The coincidence problem for shifted lattices and crystallographic point packings

Abstract

A coincidence site lattice is a sublattice formed by the intersection of a lattice $\Gamma$ in ${\mathbb{R}}^d$ with the image of $\Gamma$ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of $\Gamma$. Here, we consider the more general case allowing any affine isometry. Consequently, general results on coincidence isometries of shifted copies of lattices, and of crystallographic point packings are obtained. In particular, we discuss the shifted square lattice and the diamond packing in detail.

arXiv comment22 pages, 2 figures

References in corpus (6)

• Planar coincidences for N-fold symmetry
• Coincidence rotations of the root lattice $A_4$
• Coincidence site modules in 3-space
• Multiplicativity in the theory of coincidence site lattices
• Coincidence isometries of a shifted square lattice
• Similar Sublattices and Coincidence Rotations of the Root Lattice A4 and its Dual

Cited by in corpus (3)

• Coincidences of a shifted hexagonal lattice and the hexagonal packing
• Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules
• Coincidence indices of sublattices and coincidences of colorings