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ARS MATHEMATICA CONTEMPORANEA 6 (2013) 305–322
On stratifications for planar tensegrities
with a small number of vertices
Oleg Karpenkov
TU Graz, Kopernikusgasse 24, A8010, Graz, Austria
Jan Schepers
Departement Wiskunde, Katholieke Universiteit Leuven
Celestijnenlaan 200B, 3001, Leuven, Belgium
Brigitte Servatius
Mathematics Department, Worcester Polytechnic Institute
100 Institute Road, Worcester, MA 01609-2280, USA
Received 16 January 2012, accepted 30 November 2012, published online 8 December 2012
Abstract
In this paper we discuss several results about the structure of the configuration space
of two-dimensional tensegrities with a small number of points. We briefly describe the
technique of surgeries that is used to find geometric conditions for tensegrities. Further we
introduce a new surgery for three-dimensional tensegrities. Within this paper we formulate
additional open problems related to the stratified space of tensegrities.
Keywords: Tensegrities, equilibrium, surgeries.
Math. Subj. Class.: 52C30, 05C10
1 Introduction
In this paper we study the stratified spaces of tensegrities with a small number of points.
We work mostly with planar tensegrities. In the case of 4 and 5 point configurations we
give an explicit description of all the strata and present a visualization of the entire stratified
space. Further we give a geometric description of the strata for 6 and 7 points and use the
technique of surgeries to find new geometric conditions adding to the list of already known
ones. In particular, we introduce a new surgery for tensegrities in R3.
E-mail addresses: karpenkov@tugraz.at (Oleg Karpenkov), janschepers1@gmail.com (Jan Schepers),
bservat@math.wpi.edu (Brigitte Servatius)
Copyright c© 2013 DMFA Slovenije
306 Ars Math. Contemp. 6 (2013) 305–322
1.1 Configuration space of tensegrities
The first steps in the study of rigidity and flexibility of tensegrities were made by B. Roth
and W. Whiteley in [9] and further developed by R. Connelly and W. Whiteley in [3], see
also the survey about rigidity in [13]. N. L. White and W. Whiteley in [12] started the
investigation of geometric conditions for a tensegrity with prescribed bars and cables. In
the preprint [7] M. de Guzmán describes several other examples of geometric conditions
for tensegrities.
Let us recall standard definitions of tensegrities (as in [2], [4], etc.). See also [10] for a
collection of open problems and a good bibliography.
Definition 1.1. Fix a positive integer d. Let G = (V,E) be an arbitrary graph without
loops and multiple edges. Let it have n vertices v1, . . . , vn.
• A configuration is a finite collection P of n labeled points (p1, p2, . . . , pn), where
each point pi (also called a vertex) is in a fixed Euclidean space Rd.
• The embedding of G with straight edges, induced by mapping vj to pj is called a
tensegrity framework and it is denoted as G(P ).
• We say that a load or force F acting on a framework G(P ) in Rd is an assignment
of a vector fi in Rd to each vertex i of G.
• We say that a stressw for a frameworkG(P ) in Rd is an assignment of a real number
wi,j = wj,i (we call it an edge-stress) to each edge pipj of G. An edge-stress is
regarded as a tension or a compression in the edge pipj . For simplicity reasons we
put wi,j = 0 if there is no edge between the corresponding vertices. We say that w
resolves a load F if the following vector equation holds for each vertex i of G:
fi +
∑
{j|j 6=i}
wi,j(pj − pi) = 0.
By pj−pi we denote the vector from the point pi to the point pj .
• A stress w is called a self-stress if, the following equilibrium condition is fulfilled at
every vertex pi: ∑
{j|j 6=i}
wi,j(pj − pi) = 0.
• A couple (G(P ), w) is called a tensegrity if w is a self-stress for the framework
G(P ).
• If wi,j < 0 then we call the edge pipj a cable, if wi,j > 0 we call it a strut.
Let W (n) denote the linear space of dimension n2 of all edge-stresses wi,j . Consider a
framework G(P ) and denote by W (G,P ) the subset of W (n) of all possible self-stresses
for G(P ). By definition the set W (G,P ) is a linear subspace of W (n).
Definition 1.2. The configuration space of tensegrities corresponding to the graph G is the
set
Ωd(G) :=
{
(G(P ), w) |P ∈ (Rd)n, w ∈W (G,P )
}
.
The set {G(P ) |P ∈ (Rd)n} is said to be the base of the configuration space, we denote it
by Bd(G).
Oleg Karpenkov et al.: On stratifications for planar tensegrities. . . 307
1.2 Stratification of the base of a configuration space of tensegrities
Suppose we have some framework G(P ) and we want to find a cable-strut construction on
it. Then which edges can be replaced by cables, and which by struts? What is the geometric
position of points for which given edges may be replaced by cables and the others by struts?
