Proceedings of the International Congress of Mathematicians
Helsinki, 1978
Conjectures and Open Questions in Rigidity
Robert Connelly*
I. Introduction. The study of the rigidity of frameworks and surfaces is fairly old,
going back at least to the time of Euler, but progress has been slow and painful.
Even today knowledge is meager, and many basic simple questions are still unanswered. One gets the feeling that in the past these questions were considered
more a source of embarrassment than a source of conjectures or goals for future
study. My feeling is that reasonable conjectures and problems are useful and essential
to progress in mathemtics. Even if such questions turn out to be poorly stated,
ambiguous, or ultimately uninteresting, they usually have been useful, at least for
inspiration if nothing else.
Rigidity is a subject very close to the reality of our world. If a structure is proved
to be rigid one can always build it and see if it collapses. Theorems had better
hold up. The opportunities for beneficial interaction among engineers, architects,
and mathematicians seem very attractive. The subject is versatile and sufficiently undeveloped that contributions can be made at all levels.
I have chosen five categories of questions about which I think it would be nice
to know more information. Naturally, some are ambiguous and not too detailed.
If I knew the answers, I would not be asking.
H. Definitions and history. It is useful to state rigidity in terms of things called
frameworks. The name is probably inspired from structural engineering, and
physically one should think of a framework as a gadget constructed out of dowel
rods with their ends stuck in small flexible rubber connectors. Mathematically,
* Supported by a grant from the National Science Foundation of the United States.
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Robert Connelly
a framework F is just a finite ordered collection of points p = ( P i , ...9pv)9 p, in
Euclidean «-space (n is usually only 2 or 3), together with a collection of certain
pairs of points ê called the rods. For example,
Pi • -
a triangle
a square
' = {(Pi> P2), (P2> Pi), (Ps, Pi)} S = {(pl9 p2), Q?2, j>a), (p 3 , P4), (p 4 , Pi)}
A yfoc of F is a continuous motion of all the vertices p(t)=(p^t), p2(t), •••,/?„ (t))9
0<f<l,
so that p(0)=p
and the length of any rod (pi9Pj) in jp,
|Pi(0—Pj(0l> is constant for all t. The flex p(t) is ngzrf if a// the distances
\Pi(t)—pj(t)\ are constant. In the latter case it turns out that p(t) is obtained by
applying a common rigid motion of all of En to each pt{0). Thus p(t) =
(gtPi(Q)>8tPo(P)> •••> gtPv(P))> ^ = r i g i d motion of En. The framework F is rigid if
every flex of it is rigid. In the examples above, the triangle is rigid, and the square
is not rigid, even in the plane.
The natural question at this stage is to determine what frameworks are rigid. The
first nontrivial result in this direction was by Cauchy in 1813 [6].
THEOREM 1. If a framework F is obtained from the natural edges and vertices of
the boundary of a convex polyhedral surface in E\ then F is rigid under the more
restrictive condition that every flex holds each natural face rigid.
Here a natural vertex, edge, or face is a 0, 1 or 2 dimensional intersection of
a support plane with the convex surface. Note that if all the natural faces are
triangles, then they are automatically "held" rigid. Actually Cauchy tried to prove
a bit more than this theorem and his proof had a gap, but his proof was essentially
correct and the gap was closed by Steinitz. (See Steinitz [25], Stoker [26], Lyusternik [20].)
In any case this very nice result reinforced the notion ("the rigidity conjecture")
that any triangulated two-dimensional closed surface in E3—convex or not—was
rigid, where the framework consists of the vertices and the edges of the triangulation.
Bricard [5] in 1897 "classified" all flexible octahedra and it was clear that none of
them were embedded. Cauchy's techniques were applied and transformed to differential geometry and in the 1940s and 1950s A. D. Alexandrov and A. V. Pogorelov
extended and applied Cauchy's techniques (see [1]). In 1973 Gluck [13] blatantly
stated the rigidity conjecture. After showing that certain embedded surfaces were
rigid [7], I found a counterexample. Thus a flexible triangulation of an embedded
sphere exists [8].
So the question remains: If the rigidity conjecture is false, what is true?
Conjectures and Open Questions in Rigidity
409
III. The questions.
1. The bellows conjecture. Although I do not know of any spectacular consequences
of the following conjecture, I find it one of the most intriguing in the subject. The
problem is to construct a mathematical bellows. This is a closed, polyhedral, flexible
surface which flexes so that the volume changes. In fact we can generalize this as
follows: For any piecewise-linear (or smooth) map / : M2-*JE3 of a closed oriented
piecewise-linear (or smooth) two-manifold into three-space there is a welldefined
notion of the volume enclosed by /, whether / is an embedding or not. In case
/ is an embedding this number is ± the usual volume enclosed by f(M) depending on the orientation chosen. In every case of aflexibleframework coming from
a "map" of a triangulated oriented two-manifold, which I know of, this generalized
volume is constant during the flex (see [7] and [8]). Thus we have:
Conjecture. Every orientable, closed, polyhedral flexible surface (even with selfintersections) flexes with constant volume.
As far as I know this conjecture is essentially due to Dennis Sullivan.
