On Bravais Colorings Associated with Periodic and Non-Periodic Crystals
Ma. Louise Antonette N. De Las Peñas* and Enrico Paolo C. Bugarin
Department of Mathematics, Ateneo de Manila University,
Loyola Heights, Quezon City 1108, Philippines
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.
The discovery of alloys with long-range orientational order and sharp diffraction images of non-crystallographic symmetries has initiated an intensive investigation of possible structures and physical properties of such systems. It was this amplified interest that established a new branch of solid state physics and also of discrete geometry called the theory of quasicrystals. To this day, since its discovery in 1984, quasicrystals have been studied intensively by metallurgists, physicists, and mathematicians. Mathematicians in particular are increasingly intrigued by the underlying mathematical principles that govern quasicrystals, the geometry of diffraction patterns, and their link to the study of the fascinating properties of non-periodic tilings. The most important manifestation of quasicrystals is their implicit long-range internal order that makes itself apparent in the diffraction patterns associated with them. The mathematics used to model such objects and to study their diffractive and their self-similar internal structures turns out to be highly interdisciplinary and includes algebraic number . . . . . . . . .
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