These questions lead to the following definition.
Definition 1.3. A set W (G,P1) is said to be equivalent to a set W (G,P2) if there exists
a homeomorphism ξ between W (G,P1) and W (G,P2), such that for any self-stress w in
W (G,P1) the self-stress ξ(w) satisfies
sgn
(
ξ(w)
)
= sgn
(
w
)
.
Henceforth we call a set W (G,P ) a linear fiber. The described equivalence relation
on linear fibers gives us a stratification of the base Bd(G) = (Rd)n. A stratum is by
definition a maximal connected set of points with equivalent linear fibers. In the paper [4]
we prove that all strata are semialgebraic sets (which implies for instance that they are path
connected).
The idea of this paper is to make the first steps in the study of particular configuration
spaces of tensegrities. We present the techniques to find geometric conditions and open
problems for further study that already arise in very simple situations of 9 point configura-
tions.
Let us, first, make the following three general remarks.
GR1. The majority of the strata of codimension k can be defined by algebraic equations
and inequalities that define the strata of codimension 1. The exceptions here are mostly in
high codimension (the simplest one is as follows: for two points connected by an edge
there is no codimension 1 stratum, but there is one codimension 2 stratum corresponding
to coinciding points; actually it is interesting to find the complete list of such exceptions).
So the most important case to study is the codimension 1 case.
GR2. A stratification of a subgraph is a substratification of the original graph (i.e., each
stratum for a subgraph is the union of certain strata for the original graph), hence below we
skip the description of B2(G) for graphs with 5 vertices other than K5.
GR3. For any stratum there exists a certain subgraph that locally identifies the stratum
(i.e., for any point x of the stratum there exists a neighborhood B(x) such that any config-
uration in B(x) has a nonzero self-stress for the subgraph if and only if this point is on the
stratum).
According to general remarks GR1 and GR2 the most interesting case is to study the
strata of codimension 1 for the complete graph on n vertices (denoted further by Kn). It is
possible to find some of the strata of Kn directly. For the other strata one, first, should find
an appropriate subgraph that locally identifies the stratum, and then find appropriate surg-
eries (explained in Section 3) to reduce the complexity of the subgraph to find geometric
conditions.
This paper is organized as follows. In Section 2 we study the stratification of configu-
ration spaces of tensegrities in the plane with a small number of vertices. In Subsections 2.1
and 2.2 we briefly describe the trivial cases of two and three point configurations. Further
in Subsections 2.3 and 2.4 we study the four and the five point cases. In each of the cases
we describe the geometry and the number of strata. In addition we introduce the adjacency
308 Ars Math. Contemp. 6 (2013) 305–322
diagram of full dimension and codimension 1 strata. In Subsections 2.5 and 2.6 we de-
scribe geometric conditions for the codimension 1 strata of 6, 7, and 8 point tensegrities. In
Section 3 we present the technique of surgeries to find geometric descriptions for the strata.
In Subsection 3.1 we describe surgeries that do not change graphs, and in Subsection 3.2
we show a couple of surgeries in the two-dimensional case. We introduce a new three-
dimensional surgery in Subsection 3.3. In conclusion, we formulate several open questions
in Subsection 3.4.
2 Stratification of the space B2(Kn) for small n
In this section we study the geometry of tensegrity stratifications for graphs with a small
number of vertices. The cases of n = 2, 3, 4, 5 are studied in full detail. Starting from
n = 6 there are some gaps in the understanding of tensegrities. Still for n = 6, 7, 8 the
complete description of the geometric conditions for the strata is known, we briefly describe
several results on them here (see [4] for more information).
2.1 Case of two points
Consider, first, the case of two points (n = 2). There are only two graphs on two points: a
complete one K2 and a graph without edges (denote it by G0,2).
All the fibers of the base B2(G0,2) = R4 are of dimension 0, and, therefore, they are
equivalent. Hence the stratification is trivial.
The complete graph K2 here has only one edge. If two points of the graph do not
coincide then the stress at this edge should be zero. When two points coincide then the
stress at the edge can be arbitrary, and we have a one-dimensional set of solutions (i.e., a
fiber). So the base B2(K2) = R4 has a codimension 2 stratum (a 2-dimensional plane).
The complement to this stratum is a stratum of codimension 0.
2.2 Three point configurations
There are four different types of graphs here: let Gi,3 be the graph with i edges for i =
0, 1, 2, 3.
In cases G0,3 and G1,3 the base stratifications are the following direct products:
B2(G0,3) = B2(G0,2)× R2 and B2(G1,3) = B2(K2)× R2.
