Coincidence rotations of the root lattice $A_4$

Abstract

The coincidence site lattices of the root lattice $A_4$ are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of $A_4$ into the icosian ring with its rich arithmetic structure, which recently (arXiv:math.MG/0702448) led to the classification of the similar sublattices of $A_4$.

13 pages, 1 figure

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• Solution of the coincidence problem in dimensions $d\le 4$
• A radial analogue of Poisson's summation formula with applications to powder diffraction and pinwheel patterns
• A Note on Shelling
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• Coincidence site modules in 3-space
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• Coincidences in 4 dimensions
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Cited by in corpus (12)

• A Note on Coincidence Isometries of Modules in Euclidean Space
• Multiplicativity in the theory of coincidence site lattices
• Similarity and Coincidence Isometries for Modules
• The coincidence problem for shifted lattices and crystallographic point packings
• Coincidence isometries of a shifted square lattice
• Similar submodules and coincidence site modules
• Colourings of lattices and coincidence site lattices
• Similar Sublattices and Coincidence Rotations of the Root Lattice A4 and its Dual
• Similarity Isometries of Point Packings
• CSLs of the root lattice ${\mathbf{A}}_4$
• Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules
• Ten colours in quasiperiodic and regular hyperbolic tilings