Geometriae Dedicata

, Volume 203, Issue 1, pp 337–346 | Cite as

Almost all circle polyhedra are rigid

  • John C. Bowers
  • Philip L. BowersEmail author
  • Kevin Pratt
Original Paper


We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane \({\mathbb {E}}^{2}\), as well as the infinitesimal inversive rigidity of tangency circle packings on the 2-sphere \({\mathbb {S}}^{2}\). From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck’s proof (Geometric Topology, volume 238 of Lecture Notes in Mathematics, pp 225–239, 1975) of the rigidity of almost all Euclidean polyhedra to the setting of circle polyhedra, where inversive distances replace Euclidean distances and Möbius transformations replace rigid Euclidean motions.


Circle packing Circle polyhedron Inversive rigidity 



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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • John C. Bowers
    • 1
  • Philip L. Bowers
    • 2
    Email author
  • Kevin Pratt
    • 3
  1. 1.Department of Computer ScienceJames Madison UniversityHarrisonburgUSA
  2. 2.Department of MathematicsThe Florida State UniversityTallahasseeUSA
  3. 3.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

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