TY - JOUR
T1 - Coincidence lattices in the hyperbolic plane
AU - Rodríguez-Andrade, M. A.
AU - Aragón-González, G.
AU - Aragón, J. L.
AU - Gómez-Rodríguez, A.
PY - 2011/1
Y1 - 2011/1
N2 - The problem of coincidences of lattices in the space ℝp,q, with p + q = 2, is analyzed using Clifford algebra. We show that, as in ℝn, 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 ℝp,q, with p + q = 2, that includes an algorithm to decompose an orthogonal transformation into a product of reflections.
AB - The problem of coincidences of lattices in the space ℝp,q, with p + q = 2, is analyzed using Clifford algebra. We show that, as in ℝn, 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 ℝp,q, with p + q = 2, that includes an algorithm to decompose an orthogonal transformation into a product of reflections.
UR - http://www.scopus.com/inward/record.url?scp=78650651887&partnerID=8YFLogxK
U2 - 10.1107/S0108767310042431
DO - 10.1107/S0108767310042431
M3 - Artículo
SN - 0108-7673
VL - 67
SP - 35
EP - 44
JO - Acta Crystallographica Section A: Foundations of Crystallography
JF - Acta Crystallographica Section A: Foundations of Crystallography
IS - 1
ER -