Discrete & Computational Geometry

, Volume 32, Issue 3, pp 309–315

Unflippable Tetrahedral Complexes


DOI: 10.1007/s00454-004-1097-3

Cite this article as:
Dougherty, R., Faber, V. & Murphy, M. Discrete Comput Geom (2004) 32: 309. doi:10.1007/s00454-004-1097-3


We present a 16-vertex tetrahedralization of S3 (the 3-sphere) for which no topological bistellar flip other than a 1-to-4 flip (i.e., a vertex insertion) is possible. This answers a question of Altshuler et al. which asked if any two n-vertex tetrahedralizations of S3 are connected by a sequence of 2-to-3 and 3-to-2 flips. The corresponding geometric question is whether two tetrahedralizations of a finite point set S in ℝ3 in “general position” are always related via a sequence of geometric 2-to-3 and 3-to-2 flips. Unfortunately, we show that this topologically unflippable complex and others with its properties cannot be geometrically realized in ℝ3.

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.San Diego, CAUSA
  2. 2.33740 NE 84th Pl., Carnation, WA 98014USA
  3. 3.Fusion Numerics Inc., 1320 Pearl Street, Boulder, CO 80302USA

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