Almost all circle polyhedra are rigid

The infinitesimal rigidity theory of bar-and-joint frameworks traces its origins to the investigations of James Clerk Maxwell in 1864. Before Maxwell’s investigations, Legendre and Cauchy investigated the global rigidity of polyhedra, culminating in the famous Cauchy Rigidity Theorem [4] of 1813 that avers that convex, bounded polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ are globally rigid. In 1916 Max Dehn [6] generalized in one direction the Cauchy Theorem by proving the infinitesimal rigidity of strictly convex polyhedra in $\mathbb{E}^{3}$, and in the decade of the 1950’s, A.D. Alexandrov [1] greatly extended Dehn’s insights in his articulation of a rather broad theory of rigidity for $3$-dimensional polyhedra. Nonetheless, the question of whether or not all polyhedra, even non-convex ones, in $\mathbb{E}^{3}$ were rigid remained open. In 1975 Herman Gluck [7], in the vein of Maxwell and Dehn, proved that almost all of them are infinitesimally rigid, and therefore rigid. At that time, many believed that all polyhedra in $\mathbb{E}^{3}$ were rigid, but in 1977 Robert Connelly [5] surprised the community by constructing a flexible polyhedron in $\mathbb{E}^{3}$, necessarily non-convex, and by Gluck’s result, a rather rare example among polyhedra.

In [2], the authors of the present work began a study of the rigidity theory of circle frameworks, and in particular of circle polyhedra. There we showed that an analogue of the Cauchy Rigidity Theorem remains true when vertices in $\mathbb{E}^{3}$ are replaced by circles in the $2$-sphere $\mathbb{S}^{2}$ that are placed in the pattern of a Euclidean polyhedron. There global rigidity adheres when the circle polyhedron is convex and proper, and an example of Ma and Schlenker [9] shows that this fails for non-convex ones, which mirrors the classical case for Euclidean polyhedra. Of course for circle polyhedra, Euclidean distance is replaced by inversive distance and rigidity is understood with respect to Möbius transformations. In the present paper, we show that an analogue of Gluck’s Theorem on the infinitesimal rigidity as well as the rigidity of almost all polyhedra holds in the setting of circle polyhedra in the plane $\mathbb{E}^{2}$.

The proof, not surprisingly, mirrors the plan developed over the last couple of centuries by Cauchy [4], Dehn [6], Alexandrov [1], and Gluck [7] in their investigations of rigid Euclidean polyhedra, with Gluck’s paper in particular serving as a valuable guide for our development. The first task is identifying appropriate rigidity matrices and stress matrices in the setting of circle frameworks, where the trivial motions are by Möbius transformations and inversive distance between circles is the desired preserved parameter. This is accomplished in Sections 1 and 2. With the right notions of these matrices in place, the argument proceeds as in Gluck [7]. The infinitesimal rigidity of planar circle polyhedra is related to inversive stresses in Section 3. In the classical Euclidean development, Dehn’s Theorem plays the role of showing that the collection of infinitesimally rigid polyhedra is non-empty, from which Gluck argues that the collection is dense in the space of all polyhedra. The role played by Dehn’s Theorem in our development is of independent interest and appears in Section 4 where we show that univalent tangency circle packings of the $2$-sphere are infinitesimally rigid. The four preceding sections come together in the proof of the infinitesimal rigidity of almost all circle polyhedra in the final section, Section 5, as well as their rigidity.

Because of the well-known, intimate connection between the hyperbolic geometry of the unit ball in $\mathbb{E}^{3}$ (the Beltrami-Klein model), the inversive geometry of its $2$-sphere ideal boundary $\mathbb{S}^{2}$, and the projective geometry of $\mathbb{RP}^{3}$ in which these models reside, (infinitesimal) rigidity results for circle polyhedra and frameworks have implications for the (infinitesimal) rigidity of hyperbolic polyhedra. These have been articulated in works of Thurston, Rivin, Hogdsen-Rivin, Bao-Bonahon, and our previous paper [2]. We do not take the time to translate the results of this paper to the setting of, for example, strictly hyperideal hyperbolic polyhedra in $\mathbb{H}^{3}$ (see the final section of [2]) as we are content with this mere mention of the connection.

