a re o n a conic GEOMETR Y OF CONFIGURA TION SP A CES OF TENSEGRI TIES 25 Graph (8 v ert .) Geometric conditions v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oints v 1 , v 2 , v 4 , v 5 , p , and q , whe re p = [ v 1 , v 6 ; v 5 , v 8 ] and q = [ p, v 7 ; v 2 , v 3 ], are on a conic v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the three p o in ts [ v 1 , v 4 ; v 2 , v 3 ], [ v 1 , v 5 ; v 2 , v 6 ], and [ v 5 , v 8 ; v 6 , v 7 ] are o n one line Graph (8 v ert .) Sufficien t geometric conditions v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the three p oin ts [ v 1 , v 2 ; v 6 , v 7 ], [ v 1 , p ; v 6 , v 8 ], and [ p, q ; v 3 , v 8 ], where p = [ v 2 , v 4 ; v 5 , v 8 ] and q = [ v 1 , v 5 ; v 3 , v 4 ], are on one line, AND the lines p ′ v 2 , q ′ v 3 , and v 6 v 7 ha v e a common nonempty in tersection, where p ′ = [ r ′ , s ′ ; v 1 , v 6 ], q ′ = [ r ′ , s ′ ; v 6 , v 8 ], r ′ = [ v 1 , v 4 ; v 2 , v 5 ], and s ′ = [ v 3 , v 4 ; v 5 , v 8 ] v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oints v 1 , v 4 , v 7 , v 8 , p , and q , where p = [ r, s ; v 3 , v 4 ], q = [ r, s ; v 5 , v 8 ], r = [ v 1 , v 2 ; v 5 , v 6 ], and s = [ v 2 , v 3 ; v 6 , v 7 ], are on a conic, AND the six p oin ts v 1 , v 2 , v 6 , v 7 , p ′ , and q ′ , where p ′ = [ r ′ , s ′ ; v 2 , v 3 ], q ′ = [ r ′ , s ′ ; v 5 , v 6 ], r ′ = [ v 1 , v 4 ; v 5 , v 8 ], and s ′ = [ v 3 , v 4 ; v 7 , v 8 ], are on a conic v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 the six p oin ts v 1 , v 3 , v 4 , v 5 , v 7 , and p , where p = [ v 1 , q ; v 7 , v 8 ], q = [ r, s ; v 2 , v 3 ], r = [ v 3 , v 6 ; v 7 , v 8 ], and s = [ v 1 , v 6 ; v 2 , v 8 ], are on a conic, AND the six p oints v 1 , v 2 , v 3 , v 6 , v 8 , and p ′ , where p ′ = [ v 3 , q ′ ; v 7 , v 8 ], q ′ = [ r ′ , s ′ ; v 1 , v 4 ], r ′ = [ v 1 , v 5 ; v 7 , v 8 ], and s ′ = [ v 3 , v 5 ; v 4 , v 7 ], are on a conic R emark 6.5 . 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E-mail addr ess , F r anck Doray: dor ay@ma th.lei denuniv.nl E-mail addr ess , Oleg Ka rp enko v : karpenk@ mccme .ru E-mail addr ess , Jan Schepers : jansc hepers 1@gmail.com (F ranck Doray , Oleg Karp enkov, Jan Schepers) Ma thema tisch Instituut, U niversiteit Leiden, P.O. Box 9512, 23 00 RA Leiden, The N etherlands