So B2(G0,3) is trivial and B2(G1,3) has a 4-dimensional subspace and its complement as
strata.
The base B2(G2,3) contains five strata. One of them corresponds to the configuration
where three points coincide: the fiber here is 2-dimensional, this stratum is isometric to R2.
There are three strata where one of the edges of the graph vanishes: they are isometric to
R4\R2. Finally, the complement to the union of these strata is the only stratum of maximal
dimension. There are no nonzero tensegrities for a configuration in this stratum.
For the complete graph on three vertices we have, for the first time, codimension 1
strata. There are three codimension 1 strata, all of them correspond to the following config-
uration: three points are in one line. Different strata correspond to having a different point
between the two others.
Let us briefly describe one of such strata. Let Pi = (xi, yi) be the points of the graph
(i = 1, 2, 3). Then the condition that the three points are in a line is defined by a quadratic
Oleg Karpenkov et al.: On stratifications for planar tensegrities. . . 309
equation:
(x2 − x1)(y3 − y1)− (x3 − x1)(y2 − y1) = 0
This quadric divides the space into two connected components: corresponding to positively
and negatively oriented triangles.
To sum up we present for B2(K3) the following table.
Dimension of a stratum 0 1 2 3 4 5 6
Number of such strata 0 0 1 0 3 3 2
2.3 Stratification of B2(K4)
In this subsection we restrict ourselves to the complete graph K4 (for its subgraphs we
apply the reasoning of GR2 above). A plane configuration of four points in general position
admits a unique tensegrity (up to a multiplicative constant), which is called an atom. In [8]
it was proved that any self-stress for Kn is a sum of self-stressed atoms in Kn (i.e., a sum
of certain K4 ⊂ Kn with scalars). For K4 there are exactly 14 strata of general position.
The strata of codimension 1 correspond to three of four points of the graph lying in a
line. Actually in this case there is no jump of dimension of the fiber: there is also a unique
(up to scalar) solution corresponding to the three points in a line. But the stresses on the
edges from the fourth point are all zero, and hence a fiber of this stratum is not equivalent
to general fibers. The number of such strata is 24.
In codimension 2 we have two different types of strata corresponding to
• four points in a line: the dimension of a fiber is 2 (twelve strata);
• two points coincide: the dimension of a fiber is 1 (twelve strata).
In codimension 3 there is one type of strata with configurations of four points in a line,
two of which coincide. Six of them with the double point in the middle and twelve of them
with the double point not in the middle.
In codimension 4, there are two types of strata:
• three points coincide (4 strata);
• two pairs of points coincide (3 strata).
And, finally, there is a codimension 6 stratum when all four points coincide. We remark
that for none of the strata the fiber is 3-dimensional.
The cardinalities of strata are shown in the following table.
Dimension of a stratum 0 1 2 3 4 5 6 7 8
Number of strata 0 0 1 0 7 18 24 24 14
2.3.1 The space of formal configurations
Let us draw schematically the adjacency of the strata of maximal dimension via strata
of codimension 1. The dimension of the stratified space is 8, let us reduce it to two via
factoring by proper affine transformations. We will use the following simple proposition.
Proposition 2.1. Invertible affine transformations of the plane do not change the equiva-
lence class of a fiber W (G,P ). In other words if P is a configuration and T an invertible
affine transformation of the plane then
W (G,P ) 'W (G,T (P )).
310 Ars Math. Contemp. 6 (2013) 305–322
So instead of studying the stratification itself we restrict to the set of formal configura-
tions with respect to proper affine transformations of the plane.
Definition 2.2. We say that a four point configuration v1, v2, v3, v4 is formal in one of the
following cases:
i) nondegenerate case: a configuration Px,y,+ with vertices v1 = (0, 0), v2 = (1, 0),
v3 = (x, y), v4 = (x, y+1) for arbitrary (x, y).
ii) nondegenerate case: a configuration Px,y,− with v1 = (0, 0), v2 = (1, 0), v3 =
(x, y), v4 = (x, y−1) for arbitrary (x, y).
iii) degenerate case: a configuration P∆,+ with v1 = (0, 0), v2 = (1, 0), v3 = (0, 1),
v4 = (∆, 1) for an arbitrary ∆.
iv) degenerate case: a configuration P∆,− with v1 = (0, 0), v2 = (1, 0), v3 = (0,−1),
v4 = (∆,−1) for an arbitrary ∆.
v) closure: we add two formal configurations P±∞ with vertices v1 = (0, 0), v2 =
(1, 0), v3 = (1, 0), v4 = (1,±∞).
We denote the set of all formal configurations by Λ4.
