Discrete & Computational Geometry

, Volume 32, Issue 3, pp 309-315

First online:

Unflippable Tetrahedral Complexes

  • Randall DoughertyAffiliated withSan Diego, CA Email author 
  • , Vance FaberAffiliated with33740 NE 84th Pl., Carnation, WA 98014 Email author 
  • , Michael MurphyAffiliated withFusion Numerics Inc., 1320 Pearl Street, Boulder, CO 80302 Email author 

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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.