Circle Packings and Polyhedra

Figure 1. A Ma-Schlenker Octahedron.

The goal of this project is to study questions related to circle packing and circle patterns. Our first results are related to c-Polyhedra, which are certain circle patterns drawn on the surface of the 2-sphere. This purpose of this page is to give an overview of some of our interests and results and point to software, papers, and talks we have given where the reader can obtain more information on what we’re up to.


Start with a triangulation T of a topological sphere. Label each edge uv of T  with a positive real weight w(uv). Now, suppose for each vertex v of T we draw a circle c_v on the unit sphere such that for each edge uv of T the inversive distance between c_u and c_v is w(uv). The resulting pattern of circles is the c-polyhedron C_T. We say that C_T realizes (T,w).

One interesting question is on the uniqueness of c-polyhedra. If we fix T and the weight function w, are all of the c-polyhedra realizing T equivalent up to Möbius transformations on the sphere? Somewhat surprisingly, this turns out to be false, even though similar uniqueness results are true if we switch our domain from the sphere to either the plane or hyperbolic space! Ma and Schlenker showed this originally using some sophisticated and beautiful geometric tools (Pogorolov maps and de Sitter space) [7]. They gave a construction for generating two different c-polyhedra with the same underlying weighted triangulation  (T,w) that were not Möbius equivalent. Subsequently, we found many examples using only elementary spherical geometry. One such example is based on insights obtained from reading [7], which we call Ma-Schlenker octahedra. Figure 1 shows an example. The circles are drawn in blue and the triangulation is drawn in red. See the software section below for Mathematica notebooks we created to generate examples for [2].


  • Elliptical Ma-Schlenker c-Octahedra. This visualization is a companion to [2]. It allows for various Ma-Schlenker octahedra with elliptical flow to be generated and analyzed.
  • Hyperbolic Ma-Schlenker c-Octahedra. Coming soon.
  • Parabolic Ma-Schlenker c-Octahedra. Coming soon.


  • [1] Seokpum Kim, Xiang Chen, Gregory Dreifus, John Lindahl, Inseung Kang, Andy Kim, Mohamed Selim, David Nuttal, Andrew Messing, Andrzej Nycz, Robert Minneci, John C. Bowers, Brittany Braswell, Ahmed Arabi Hassan, Byron Pipes, Vlastimil Kunc. An integrated design approach for infill patterning of fused deposition modeling and its application to an airfoil.Preprint. February, 2017.
  • [2] J. C. Bowers and P. L. Bowers. Ma-Schlenker c-Octahedra in the 2-sphere. Preprint. June, 2016.


Other references

  • [7] J. Ma and J.-M. Schlenker. Non-rigidity of spherical inversive distance circle packings. Discrete & Computational Geometry, 47(3):610–617, Feb. 2012.

Colleagues and collaborators

  • Philip L. Bowers (The Florida State University)
  • Brittany Braswell (James Madison University)
  • Xiang Chen (James Madison University)
  • Gregory Dreifus (Oak Ridge National Labs)
  • Kevin Pratt (University of Connecticut)
  • Don Sheehy (University of Connecticut)
  • Ken Stephenson (The University of Tennessee Knoxville)