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Coincidence lattices in the hyperbolic plane.

Authors
  • Rodríguez-Andrade, M A
  • Aragón-González, G
  • Aragón, J L
  • Gómez-Rodríguez, A
Type
Published Article
Journal
Acta Crystallographica Section A Foundations of Crystallography
Publisher
International Union of Crystallography
Publication Date
Jan 01, 2011
Volume
67
Issue
Pt 1
Pages
35–44
Identifiers
DOI: 10.1107/S0108767310042431
PMID: 21173471
Source
Medline
License
Unknown

Abstract

The problem of coincidences of lattices in the space R(p,q), with p + q = 2, is analyzed using Clifford algebra. We show that, as in R(n), any coincidence isometry can be decomposed as a product of at most two reflections by vectors of the lattice. Bases and coincidence indices are constructed explicitly for several interesting lattices. Our procedure is metric-independent and, in particular, the hyperbolic plane is obtained when p = q = 1. Additionally, we provide a proof of the Cartan-Dieudonné theorem for R(p,q), with p + q = 2, that includes an algorithm to decompose an orthogonal transformation into a product of reflections.

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