Circle packing theorem

"Coin graph" redirects here. For contact graphs of equal-radius disks, see penny graph.

The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a collection of circles (in general, on any Riemann surface) whose union is connected and whose interiors are disjoint. The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph.^[1] More generally, intersection graphs of interior-disjoint geometric objects are called tangency graphs^[2] or contact graphs.^[3] Coin graphs are always connected, simple, and planar. The circle packing theorem states that these are the only requirements for a graph to be a coin graph: every finite connected simple planar graph ${\displaystyle G}$ has a circle packing in the plane whose intersection graph is isomorphic to ${\displaystyle G}$.

A circle packing for a five-vertex planar graph

Existence and uniqueness

Theorem statement

A maximal planar graph ${\displaystyle G}$ is a finite simple planar graph to which no more edges can be added while preserving planarity. It may be embedded in the plane with different choices of the outer face, but all such embeddings share the same set of faces (including the outer face) which must all be triangles. In other words, every maximal planar graph ${\displaystyle G}$ is the 1-skeleton of a simplicial complex which is homeomorphic to the sphere. The circle packing theorem guarantees the existence of a circle packing with finitely many circles whose intersection graph is isomorphic to ${\displaystyle G}$. As the following theorem states more formally, every maximal planar graph can have at most one packing.

Koebe–Andreev–Thurston circle packing theorem: If ${\displaystyle G}$ is a finite maximal planar graph, then a circle packing in the plane whose tangency graph is isomorphic to ${\displaystyle G}$ exists and is unique, up to Möbius transformations and reflections in lines.

Existence

Paul Koebe's original proof of the existence of a circle packing for any planar graph is based on his conformal uniformization theorem saying that a finitely connected planar domain is conformally equivalent to a circle domain.^[4] A circle domain is a domain, a connected open subset of the plane, whose boundary components are circles or points. It is finitely connected when it has finitely many boundary components, and hence finite first Betti number (the number of generators of its fundamental group). The circle packing is a limiting case of Koebe's result, for domains complementary to the union of disks of the desired circle packing. Koebe conjectured that the finite connectivity assumption was unnecessary in his theorem but was unable to prove it. He & Schramm (1993) extend Koebe's theorem to countably connected domains, and to certain packings of countably many circles.^[5]

Thurston's proof is based on the Brouwer fixed point theorem. A former student of Thurston, Oded Schramm, wrote of Thurston's proof that "One can just see the terrible monster swinging its arms in sheer rage, the tentacles causing a frightful hiss, as they rub against each other."^[6] Another proof uses a discrete variant of Perron's method of constructing solutions to the Dirichlet problem.^[7] Yves Colin de Verdière proved the existence of the circle packing as a minimizer of a convex function on a certain configuration space.^[8]

Uniqueness

Thurston observes that this uniqueness is a consequence of the Mostow rigidity theorem. To see this, let ${\displaystyle G}$ be represented by a circle packing. Then the plane in which the circles are packed may be viewed as the boundary of a halfspace model for three-dimensional hyperbolic space; with this view, each circle is the boundary of a plane within the hyperbolic space. One can define a set of disjoint planes in this way from the circles of the packing, and a second set of disjoint planes defined by the circles that circumscribe each triangular gap between three of the circles in the packing. These two sets of planes meet at right angles, and form the generators of a reflection group whose fundamental domain can be viewed as a hyperbolic manifold. By Mostow rigidity, the hyperbolic structure of this domain is uniquely determined, up to isometry of the hyperbolic space; these isometries, when viewed in terms of their actions on the Euclidean plane on the boundary of the half-plane model, translate to Möbius transformations.^[9]

