A pedestrian's introduction to spacetime crystallography

  • Authors:

  • T. Toffoli

  • Affiliations:

  • Boston University, Electrical and Computer Engineering Department, 8 Saint Mary's Street, Boston, Massachusetts 02215

  • Venue:
  • IBM Journal of Research and Development
  • Year:
  • 2004

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Abstract

Ordinary crystallography deals with regular, discrete, static arrangements in space. Of course, dynamic considerations--and thus the additional dimension of time--must be introduced when one studies the origin of crystals (since they are emergent structures) and their physical properties such as conductivity and compressibility. The space and time of the dynamics in which the crystal is embedded are assumed to be those of ordinary continuous mechanics. In this paper, we take as the starting point a spacetime crystal, that is, the spacetime structure underlying a discrete and regular dynamics. A dynamics of this kind can be viewed as a "crystalline computer." After considering transformations that leave this structure invariant, we turn to the possible states of this crystal, that is, the discrete spacetime histories that can take place in it and how they transform under different crystal transformations. This introduction to spacetime crystallography provides the rationale for making certain definitions and addressing specific issues; presents the novel features of this approach to crystallography by analogy and by contrast with conventional crystallography; and raises issues that have no counterpart there.