In some sense the space Λ4 is the space of all codimension 0 and codimension 1 con-
figurations factored by the group of proper affine transformations.
Proposition 2.3. For any codimension 0 and codimension 1 configuration there exists a
unique formal configuration to which the first configuration can be affinely deformed.
The space Λ4 is endowed with a natural topology of a quotient space.
Proposition 2.4. There is a natural topology of a sphere S2 for the set Λ4.
Proof. Let us introduce a topology of the unit sphere S2 for Λ4. Consider the configura-
tions of case i) on the plane z = 1: we identify the point Px,y,+ with the point (x, y, 1).
Consider the projection of this plane to the upper unit hemisphere S2 from the origin. So
we have a one to one correspondence between the configurations of case i) and the upper
hemisphere.
Similarly we take the plane z = −1 for the case ii) identifying the point (−x,−y,−1)
with the configuration Px,y,− and projecting it to the lower hemisphere.
For the equator of the unit sphere we use all the other cases as asymptotic directions.
First, we associate the configuration P∆,+ with the point
(cos(π − arccotan ∆), sin(π − arccotan ∆), 0).
Let us explain the topology at one of such points of the equator. Suppose we start with
Px,y,+. The transformation sending the first three points to (0, 0), (1, 0), and (0, 1) is
linear with matrix (
1 −x/y
0 1/y
)
.
Then the image of the fourth point of Px,y,+ is (−x/y, 1+1/y). While x tends to infinity
and x/y tends to ∆ the last point tends to (−∆, 1), and hence the configuration Px,y,+
tends to P−∆,+, as in the above formula.
Oleg Karpenkov et al.: On stratifications for planar tensegrities. . . 311
Figure 1: Stratification of B2(K4).
Secondly, we associate P∆,− with the point
(cos(− arccotan ∆), sin(− arccotan ∆), 0)
in a similar way.
Finally, we glue P+∞ and P−∞ to the points (1, 0, 0) and (−1, 0, 0) respectively.
So, the codimension 0 and 1 stratification of B2(K4) can be derived from the stratifi-
cation of the sphere. We show the stereographic projection of Λ4 from the point (0, 0,−1)
to the plane z = 1 on Figure 1. There are four types of strata of codimension 1, they
correspond to the fact that certain three points are in a line. They separate the plane into
14 connected components. In each of the connected components we draw a typical type of
configuration: (v1, v2, v3, v4). Here v1 is blue, v2 is purple, v3 is red and v4 is green.
Remark 2.5. Different geometric conditions are represented by different colors in the pic-
ture, the correspondence is as follows.
• Light blue strata (6 strata forming a circle) correspond to configurations with v1, v2,
and v3 in a line.
312 Ars Math. Contemp. 6 (2013) 305–322
• Dark blue strata (6 strata) contain configurations with v1, v2, and v4 in a line.
• Light green strata (6 strata) contain configurations with v1, v3, and v4 in a line.
• Dark green strata (6 strata) correspond to configurations with v2, v3, and v4 in a line.
We have 24 strata of codimension 1 in total.
• The dashed black line is the projection of the equator. It corresponds to the degenerate
case of parallel segments. The dashed line is not a stratum, it has the same fiber as
all the points in its neighborhood. While one passes the dashed line the red-green
segment ”rotates” around the blue-purple segment.
Remark 2.6. The 14 connected components of the plane are in one-to-one correspondence
with the 14 faces of a cuboctahedron (accordingly, the 12 points on these circles correspond
to its vertices). Thus, the four circles are those circumscribed around the equatorial reg-
ular hexagons of the cuboctahedron. The vertices of this polytope lie on a sphere, hence,
through stereographic projection the four circumcircles in question project in fact to circles
in the image plane.
2.4 Stratification of B2(K5)
2.4.1 General description of the strata
We have 264 strata of general position.
As in the two previous cases the strata of codimension 1 correspond to three points of
the graph lying in a line. The number of such strata is 600.
The following strata are of codimension 2:
• twice three points in a line: 270 strata;
• four points in a line: 120 strata;
• two points coincide: 420 strata.
In codimension 3 we have the following cases:
• three points in a line and one double point: 60 strata;
• four points in a line two of which coincide: 180 strata;
• five points in a line: 60 strata.
For codimension 4 we have the following list:
• one triple point: 20 strata;
• five points in a line two of which coincide: 120 strata;
• two double points: 30 strata.
In codimension 5 we get:
• five points in a line three of which coincide: 30 strata;
• five points in a line with two pairs of points coinciding: 45 strata.
In codimension 6 there are the following strata:
• a triple point and a double point: 10 strata;
• one point and one point of multiplicity four: 5 strata.