There is also a more elementary proof of the same uniqueness property, based on existence of a maximum value in any finite set and on the observation that, in the triangle connecting the centers of three mutually tangent circles, the angle formed at the center of one of the circles is monotone decreasing in its radius and monotone increasing in the two other radii.^[10] Given two packings for the same graph ${\displaystyle G}$, one may apply reflections and Möbius transformations to make the outer circles in these two packings correspond to each other and have the same radii. Then, let ${\displaystyle v}$ be an interior vertex of ${\displaystyle G}$ for which the circles in the two packings have sizes that are as far apart as possible: that is, choose ${\displaystyle v}$ to maximize the ratio ${\displaystyle r_{1}/r_{2}}$ of the radii of its circles in the two packings. For each triangular face of ${\displaystyle G}$ containing ${\displaystyle v}$, it follows that the angle at the center of the circle for ${\displaystyle v}$ in the first packing is less than or equal to the angle in the second packing, with equality possible only when the other two circles forming the triangle have the same ratio ${\displaystyle r_{1}/r_{2}}$ of radii in the two packings. But the sum of the angles of all of these triangles surrounding the center of the triangle must be ${\displaystyle 2\pi }$ in both packings, so all neighboring vertices to ${\displaystyle v}$ must have the same ratio as ${\displaystyle v}$ itself. By applying the same argument to these other circles in turn, it follows that all circles in both packings have the same ratio. But the outer circles have been transformed to have ratio one, so ${\displaystyle r_{1}/r_{2}=1}$ and the two packings have identical radii for all circles.

Applications

The circle packing theorem is a useful tool to study various problems in planar geometry, conformal mappings and planar graphs.

Conformal mapping

Circle packings can be used to approximate conformal mappings between specified domains. Each circle on the left corresponds to a circle on the right.

A conformal map between two open sets in the plane or in a higher-dimensional space is a continuous function from one set to the other that preserves the angles between any two curves. The Riemann mapping theorem, formulated by Bernhard Riemann in 1851, states that, for any two open topological disks in the plane, there is a conformal map from one disk to the other. Conformal mappings have applications in mesh generation, map projection, and other areas. However, it is not always easy to construct a conformal mapping between two given domains in an explicit way.^[11]

At the Bieberbach conference in 1985, William Thurston conjectured that circle packings could be used to approximate conformal mappings. More precisely, Thurston used circle packings to find a conformal mapping from an arbitrary open disk ${\displaystyle A}$ to the interior of a circle; the mapping from one topological disk ${\displaystyle A}$ to another disk ${\displaystyle B}$ could then be found by composing the map from ${\displaystyle A}$ to a circle with the inverse of the map from ${\displaystyle B}$ to a circle.^[11] Thurston's idea was to pack circles of some small radius ${\displaystyle r}$ in a hexagonal tessellation of the plane, within region ${\displaystyle A}$, leaving a narrow region near the boundary of ${\displaystyle A}$, of width ${\displaystyle r}$, where no more circles of this radius can fit. He then constructs a maximal planar graph ${\displaystyle G}$ from the intersection graph of the circles, together with one additional vertex adjacent to all the circles on the boundary of the packing. By the circle packing theorem, this planar graph can be represented by a circle packing ${\displaystyle C}$ in which all the edges (including the ones incident to the boundary vertex) are represented by tangencies of circles. The circles from the packing of ${\displaystyle A}$ correspond one-for-one with the circles from ${\displaystyle C}$, except for the boundary circle of ${\displaystyle C}$ which corresponds to the boundary of ${\displaystyle A}$. This correspondence of circles can be used to construct a continuous function from ${\displaystyle A}$ to ${\displaystyle C}$ in which each circle and each gap between three circles is mapped from one packing to the other by a Möbius transformation. Thurston conjectured that, in the limit as the radius ${\displaystyle r}$ approaches zero, the functions from ${\displaystyle A}$ to ${\displaystyle C}$ constructed in this way would approach the conformal function given by the Riemann mapping theorem.^[11]

Thurston's conjecture was proven by Rodin & Sullivan (1987). More precisely, they showed that, as ${\displaystyle n}$ goes to infinity, the function ${\displaystyle f_{n}}$ determined using Thurston's method from hexagonal packings of radius-${\displaystyle 1/n}$ circles converges uniformly on compact subsets of ${\displaystyle A}$ to a conformal map from ${\displaystyle A}$ to ${\displaystyle C}$.^[11] Despite the success of Thurston's conjecture, practical applications of this method have been hindered by the difficulty of computing circle packings and by its relatively slow convergence rate.^[12] However, it has some advantages when applied to non-simply-connected domains and in selecting initial approximations for numerical techniques that compute Schwarz–Christoffel mappings, a different technique for conformal mapping of polygonal domains.^[11]