2. Classifyingflexiblesurfaces. Although it may seem ambitious, one way of
determining when a polyhedral surface is rigid is simply to classify all the flexible
surfaces. (Where the triangulation gives the framework.) Here things get a bit
vague. My feeling is that a flexible surface is the "union" of two kinds of pieces.
Thefirstis in some sense "prime" and has a "volume" of zero. The second is "rigid"
and soflexeswith constant volume. So in particular the whole surface flexes with
constant volume.
If the above is confusing, the reader is referred to the case of the suspension of
a polygonal circle (see [7]). One takes a polygonal circle in JE3 and connects two
additional points, the north and south poles, to each vertex on this equatorial circle.
It is easy to see that this gives a triangulated surface, topologically a sphere. It is
my opinion that the above "conjecture" holds for this surface. In fact if the distance
between the north and south poles moves during a flex of such a surface, all the
pieces turn out to be "prime" and the surface has zero volume. This theorem is the
only way I know of showing all such embedded suspensions are rigid.
One natural starting point here might be to classify all flexible triangulations of
a sphere with 8 vertices. It is conceivable one of these could be embedded. If not
then Klaus Steffen's very pretty example with 9 vertices would be the best possible.
(See [24] or [9].)
3. Algorithms. Short of complete success for the previous problem and also for
other similar problems, it would be very nice to have precise and hopefully efficient
algorithms for deciding various rigidity problems. In particular one is "given"
a framework and one wishes to know:
A. Whether it is rigid or flexible.
B. Whether it is infinitesimally rigid. (Infinitesimal rigidity will be defined shortly.)
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Robert Connelly
C. How many mutually noncongruent embeddings there are of this framework.
Recent work of Peter Kahn demonstrates that there are at least in principle
algorithms for A and estimates for problems like C (see [15]); and it is wellknown (see [13]) that problem B is equivalent to a large determinant being nonzero.
This is encouraging and a good first step, but all these processes are much too unwieldy for practical use, even with a very efficient computer.
It would be pleasing to apply an algorithm for problem A to several examples
that are known to be embedded but not known to be flexible. In this respect many
of the surfaces of stellated regular solids seem to be very "sloppy" when made out
of cardboard, and it would be interesting to know if they are really flexible, (e.g.
Charles Schwartz has pointed out the stellated rhomboid dodecahedron, see
Coxeter [11]).
Problem B would be of great interest to those concerned with the actual construction of structures, since generally (but not always) infinitesimal rigidity is what is
desired. In this respect the "groupe de recherche" in Montreal has made some
progress and can efficiently determine infinitesimal rigidity in many interesting
cases.
Infinitesimal rigidity is roughly, a linearized version of ordinary rigidity. It is
defined as follows: First we define an infinitesimal flex of a framework F in
E3 as a sequence of vectors p=(Pi, ..., pv) such that (P/—P/)'(pj~~P/)=0
for all edges i9 j in ê. p is called trivial if there are vectors r9 t (in E3) such that
p. = (rXpt) + t for all i (1 ^ i ^ v).
r and t can be regarded as an infinitesimal rotation and translation. F is infinitesimally rigid if all infinitesimal flexes are trivial.
THEOREM
2. If F is infinitesimally rigid, then it is rigid (see Gluck [13]).
The converse is false as one can see by the example of a triangle with a point
inside and all possible edges.
Pi
Pi = p 2 = p 3 = 0,
p4 is perpendicular to the plane of the triangle.
(It is interesting and useful to know that infinitesimal rigidity is preserved under
projective collineations of E3 (see [12]).)
A framework is generically rigid if when we consider all possible positions
=
P (Pii --iPv) of the vertices (we regard p now as a point in E3°) keeping the
Conjectures and Open Questions in Rigidity
411
same pairs of vertices as edges, then an open dense set of these positions is rigid.
We ask:
D. What frameworks are generically rigid? Namely is there an algorithm for
determining this or some reasonable characterization?
It turns out that, for JB2, Leman [19] (see also Asimow and Roth [4]) has
essentially solved this problem, but it remains open in E3 and seems quite difficult.
Also, the main theorem of Gluck [13] (see also Kuiper [18]) is that triangulated
spheres are generically rigid. In fact, any triangulated convex surface with all the
natural faces as triangles is infinitesimally rigid. (More generally if the triangulation
has no vertex in the interior of a natural face, then it is infinitesimally rigid. See
Alexandrov [1] or Asimow and Roth [4].)
4. Exotic rigidity. Despite the attention payed to the infinitesimal rigidity of
convex surfaces, there are many other situations that are also interesting. For
instance, even for a convex polyhedral surface, if a triangulation has a vertex in
the interior of a natural face, then the associated framework is not infinitesimally
rigid; but it turns out to be rigid nevertheless. In fact we have the result (see Connelly
[10]):
THEOREM 3. The framework associated to any triangulation of any convex polyhedral surface is second order rigid and thus rigid.
Note that vertices are allowed anywhere in the surface. Second order rigidity is
defined as follows : A second order flex of a framework F is a first order flex
P = (Pi» •••> Pv) together with another sequence p = (pl9 -..,pv) of vectors such
that for each rod of JF,
(Pi-Pj) • (Pi - Pi) + (Pt- Pj) • (Pi- Pj) = 0.
