A Note on Coincidence Isometries of Modules in Euclidean Space

TL;DR

This paper extends the understanding of coincidence isometries in Euclidean space, showing they can be decomposed into a finite product of reflections, thus generalizing lattice results to modules relevant in quasicrystallography.

Contribution

It introduces a decomposition of coincidence isometries of modules into a product of reflections, broadening previous lattice-focused results to quasicrystallography applications.

Findings

Coincidence isometries decompose into at most n reflections.

Generalization from lattices to modules in Euclidean space.

Relevance to quasicrystallography applications.

Abstract

It is shown that the coincidence isometries of certain modules in Euclidean $n$-space can be decomposed into a product of at most $n$ coincidence reflections defined by their non-zero elements. This generalizes previous results obtained for lattices to situations that are relevant in quasicrystallography.