And, finally, there is a codimension 8 stratum when all five points coincide.
The cardinalities of the strata are shown in the following table.
Dimension of a stratum 0 1 2 3 4 5 6 7 8 9 10
Number of strata 0 0 1 0 15 75 170 300 810 600 264
Oleg Karpenkov et al.: On stratifications for planar tensegrities. . . 313
2.4.2 Visualization of B2(K5)
Let us now describe the structure of the stratification B2(K5). Like in case of B2(K4)
we introduce a set Λ5 which represents the adjacency of strata of full dimension and of
codimension 1. By definition we put
Λ5 = Λ4 × R2,
i.e., we consider all the four point configurations of Λ4, and to each configuration we add
the fifth point. We take the product topology for Λ5.
So at each point of Λ4 we attach an R2-fiber. It will soon become clear that for any full
dimension stratum of Λ4 the corresponding fibration is trivial, but the adjacency is not.
On Figures 2 and 3 we show Λ5 in the following way. We draw the stratification of
Λ4 and inside each connected component we show the typical fiber of the component. The
first four points are represented by purple, blue, green, and red points. The lines passing
through any pair of them divide the fiber into 18 connected components, that correspond
to strata of full dimension. At each such component we write a letter of the Latin alphabet
(we consider capital and small letters as distinct).
• Two regions denoted by the same letter and lying in neighboring connected compo-
nents of Λ4 separated by light red, dark red, and black strata are in the same stratum.
• Two regions denoted by the same letter and lying in neighboring connected compo-
nents of Λ4 separated by light blue, dark blue, light green, and dark green strata are
in distinct strata which are adjacent to the same codimension 1 stratum.
• Two regions denoted by a distinct letter and lying in neighboring connected compo-
nents of Λ4 are not in one stratum and are not adjacent to the same codimension 1
stratum.
The light blue, dark blue, light green, and dark green strata represent the same geomet-
ric conditions as in Remark 2.5 above. For the remaining strata we have:
• The dark red stratum symbolizes that the line through the red and blue points is par-
allel to the line through the green and purple points.
• The light red stratum symbolizes that the line through the red and purple points is
parallel to the line through the green and blue points.
• The black stratum symbolizes that the line through the red and green points is parallel
to the line through the purple and blue points.
Remark 2.7. The configuration spaceB2(K5) has several obvious symmetries. First, there
is the group of permutations S5 that acts on the points of B2(K5); these symmetries are
hardly seen from Figures 2 and 3 since the representation is not S5-symmetric. Secondly,
there is a symmetry about the origin that sends configurations from B2(K5) to themselves,
on Figures 2 and 3 we used capital and small letters to indicate this symmetry (for instance,
the strata of ”a” contain centrally symmetric configurations to the configurations of the
strata ”A”).
As in the case of 4 point configurations we skip the subgraphs of K5, see the second
general remark above (GR2).
314 Ars Math. Contemp. 6 (2013) 305–322
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Oleg Karpenkov et al.: On stratifications for planar tensegrities. . . 315
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316 Ars Math. Contemp. 6 (2013) 305–322
2.5 Essentially new strata in B2(K6)
The stratification of B2(K6) is much more complicated, at this moment we do not even
know how many strata of distinct dimension are present in the stratification.
According to GR1 the first step in studying the stratification of B2(K6) is to study all
possible distinct types of strata of codimension 1. In the examples of Kn for n < 6 we
only have strata corresponding to the following geometric condition: three points are in a
line. For the case of 6 points we get two additional types of strata: six points on a conic,
and three lines passing through three pairs of points have a unique point of intersection.
So the following are three codimension 1 strata (appeared in [12] by N. L. White and
W. Whiteley):
• three points in a line;
• the lines v1v2, v3v4, and v5v6 meet in one point (or all parallel);
• all the six points are on a conic.
We conclude this subsection with the following problems.
Problem 2.8. Find a description of B2(K6), B3(K4) and B3(K5) similar to the ones for
B2(K4) and B2(K5) shown in the previous subsections.
2.6 A few words about the case n > 6
In [4] we have studied strata of the 7 and 8 point configurations. There are 4 distinct types
of codimension 1 strata for 7 points and 17 types for 8 points.
The 4 types of codimension 1 strata for 7 points are defined by the following geometric
conditions:
• three points in a line;
• the lines v1v2, v3v4, and v5v6 meet in one point (or all parallel);
• the lines v1v2, v3v4, and v5p (where p is the intersection of the lines v2v6 and v3v7)
have a common nonempty intersection;
• the six points v1, v2, v3, v4, v5, and p (where p is the intersection of the lines v1v6
and v3v7) are on a conic.
For the list of strata of 8 point configurations we refer to [4].
It turns out that the geometric conditions of any codimension 1 stratum can be obtained
by the following procedure. Consider the points of configuration P ; for each two pairs
of points (vi, vj) and (vk, vl) of this configuration consider the point of intersection of
the lines vivj and vkvl. This leads to a bigger configuration of points including P and
the above intersections, we denote it by U(P ). This operation can be iteratively applied
infinitely many times, which results in a universal set
U∞(P ) =
∞⋃
m=0
Um(P ).
Any condition for a codimension 1 stratum is always as follows: three certain points of
U∞(P ) are in a line (for the details, see for instance [9] and [4]).
Example 2.9. The condition the lines v1v2, v3v4, and v5v6 meet in one point in terms of
points of U1(P ) = U(P ) is as follows. The points v1, v2, and p = v3v4 ∩ v5v6 are in a
line.
Oleg Karpenkov et al.: On stratifications for planar tensegrities. . . 317
Remark 2.10. For simplicity reasons we omit discussions of cases where certain lines vivj
and vkvl are parallel, due to the fact that this situation is never generic for codimension 1
strata. In general one may think that if the lines vivj and vkvl are parallel, then their
intersection point is in the line at infinity in the projectivization of R2.
Remark 2.11. At first glance, the condition six points are on a conic is of different nature.
Nevertheless, it is a relation on the points of the configuration in U1(P ) described by
Pascal’s theorem: The intersections of the extended opposite sides of a hexagon inscribed
in a conic lie on the Pascal line. See also Example 2.15 below.
Problem 2.12. Describe all the possible different types of strata for 9 points.
Problem 2.13. How to calculate the number of different types of strata for n points with
arbitrary n?
It is also interesting to have an answer for the following question: how many iterations
does one need to perform (i.e., find the minimal m for Um(P )) to describe all conditions
for the codimension 1 strata of n-point configurations P?
Problem 2.14. Which configurations of Um(P ) define the same geometric condition?
This problem is a kind of question of finding generators and relations for the set of all
conditions. Let us show one type of such ”relations” in the following example.
Example 2.15. Consider the condition: six points v1, v2, . . . , v6 are on a conic. This
condition is described by configurations contained in U1(P ) via Pascal’s theorem:
The points p, q, r are in a line for
 p = vσ(1)vσ(2) ∩ vσ(4)vσ(5)q = vσ(2)vσ(3) ∩ vσ(5)vσ(6)
r = vσ(3)vσ(4) ∩ vσ(6)vσ(1)
,
where σ is an arbitrary permutation of the set of six elements. So, there are 60 different
configurations of U1(P ) defining the same geometric condition.
3 Further study of strata: surgeries
We now look into subgraphs contained in a particular stratum and ask the basic question
on the dimension of the fiber.
Even graphs of very low connectivity admit non-zero tensegrities, for disconnected or
one-connected graphs we may simply examine the connected or 2-connected components.
Also 2-connected graphs may be decomposed via the 2-sum, see [11]: Consider graphs G1
and G2, their configurations P1 and P2 admitting tensegrities with p1q1 a cable in G1(P1)
and p2q2 a strut in G2(P2). We form the 2-sum G1
⊕
G2 by identifying p1 with p2 and q1
with q2 and removing the identified edge. We can inherit a configuration P from P1 and
P2 by fixing P1 and properly dilating, rotating and translating P2. It is clear that
dimW (G1
⊕
G2, P ) = dimW (G1, P1) + dimW (G2, P2)− 1.
Since 2-sum decomposition is canonical, we can describe geometric conditions for 2-
connected graphs by geometric conditions on their 3-blocks. For example the geometric
condition for G in Figure 4 is that the lines v1v2, v3v4, and v5v6 meet in one point.
318 Ars Math. Contemp. 6 (2013) 305–322
v
6
v
5
v
1
v
2
v
3
v
4
v
6
v
5
v
1
v
2
v
3
v
4
v
6
v
8
v
5
v
7
v
8
v
7
=
Figure 4: The 2-sum of a triangular prism with K4
v1 v2
v3v4
v5 v6
v1 v2
v3v4
v5 v6
v1 v2
v3v4
v5 v6
Figure 5: Examples of subgraphs of K6 admitting tensegrities at codimension 1 strata of
B2(K6).
3.1 Subgraphs related to codimension 1 strata
As we have already mentioned in GR3, for any codimension 1 stratum there exists at least
one subgraph of Kn that generically does not admit tensegrities but at this stratum admits
a one-dimensional family of tensegrities. Let us show such subgraphs for the codimension
one strata of B2(K6) and B2(K7).
