Similarity and Coincidence Isometries for Modules

Abstract

The groups of (linear) similarity and coincidence isometries of certain modules in d-dimensional Euclidean space, which naturally occur in quasicrystallography, are considered. It is shown that the structure of the factor group of similarity modulo coincidence isometries is the direct sum of cyclic groups of prime power orders that divide d. In particular, if the dimension d is a prime number p, the factor group is an elementary Abelian p-group. This generalizes previous results obtained for lattices to situations relevant in quasicrystallography.

14 pages

References in corpus (10)

• Planar coincidences for N-fold symmetry
• 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 $A_4$
• A Note on Shelling
• Combinatorial problems of (quasi-)crystallography
• Multi-Component Model Sets and Invariant Densities
• A Note on Coincidence Isometries of Modules in Euclidean Space
• Coincidence site modules in 3-space
• Similar sublattices of planar lattices

Cited by in corpus (4)

• Similar sublattices of planar lattices
• Similar submodules and coincidence site modules
• Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules
• Similarity Isometries of Point Packings