F is second order rigid if every nontrivial infinitesimal flex p fails to extend to
a second order flex p9 p.
As indicated in the theorem, second order rigidity implies rigidity. It is not hard
to see how to generalize the above definition to higher order rigidity and we have the
question for n^3
A. Does /7th order rigidity imply rigidity?
Note that Efimov in [12] mentions a very similar problem.
If the answer is affirmative, this could possibly provide a method for doing 3A,
since if one has an nth order flex, the problem of finding the n + \ order flex is
entirely linear.
If we turn to the rigidity of surfaces other than the sphere we have the very basic
conjecture :
B. Every triangulated closed surface in E3 is generically rigid.
Due to methods of Gluck [13], see also Asimow and Roth [3], this amount^ to
finding one infinitesimally rigid framework corresponding to each abstract triangulation of the surface. At this point I can show that many triangulations of such surfaces
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Robert Connelly
have one infinitesimally rigid framework but I am not sure this includes all possible
triangulations. This result is inspired by similar results in the smooth category
by L. Nirenberg [21], and A. D. Alexandrov [2]. Also Stoker [26] has methods for
showing that certain non convex surfaces are rigid.
Also related to the proof of Theorem 3 is the idea of "cabled" frameworks.
Here, in addition to the rods there are certain pairs of vertices designated as cables,
which can decrease but not increase in length during a flex. In Griinbaum's notes
[14] there are many very interesting examples of such rigid cabled frameworks,
and this idea is very helpful in Theorem 3. See also Whiteley [27] for similar results.
It seems that many of these examples can be thought of as having springs for the
cables and allowing the framework to move to a position of minimum energy. It
would be very nice to have information as in 3A or 3B about such frameworks.
There is one other natural question concerning the rigidity of piecewise-linear
surfaces, for n^3.
C. Is every piecewise-linear closed «-dimensional manifold, which is embedded
in En+\ rigid?
This is the higher dimensional analogue to the rigidity conjecture. The construction
of my example works if the ambient embedding space is S 3 , the standard round
3-sphere, and the cone over this example from the center gives a nontrivial flexible
surface in E 4 , but with boundary. Thus in E4, counterexamples exist locally at
least.
5. Rigidity for smooth surfaces. The basic question for smooth surfaces seems
to be:
A. Are there flexible, closed, smooth surfaces in E3?
Here the surface (and the flex) are to have at least two continuous derivatives
(to be of class C 2 ), since methods of Kuiper [16], and [17], following Nash, give
a flexible C 1 embedding of any C 1 surface in E3. (All the above must preserve the
metric of course.) In particular even the standard round two-sphere can be embedded
strangely and flexed in a C 1 fashion in E3.
Even for immersed C 2 surfaces, the answer to the above question is unknown.
The methods of my counterexample do not apply in the C 2 case.
The notion of infinitesimal rigidity carries over into this category as well, and the
theorems about the uniqueness of convex surfaces and their infinitesimal rigidity
hold here as well. Spivak, vol. 5, [23] is a good reference for what is known here.
Unfortunately, even the following very basic questions are unanswered.
B. If a smooth (of class C 2 say) closed surface is infinitesimally rigid, is it rigid?
It is not inconceivable that the methods of piecewise-linear surfaces may be applicable to problems in the smooth category via the appropriate sort of approximation,
somewhat in the spirit of Pogorelov's work [22]. Thus we have the following questions :
C. If a piecewise-linear manifold approximates a closed smooth surface "closely
enough", is the associated framework rigid?
Conjectures and Open Questions in Rigidity
413
Here it seems appropriate to say at least that the normal to each of the triangular
faces must approximate the normals to the surface, as well as the points approximating the points on the surface. In fact it may be enough that the dihedral angles are
not too sharp in some sense.
Once again the author would like to thank the Institut Des Hautes Etudes
Scientifiques, and in particular N. H. Kuiper, for their kind support and encouragement.
References
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Polyeder, Akademie-Verlag, Berlin, 1958.
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On a class of closed surfaces, Recueil. Math. (Moscow) 4 (1938), 69—77.
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Rigidity of graphs. II J. Math. Anal. Appi, (to appear).
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148.
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(1813), Sl( = Oeuvres complètes d'Augustin Cauchy, 2nd sér., Tome 1, 1905, pp. 26—38).
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A counterexample to the rigidity conjecture for polyhedra, Inst. Hautes Études Sci.
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A flexible sphere, Math. Intellegencer; vol. 3. August 1978.
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The rigidity of certain cabled frameworks and the second order rigidity of arbitrarily
triangulated convex surfaces, preprint, Cornell University, 1978.
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On C^isometric embeddings, Indag. Math. 17 (1954), 545—556, 683—689.
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Sphères polyédriquesflexiblesdans E3, d'après Robert Connelly, Séminaire Bourbaki,
February 1978, pp. 514-01—514-21.
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24. Klaus Steffen, A symmetricflexibleConnelly sphere with only nine vertices, letter, Inst. Hautes.
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