We just posted a new paper to the ArXiV, “Rigidity of Circle Polyhedra in the 2-Sphere and of Hyperideal Polyhedra in Hyperbolic 3-Space“. This is joint work with Philip L. Bowers at FSU and Kevin Pratt, who worked on the project over the summer as a visiting undergraduate research assistant here at JMU.
This paper grew out of the curious and surprising result of Jiming Ma and Jean-Marc Schlenker, in which they construct an inversive distance circle packing on the sphere which is not globally rigid (meaning that there exist more than one realization of the same inversive distance data as a pattern of circles that are not Möbius equivalent).
Suppose you are given a triangulation of a topological sphere and a real number weight on each edge of . The inversive distance circle packing problem asks, is there a set of circles on the sphere and a bijection such that for every edge of , the inversive distance between circles and is equal to the weight ? We call an inversive distance circle packing realizing . There are really two important questions: (1) given such a and edge labeling , does there exist a packing? and (2) if one does exist, is it unique?
The Ma-Schlenker result was especially surprising because the answer to question (2) has been yes, it’s unique, in virtually every setting circle packings have been studied. (As of this post’s writing, question (1) remains very much open for general inversive distance circle packings.) Ma and Schlenker start with a Euclidean twisted octahedron and use some powerful mathematical tools (Pogorolov maps and infinite de Sitter space) to obtain their result. Recently, we provided some constructions of Ma-Schlenker style octahedra in the intrinsic inversive geometry of the sphere in another paper that may interest the reader. We can now construct lots of examples of families of circle-polyhedra where there is not uniqueness.
The Ma-Schlenker construction raises the question, “When is global rigidity of an inversive distance circle pattern on the sphere guaranteed?” This is the subject of our paper.
The questions being asked of circle patterns have an analog in the study of Euclidean polyhedra dating back to the ancient Greeks. The question might be asked, if we know the shapes of all the faces of a polyhedron and which faces are attached together along which edges (though not at what dihedral angle), does that data determine the polyhedron uniquely? In general, the answer is no–take a cube and replace its top face with a pyramid made of four equilateral triangles to obtain a house-like structure; now, invert the pyramid. However, in 1813, Cauchy proved his celebrated Rigidity Theorem, which states that if we further require that the polyhedron be convex, along with the specified faces and their combinatorics, then there is only one construction possible. This theorem (and some of its later proofs–Cauchy’s original argument had several serious bugs) is certainly one of Erdös’s proofs from the book (and in fact one proof of the theorem is in the Aigner and Ziegler book Proofs from THE BOOK).
To Our Paper
Inspired by Ma and Schlenker’s use of Euclidean polyhedra to prove their result, we set out to recreate Cauchy’s proof, except in the case of inversive distance circle packings. To do this, we first generalized packings to circle-polyhedra, which are really the natural way of talking about gluing up circle-polygons along “edges” to form patterns of circles on the sphere. In order to do this, we began to work with circle space, a space that is a partial dual to the real-projective 3-space in which circles are points, coaxial families of circles are lines, and bundles of circles are planes. Along with this space comes a notion of convexity for circle-polyhedra, and our main result is an analog to Cauchy’s: if you specify (up to Möbius transformations) a bunch of circle-polygons (to serve as the faces of a polyhedron), and how the polygons should be combined (by identifying edges), then if there exists a convex circle-polyhedron satisfying your specifications, it is the unique convex circle-polyhedron satisfying your specifications.
Our proof of this theorem follows the same general outline as Cauchy’s original proof, though with some really lovely forays into hyperbolic geometry. (A quick preview: if a bunch of circles intersect some other circle orthogonally, we can take the parts of each of the intersecting circles lying on the interior of the one circle to be hyperbolic lines in the Poincare disk model of hyperbolic 2-space. From there we build some nice hyperbolic polygons with some interesting properties, and derive a lemma about certain hyperbolic robot arms constructed out of revolute joints with the occasional piston thrown in. It’s all very fun.)