2010/10/19 by Ma. Louise Antonette N. De Las Peñas, Peñas, Ma. Louise Antonette N. De Las, Enrico Paolo Bugarin +2
Materials Science · Mathematics · #FOS: Mathematics #Quasicrystal Structures and Properties #Rings and Algebras (math.RA) #math.RA
paper · pdf · doi:10.48550/arxiv.1010.3908
29 pages, 4 figures, 2 tables
openalex publication_date 2010/10/19 · arxiv created 2011/04/08 · arxiv updated 2015/03/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this work, a theory of color symmetry is presented that extends the ideas of traditional theories of color symmetry for periodic crystals to apply to non-periodic crystals. The color symmetries are associated to each of the crystalline sites and may correspond to different chemical species, various orientations of magnetic moments and colorings of a non-periodic tiling. In particular, we study the color symmetries of periodic and non-periodic structures via Bravais colorings of planar modules that emerge as the ring of integers in cyclotomic fields with class number one. Using an approach involving matrices, we arrive at necessary and sufficient conditions for determining the color symmetry groups and color fixing groups of the Bravais colorings associated with the modules Mn = Z[exp(2πi/n)], and list the findings for M15 = Z[exp(2πi/15)] and M16 = Z[exp(πi/8)]. In the second part of the paper, we discuss magnetic point groups of crystal and quasicrystal structures and give some examples of structures whose magnetic point group symmetries are described by Bravais colorings of planar modules.