Primal–dual packing, polyhedral realization, and planar separators

A polyhedron and its midsphere. The red circles on the sphere (its horizons as viewed from the polyhedron vertices), and the blue domes where the midsphere protrudes through the polyhedron, form orthogonal spherical circle packings of the graph of the polyhedron and its dual graph.

A stronger form of the circle packing theorem asserts that any polyhedral graph and its dual graph can be represented by two circle packings, such that the two tangent circles representing a primal graph edge and the two tangent circles representing the dual of the same edge always have their tangencies at right angles to each other at the same point of the plane. A primal–dual packing of this type can be lifted from the plane to a sphere by stereographic projection.^[13]

A primal–dual packing on a sphere can be used to construct a convex polyhedron that represents the given graph and that has a midsphere, a sphere tangent to all of the edges of the polyhedron. Each vertex of the polyhedron lies outside the sphere, at the apex of a cone tangent to the sphere on the circle corresponding to that vertex. Each face of the polyhedron lies on the plane through its corresponding circle. Each edge of the polyhedron passes from one cone apex to another through the point of tangency of two circles, where both cones are tangent to the sphere. Conversely, if a polyhedron has a midsphere, then the circles formed by the intersections of the sphere with the polyhedron faces and the circles formed by the horizons on the sphere as viewed from each polyhedron vertex form a dual packing of this type.^[13]

Möbius transformations of the sphere lead to geometrically different polyhedron realizations. One particular realization, called the canonical polyhedron, is obtained by using a unit sphere and choosing a Möbius transformation that causes the center of the sphere to coincide with the centroid of the points of tangency of the polyhedron edges. All combinatorially equivalent polyhedra will produce in this way the same canonical polyhedron, up to rotations of the sphere.^[14] An alternative proof of the planar separator theorem, originally proved in a different way by Lipton and Tarjan,^[15] uses a similar idea of applying a Möbius transformation to a spherical circle packing. The construction obtains a circle packing on a sphere, representing the given graph. Next, a Möbius transformation causes the center of the sphere to coincide with a centerpoint of the circle centers, a point in space such that every plane through the centerpoint divides the ${\displaystyle n}$ centers into two subsets that each have size ${\displaystyle 3n/4}$. After this step, a random plane through the center of the sphere intersects ${\displaystyle O({\sqrt {n}})}$ of the circles in expectation. The vertices corresponding to these intersected circles form a separator that, when removed from the graph, leaves connected subgraphs of at most ${\displaystyle 3n/4}$ vertices.^[16]

Graph drawing

Circle packing has been used as a frequent tool in graph drawing, the study of methods for visualizing graphs. Fáry's theorem, that every graph that can be drawn without crossings in the plane using curved edges can also be drawn without crossings using straight line segment edges, follows as a simple corollary of the circle packing theorem: by placing vertices at the centers of the circles and drawing straight edges between them, a straight-line planar embedding is obtained.^[17]

Construction for rings of circles achieving the most extreme ratio of inner radius to surrounding radius in the ring lemma

An important tool in this application is the ring lemma, which controls the sizes of adjacent circles in a Euclidean circle packing. According to this result, if one circle in a packing is surrounded by a ring of ${\displaystyle d}$ others, then the ratios of the inner circle's radius to the radius of each surrounding circle is at most exponential in ${\displaystyle d}$. More precisely, this ratio is at most ${\displaystyle F_{2d-1}-1}$, where ${\displaystyle F_{i}}$ denotes the ${\displaystyle i}$th Fibonacci number.^[18]