For the graph $G$ with vertex set $V(G)=\{1,\dots,n\}$, the set $\mathfrak{p}=\{p_{1},\dots,p_{n}\}$ of points in $\mathbb{R}^{d}$ indexed by the vertex set $V(G)=\{1,\dots,n\}$ is called a Euclidean framework and denoted as $G(\mathfrak{p})$. A key concept that has found many uses in the rigidity theory of Euclidean frameworks is that of a stress on a framework. Informally, a stress is a real number $\omega_{ij}$ attached to each edge $ij$ of a framework and is used to compute force vectors $F_{ij}=-F_{ji}$ acting on $i$ and $j$ so that a positive stress attracts $i$ and $j$ towards each other, a negative stress repels, and a zero stress results in $F_{ij}=F_{ji}=0$. For each vertex $i$ a force $F_{i}$ is computed by the sum $F_{i}=\sum_{j:ij\in E(G)}F_{ij}$. If for all $i$, $F_{i}=0$, the stress is said to be an equilibrium stress. An equilibrium stress is non-trivial whenever there exists at least one edge $ij$ such that $F_{ij}\neq 0$. Alexandrov [1] proved that the only equilibrium stress on the edge-framework of a convex polyhedron in $\mathbb{R}^{3}$ is the trivial one if and only if the framework is infinitesimally rigid. Our aim in this section is to develop a similar notion of stress for c-frameworks.

Before specializing to circle polyhedra, which are the c-frameworks of interest in the remainder of the paper, we make some observations connecting the inversive rigidity matrix $R$ to the inversive stress matrix $V$. For each edge $ij$ of $G$, let $\lambda_{ij}=r_{i}^{2}r_{j}^{2}$ and let $\widehat{V}$ be the scaled inversive stress matrix scaled by the non-zero constants $\Lambda=\{\lambda_{ij}:ij\in E(V)\}$. A moment’s inspection reveals that

the transpose of the scaled inversive stress matrix is precisely the inversive rigidity matrix. In particular, the ranks of the matrices $\widehat{V}$ and $R$ agree, and by Lemma 2.1 so too does the rank of $V$. The next lemma now follows from Lemma 1.1.

We now specialize to those c-frameworks whose graphs are polyhedral graphs. A c-framework $P(\mathcal{C})$ is a c-polyhedron if the graph $P$ is the 1-skeleton of an abstract triangulated polyhedron, equivalently, the $1$-skeleton of a simplicial triangulation of the $2$-sphere.²²2This is more general than the development of c-polyhedra in [2] since we do not require the existence of ortho-circles for each face, as required there, and at the same time less general in that we do require the polyhedra to be triangulated, unlike there. The importance of the triangulation assumption is seen in the lemma following.

Note that this proof works precisely because the number of edges $m$ is equal to $3n-6$ and the Möbius group is a $6$-dimensional Lie group.

A tc-polyhedron is a tc-framework that is also a c-polyhedron, so a configuration of circles in the pattern of a triangulated polyhedron whose adjacent circles are tangent. Recall that a c-framework is univalent if the open disks bounded by the circles are pairwise disjoint.

The arguments of Gluck [7], adapted to the setting of c-polyhedra, now apply to verify our main theorem. Here are the details.

Let $P$ be the $1$-skeleton of a simplicial triangulation of the $2$-sphere with vertices labeled as $1,\dots,n$. Identify the c-polyhedron $P(\mathcal{C})$ where $\mathcal{C}$ is the circle collection $\{C_{1},\dots,C_{n}\}$ with the point $p\in\mathbb{R}^{3n}$ whose coordinates are $p_{3i-2}=x_{i}$, $p_{3i-1}=y_{i}$, and $p_{3i}=r_{i}$ when $C_{i}=(x_{i},y_{i},r_{i})$ under our parameterization of circles. The collection of all c-polyhedra in the pattern of $P$ is then parameterized by the points of the open subspace $\mathbb{O}$ of $\mathbb{R}^{3n}$ determined by the inequalities $r_{i}>0$ for $i=1,\dots,n$.

By Lemma 1.1, the c-polyhedron $P(\mathcal{C})$ is infinitesimally flexible precisely when the rank of the inversive rigidity matrix $R=R_{P(\mathcal{C})}$ is less than $3n-6$. Since the rank of a matrix is the greatest integer $d$ such that some $d\times d$ sub-matrix has non-zero determinant, this occurs when every $(3n-6)\times(3n-6)$ sub-matrix of the rigidity matrix $R$ has zero determinant. By Equations 1.2, the coefficients of the rigidity matrix $R$ are polynomials in the coordinates of the parameter point $p\in\mathbb{R}^{3n}$ that represents $P(\mathcal{C})$, and this implies that the determinant of any $(3n-6)\times(3n-6)$ sub-matrix of $R$ is a polynomial in the coordinates of $p$, i.e., in the variables $x_{i},y_{i},r_{i}$, $i=1,\dots,n$. It follows that the c-polyhedron $P(\mathcal{C})$ is infinitesimally flexible if and only if the point $p$ representing $P(\mathcal{C})$ lies in the real algebraic variety $\mathbb{V}$ of $\mathbb{R}^{3n}$ determined by the polynomials $\det D=0$, as $D$ ranges over the $(3n-6)\times(3n-6)$ sub-matrices of $R$.