Planar coincidences for N-fold symmetry

Abstract

The coincidence problem for planar patterns with $N$-fold symmetry is considered. For the N-fold symmetric module with $N<46$, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case $N\ge 46$ is briefly discussed and N=46 is described explicitly. The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.

38 pages, 4 figures; updated and slightly corrected version of a paper published in J. Math. Phys. (see below)

References in corpus (1)

• Solution of the coincidence problem in dimensions $d\le 4$

Cited by in corpus (19)

• Solution of the coincidence problem in dimensions $d\le 4$
• Bravais colourings of planar modules with N-fold symmetry
• Coincidence rotations of the root lattice $A_4$
• Combinatorial problems of (quasi-)crystallography
• A Note on Coincidence Isometries of Modules in Euclidean Space
• Coincidence site modules in 3-space
• Multiple planar coincidences with N-fold symmetry
• Coincidences in 4 dimensions
• Similarity and Coincidence Isometries for Modules
• Multiplicativity in the theory of coincidence site lattices
• 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
• Multiple CSLs for the body centered cubic lattice
• On the Existence of Similar Sublattices
• Coincidences of a shifted hexagonal lattice and the hexagonal packing
• Coincidence indices of sublattices and coincidences of colorings
• Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules