Colourings of cyclotomic integers with class number one

Abstract

This paper continues the study of colourings of the sets of cyclotomic integers M n ¼ Z[ n ] ( n ¼ e 2i/n , a primitive nth root of unity) with class number one. We present results for the colour symmetry group and colour preserving group for a given ideal colouring of M n , with (n) ¼ 8 and 10, thus completing the characterisation of the colour preserving group for the cases (n) 10, where is Euler's totient function.

Quasicrystals: diversity and complexity

“…Aperiodic tilings are now studied and classified using advanced topological tools, as explained by Franz Ga¨hler [24]; and new methods are being devised for the generation of icosahedral tilings, as described by Nobuhisa Fujita [25]. Traditional theories of color symmetry [26] and magnetic order [27,28] have been generalized to deal with imperfect and partial coloring, as we saw in the presentations of Peter Zeiner [29] and Louise de Las Pen˜as [30], and in a new model of frustrated antiferromagnetic order on the Penrose tiling that was presented by Anu Jagannathan [31]. But probably most exciting is the fact that diffraction theory has gained much progress, bringing us closer than ever to the point where we can understand what constitutes a crystal [32,33], namely, what sets of points in space give rise to Bragg peaks in their diffraction [34].…”

Symmetry groups of two-way twofold and three-way threefold fabrics

“…Color symmetry has been studied in previous works in the context of periodic structures (Senechal, 1979;Schwarzenberger, 1984;Lifshitz, 1997;De Las Pen ˜as & Basilio, 2004;De Las Pen ˜as & Felix, 2007), quasicrystal and non-periodic structures (Lifshitz, 1997;Baake et al, 2002;Baake & Grimm, 2004;De Las Pen ˜as et al, 2011;Bugarin et al, 2008) and monoperiodic/rod structures (Loyola et al, 2012;De Las Pen ˜as et al, 2014). In this article, color symmetry is employed to study diperiodic or layer group structures.…”