Example 3.1. In the case of K6 we have three strata of different geometrical nature. The
first triangular subgraph (Figure 5, left) is related to the strata with three points in a line.
The second (Figure 5, middle) corresponds to the strata whose three pairs of points gen-
erate lines passing through one point. The last one (Figure 5, right) corresponds to the
configurations of six points on a conic.
Example 3.2. In the case of K7 there are the following new examples of subgraphs, cor-
responding to the main 4 different types of strata.
From the left to the right we have the following geometric conditions
• v1, v2, and v3 are in a line;
• the lines v1v2, v3v4, and v5v6 meet in one point;
• the lines v1v2, v3v4, and v5p (where p = v2v6 ∩ v3v7) have a common point;
• the six points v1, v2, v3, v4, v5, and p (where p = v1v6 ∩ v3v7) are on a conic.
Note that the example for three points in a line is actually the 2-sum of a triangle with
two atoms, so the only way for a non-zero self-stress on the edges is to have v1, v2, and v3,
v1 v2
v3 v4v5
v6 v7
v1 v2
v3v4
v5 v6
v7
v1 v2
v3v4
v5
v6
v7
v1 v2
v3v4
v5
v6
v7
Figure 6: Examples of subgraphs of K7 admitting tensegrities at codimension 1 strata of
B2(K7).
Oleg Karpenkov et al.: On stratifications for planar tensegrities. . . 319
the vertices of the triangle, in a line.
Remark 3.3. Geometric conditions for the graphs with 8 and fewer vertices are given
in [4]. Several of those geometric conditions were described before in terms of bracket
polynomials in [12] by N. L. White and W. Whiteley. We also refer to the paper [1] by
E. D. Bolker and H. Crapo for the relation of bipartite graphs with rectangular bar con-
structions.
3.2 Surgeries on subgraphs that change geometric conditions in a predictable way
In this subsection we present several surgeries that allow to guess the geometric conditions
for new strata (characterized by certain subgraphs) via other strata (characterized by these
graphs modified in a certain way). We call such modifications of graphs surgeries.
3.2.1 Surgeries that do not change geometric conditions
Let G be a graph, denote by Ge the graph with an edge e removed.
Proposition 3.4. (Edge exchange) Consider a graph G and a subgraph H , and let e1 and
e2 be two edges of H . Let P be a configuration for which dimW (H,P ) = 1. Suppose
also that the self-stresses of H do not vanish at the edges e1 and e2. Then we have
dimW (Ge1 , P ) = dimW (Ge2 , P ).
In the situation of Proposition 3.4 the strata of Ge1(P ) and Ge2(P ) are defined by the
same geometrical conditions.
3.2.2 Two two-dimensional surgeries that change geometric conditions
The first surgery is described in the following proposition.
Proposition 3.5. Consider the frameworks G(P ), GI1(P I1 ), and GI2(P I2 ) as on the figure:
v1
v2 v3
v4
p
q
G(P )
v1
v4
p
q
GI1(P
I
1 )
v2 v3
v4
p
q
GI2(P
I
2 )
If none of the triples of points (p, v2, v3), (q, v2, v3), (p, v2, v4), (q, v3, v4) and (v2, v3, v4)
are on a line then we have
dimW (GI1, P
I
1 ) = dimW (G
I
2, P
I
2 ).
Example 3.6. Let us consider a simple example of how to get a geometric condition for
the graph
v1 v2
v3v4
v5
v6
v7
320 Ars Math. Contemp. 6 (2013) 305–322
to admit a tensegrity knowing all geometric conditions for 6-point graphs. Let us apply
Surgery I to the points v5, v6, v7. We have:
v1 v2
v3v4
v5
v6
v7
v1 v2
v3v4
v5 p
.
The geometric condition to admit a tensegrity for the graph on the right is:
the lines v1v2, v3v4 and v5p intersect in a point.
Hence the geometric condition for the original graph is:
the lines v1v2, v3v4 and v5p intersect in a point, where p = v2v6 ∩ v3v7.
Now let us show the second surgery.
Proposition 3.7. Consider the frameworks G(P ), GII1 (P II1 ), and GII2 (P II2 ) as on the
following figure:
v1
v2 v3
v4
p
q
rs
G(P )
v2 v3
p
q
rs
GII1 (P
II
1 )
v1
v4
p
q
rs
GII2 (P
II
2 )
If none of the triples of points (p, q, v1), (p, v1, v4), (r, v1, v4), (q, v1, v4), (s, v1, v4), or
(r, s, v4) lie on a line then we have
dimW (GII1 , P
II
1 ) = dimW (G
II
2 , P
II
2 ).