Malitz & Papakostas (1994) derive a version of the ring lemma and use it to prove that the angular resolution of the drawing in which each vertex is placed at the center of its circle is bounded from below as a function of the maximum degree of the graph: it is at least an inverse-exponential function of this degree.^[17] Keszegh, Pach & Pálvölgyi (2013) use a more complicated strategy for placing vertices, near the circle centers on subdivisions of integer lattices, in order to find drawings where the number of distinct edge slopes (the slope number) is at most an exponential function of the degree.^[19]

Other

Another application of the circle packing theorem is that unbiased limit graphs of bounded-degree planar rooted graphs are almost surely recurrent, meaning that random walks on these graph limits almost surely return to the root infinitely often.^[20] Jonnason & Schramm (2000) used circle packings to prove that the expected cover time of a random walk on every ${\displaystyle n}$-vertex planar graph (the average number of steps of the walk before all vertices are visited) is ${\displaystyle \Omega (n\log ^{2}n)}$.^[21] Other applications include approximating Brownian motion,^[2] implications for the cover time of random walks on graphs,^[21] and estimates for the largest eigenvalue of bounded-genus graphs.^[22]

Algorithmic aspects

Collins & Stephenson (2003) describe a numerical relaxation algorithm for finding circle packings, based on ideas of William Thurston. The version of the circle packing problem that they solve takes as input a planar graph, in which all the internal faces are triangles and for which the external vertices have been labeled by positive numbers. It produces as output a circle packing whose tangencies represent the given graph, and for which the circles representing the external vertices have the radii specified in the input. As they suggest, the key to the problem is to first calculate the radii of the circles in the packing; once the radii are known, the geometric positions of the circles are not difficult to calculate. They begin with a set of tentative radii that do not correspond to a valid packing, and then repeatedly perform the following steps:

1. Choose an internal vertex ${\displaystyle v}$ of the input graph.
2. Calculate the total angle ${\displaystyle \theta }$ that its ${\displaystyle k}$ neighboring circles would cover around the circle for ${\displaystyle v}$, if the neighbors were placed tangent to each other and to the central circle using their tentative radii.
3. Determine a representative radius ${\displaystyle r}$ for the neighboring circles, such that ${\displaystyle k}$ circles of radius ${\displaystyle r}$ would give the same covering angle ${\displaystyle \theta }$ as the neighbors of ${\displaystyle v}$ give.
4. Set the new radius for ${\displaystyle v}$ to be the value for which ${\displaystyle k}$ circles of radius ${\displaystyle r}$ would give a covering angle of exactly ${\displaystyle 2\pi }$.

Each of these steps may be performed with simple trigonometric calculations, and as Collins and Stephenson argue, the system of radii converges rapidly to a unique fixed point for which all covering angles are exactly ${\displaystyle 2\pi }$. Once the system has converged, the circles may be placed one at a time, at each step using the positions and radii of two neighboring circles to determine the center of each successive circle.^[23]

Mohar (1993) describes a similar iterative technique for finding simultaneous packings of a polyhedral graph and its dual, in which the dual circles are at right angles to the primal circles. He proves that the method takes time polynomial in the number of circles and in ${\displaystyle \log 1/\varepsilon }$, where ${\displaystyle \varepsilon }$ is a bound on the distance of the centers and radii of the computed packing from those in an optimal packing.^[24]

Generalizations

Infinite periodic locally finite circle packing

A Doyle spiral, not locally finite at the spiral center

A version of the circle packing applies to some infinite graphs. In particular, an infinite planar triangulation of an open disk has a locally finite packing in either the Euclidean plane or the hyperbolic plane (but not both). Here, locally finite means that every point of the plane has a neighborhood that intersects only finitely many circles. In the Euclidean case, the packing is unique up to similarity; in the hyperbolic case, it is unique up to isometry.^[25]

