Almost all circle polyhedra are rigid


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.

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  1. 1.

    For example, non-univalent tangency circle packings are circle frameworks, and their corresponding disks can cover the 2-sphere multiple times.

  2. 2.

    This 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.

  3. 3.

    This holds more generally, for planar graphs with no loops and no multiple edges.

  4. 4.

    This was proved first in Koebe [8], rediscovered by Thurston [10], and now is a part of what is known as the Koebe-Andre’ev-Thurston Theorem. There are many proofs in the literature. See Bowers [3] for a fairly quick proof and a survey of results relating to the theorem, as well as a bibliography of literature surrounding the theorem.


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Correspondence to Philip L. Bowers.

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Bowers, J.C., Bowers, P.L. & Pratt, K. Almost all circle polyhedra are rigid. Geom Dedicata 203, 337–346 (2019).

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  • Circle packing
  • Circle polyhedron
  • Inversive rigidity