Remark 3.8. Both surgeries were shown in [4]. There is a certain analogy of the first
surgery to ∆Y exchange in matroid theory (see for instance [13] and [5] for the connections
between matroids and rigidity theory), but it is not exactly the same.
Remark 3.9. Actually these surgeries are valid in the multidimensional case as well under
the condition that certain points are in one plane.
3.3 A new tensegrity surgery in R3
We conclude this paper with a single surgery for tensegrities in R3.
Proposition 3.10. Consider a graph G and frameworks G(P ), G1(P1), and G2(P2) as
follows:
v1
v2 v3
v4
e1
e2
e3
e4
e5 e6
G(P )
v2 v3
v4
e1
e2
e3
e4
e5 e6
G1(P1)
v1
v2 v3
v4
e1
e2
e3
e4
e5 e6
G2(P2)
Oleg Karpenkov et al.: On stratifications for planar tensegrities. . . 321
Denote the plane v2v3v4 by π1. Suppose that the couples of edges e1 and e2, e3 and e4, e5
and e6 define planes π2, π3, and π4, different from π1. Assume that π2 ∩ π3 ∩ π4 is a one
point intersection.
If G1(P1) and G2(P2) have nonzero stress on the edges connecting v1, v2, v3, and v4
then
π1 ∩ π2 ∩ π3 ∩ π4 = v1.
In this case we additionally have
dimW (G1, P1) = dimW (G2, P2).
Proof. The first statement follows since v1 only has valency 3 in G2(P2), so v1, v2, v3,
and v4 need to be coplanar to have a nonzero edge-stress. Now we explain how to map
W (G1, P1) to W (G2, P2). The inverse map is simply given by the reverse construction.
By the conditions v1 is the intersection point of the planes π1, π2, and π3. We add the
uniquely defined plane atom on v1, v2, v3, v4 to G1(P1) that cancels the edge-stress on
v2v3. Since the plane π1 does not coincide with the plane π2 spanned by the forces on e1
and e2, the edge-stress on v2v4 is also canceled. By the same reasons the edge-stress on
v3v4 is canceled as well. This uniquely defines a self-stress on G2(P2).
3.4 Some related open problems
The next goal in this approach is to continue to study the geometry of the strata. Ideally
one would like to find techniques that will give geometric conditions for a graph via its
combinatorics. This question seems to be a very hard open problem. The study of surgeries
is the first step to solve it at least in codimension 1.
For a start we propose the following open question.
Problem 3.11. Find all geometric conditions for the strata of 9 point tensegrities.
The surgeries introduced in this section were extremely useful for the study of 8 point
configurations (see in [4]). We think that it is not enough to know only these surgeries to
find all the geometric conditions. This gives rise to another question.
Problem 3.12. Find other surgeries on graphs that predictably change the geometric con-
ditions.
As far as we know there is no systematic study of strata for tensegrities in R3 or higher
dimensions: these cases are much more complicated than the planar case. At least the
stratification of B3(K5) should have a description similar to that of B2(K4), since 5 points
in general position in R3 admit a unique non-zero self-stress.
Additionally one should examine the rigidity properties of subgraphs in a stratum. For
K4 we have 14 strata of full dimension. For 8 of them the convex hull is a triangle, in 5 of
the strata the points are in convex position. A tensegrity for the convex position has 4 struts
(cables) and two cables (struts), while in the non-convex case there are three cables and
three struts. All of these tensegrities are (infinitesimally) rigid and struts and cables may
be exchanged without destroying rigidity. However, when viewed as graphs embedded in
R3 only half of them are rigid. For the convex case, there must be cables on the convex
hull and two struts. In the non-convex case there must be a triangle of struts on the convex
hull and three cables in the interior, termed a spider web by R. Connelly. None of these
322 Ars Math. Contemp. 6 (2013) 305–322
are proper forms in the sense of B. Grünbaum. They are minimally rigid, but in the convex
case they have members intersecting in a vertex other than a vertex of the graph, in the
non-convex case there is a vertex without a strut. B. Grünbaum in his lectures on lost
mathematics [6] asks about the number of proper forms given n struts. On 3 struts, there
is only one tensegrity which is minimally rigid with edges only intersecting at vertices and
such that every vertex is endpoint of at least one strut. For 4 struts there are at least 20
forms, but it is not known how many there are. The number of forms on n struts is bounded
by the number of strata on B3(Kn). For the hierarchies of the various kinds of rigidity
see [3].
Acknowledgments. We are grateful to the unknown reviewer for Remark 2.6 and other
useful suggestions. Oleg Karpenkov is supported by the Austrian Science Fund (FWF),
grant M 1273-N18. Jan Schepers is a Postdoctoral Fellow of the Research Foundation –
Flanders (FWO).
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