The circle packing theorem generalizes to graphs that are not planar. If ${\displaystyle G}$ is a graph that can be embedded on a surface ${\displaystyle S}$ (more precisely, an orientable 2-manifold), then there is a Riemannian metric of constant curvature ${\displaystyle d}$ on ${\displaystyle S}$ and a circle packing on ${\displaystyle (S,d)}$ whose contact graph is isomorphic to ${\displaystyle G}$. If ${\displaystyle S}$ is closed (compact and without boundary) and ${\displaystyle G}$ is a triangulation of ${\displaystyle S}$, then ${\displaystyle (S,d)}$ and the packing are unique up to conformal equivalence. If ${\displaystyle S}$ is the sphere, then its geometry is spherical geometry and the equivalence is up to Möbius transformations. If it is a torus, the geometry is locally Euclidean (as a flat torus), and the equivalence is up to scaling by a constant and isometries. If ${\displaystyle S}$ has genus at least 2, then the geometry is locally hyperbolic and the equivalence is up to isometries.

Another generalization of the circle packing theorem involves replacing the condition of tangency with a specified intersection angle between circles corresponding to neighboring vertices. One version is as follows. Suppose that ${\displaystyle G}$ is a finite 3-connected planar graph (that is, a polyhedral graph), then there is a pair of circle packings, one whose intersection graph is isomorphic to ${\displaystyle G}$, another whose intersection graph is isomorphic to the planar dual of ${\displaystyle G}$, and for every vertex in ${\displaystyle G}$ and face adjacent to it, the circle in the first packing corresponding to the vertex intersects orthogonally with the circle in the second packing corresponding to the face.^[1] For instance, applying this result to the graph of the tetrahedron gives, for any four mutually tangent circles, a second set of four mutually tangent circles each of which is orthogonal to three of the first four.^[26] A further generalization, replacing intersection angle with inversive distance, allows the specification of packings in which some circles are required to be disjoint from each other rather than crossing or being tangent.^[27]

Yet another variety of generalizations allow shapes that are not circles.^[3] Suppose that ${\displaystyle G=(V,E)}$ is a finite planar graph, and to each vertex ${\displaystyle v}$ of ${\displaystyle G}$ corresponds a shape ${\displaystyle K_{v}\subset \mathbb {R} ^{2}}$, which is homeomorphic to the closed unit disk and whose boundary is smooth. Then there is a packing ${\displaystyle P=(K'_{v}:v\in V)}$ in the plane such that ${\displaystyle K'_{v}\cap K'_{u}\neq \varnothing }$ if and only if ${\displaystyle [v,u]\in E}$ and for each ${\displaystyle v\in V}$ the set ${\displaystyle K'_{v}}$ is obtained from ${\displaystyle K_{v}}$ by translating and scaling. (In the original circle packing theorem, there are three real parameters per vertex, two of which describe the center of the corresponding circle and one of which describe the radius, and there is one equation per edge. This also holds in this generalization.) This result, from Oded Schramm's 1990 Ph.D. thesis,^[28] has become known as Schramm's "monster packing theorem".^[29] One proof of this generalization can be obtained by applying Koebe's original proof^[30] and the theorem of Brandt^[31] and Harrington^[32] stating that any finitely connected domain is conformally equivalent to a planar domain whose boundary components have specified shapes, up to translations and scaling.^[28]

History

Circle packings were studied as early as 1910, in the work of Arnold Emch on Doyle spirals in phyllotaxis (the mathematics of plant growth).^[33] The circle packing theorem was first proved by Paul Koebe.^[30] William Thurston^[9] rediscovered the circle packing theorem, and noted that it followed from the work of E. M. Andreev.^[34] Thurston also proposed a scheme for using the circle packing theorem to obtain a homeomorphism of a simply connected proper subset of the plane onto the interior of the unit disk. The Thurston Conjecture for Circle Packings is his conjecture that the homeomorphism will converge to the Riemann mapping as the radii of the circles tend to zero. The Thurston Conjecture was later proved by Burton Rodin and Dennis Sullivan.^[35]

See also

• Apollonian gasket, an infinite packing formed by repeatedly filling triangular gaps
• Circle packing, dense arrangements of circles without specified tangencies
• Ford circles, a packing of circles along the rational number line
• Penny graph, the coin graphs whose circles all have equal radii