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Collisions and their Catenations: Ultimately Periodic Tilings of the Plane

Abstract

Motivated by the study of cellular automata algorithmics and dynamics, we investigate an extension of ultimately periodic words to two-dimensional infinite words: collisions. A natural composition operation on tilings leads to a catenation operation on collisions. By existence of aperiodic tile sets, ultimately periodic tilings of the plane cannot generate all possible tilings but exhibit some useful properties of their one-dimensional counterparts: ultimately periodic tilings are recursive, very regular, and tiling constraints are easy to preserve by catenation. We show that, for a given catenation scheme of finitely many collisions, the generated set of collisions is semi-linear.

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Other [cs.OH]
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Dates and versions

hal-00175397 , version 1 (27-09-2007)
hal-00175397 , version 2 (17-12-2007)
hal-00175397 , version 3 (07-08-2008)

Identifiers

  • HAL Id : hal-00175397 , version 3

Cite

Nicolas Ollinger, Gaétan Richard. Collisions and their Catenations: Ultimately Periodic Tilings of the Plane. 2008. ⟨hal-00175397v3⟩
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