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Rigidity of spherical codes

Henry Cohn, Yang Jiao, Abhinav Kumar and Salvatore Torquato

Geometry & Topology 15 (2011) 2235–2273

A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter–Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter–Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes–Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices.

Additional material
rigidity, jamming, packing, spherical codes, kissing problem
Mathematical Subject Classification 2000
Primary: 52C25
Secondary: 52C17
Received: 24 February 2011
Revised: 23 May 2011
Accepted: 3 June 2011
Published: 23 November 2011
Proposed: Rob Kirby
Seconded: Dmitri Burago, Joan Birman
Henry Cohn
Microsoft Research New England
One Memorial Drive
Cambridge, MA 02142
Yang Jiao
Physical Sciences–Oncology Center
Princeton University
Princeton, New Jersey 08544
Abhinav Kumar
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
Salvatore Torquato
Department of Chemistry
Princeton University
Princeton, New Jersey 08544