On Bravais Colorings Associated with Periodic and Non-Periodic Crystals

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, vario...

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Main Authors: De Las Peñas, Ma. Louise Antonette N, Bugarin, Enrico Paolo
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Published: Archīum Ateneo 2011
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Online Access:https://archium.ateneo.edu/mathematics-faculty-pubs/99
http://philjournalsci.dost.gov.ph/33-vol-140-no-1-june-2011/438-on-bravais-colorings-associated-with-periodic-and-non-periodic-crystals
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spelling ph-ateneo-arc.mathematics-faculty-pubs-10992020-06-27T02:22:39Z On Bravais Colorings Associated with Periodic and Non-Periodic Crystals De Las Peñas, Ma. Louise Antonette N Bugarin, Enrico Paolo 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 M_n = Z[exp(2{\pi}i/n)], and list the findings for M_{15} = Z[exp(2{\pi}i/15)] and M_{16} = Z[exp({\pi}i/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. 2011-01-01T08:00:00Z text https://archium.ateneo.edu/mathematics-faculty-pubs/99 http://philjournalsci.dost.gov.ph/33-vol-140-no-1-june-2011/438-on-bravais-colorings-associated-with-periodic-and-non-periodic-crystals Mathematics Faculty Publications Archīum Ateneo Mathematics
institution Ateneo De Manila University
building Ateneo De Manila University Library
country Philippines
collection archium.Ateneo Institutional Repository
topic Mathematics
spellingShingle Mathematics
De Las Peñas, Ma. Louise Antonette N
Bugarin, Enrico Paolo
On Bravais Colorings Associated with Periodic and Non-Periodic Crystals
description 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 M_n = Z[exp(2{\pi}i/n)], and list the findings for M_{15} = Z[exp(2{\pi}i/15)] and M_{16} = Z[exp({\pi}i/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.
format text
author De Las Peñas, Ma. Louise Antonette N
Bugarin, Enrico Paolo
author_facet De Las Peñas, Ma. Louise Antonette N
Bugarin, Enrico Paolo
author_sort De Las Peñas, Ma. Louise Antonette N
title On Bravais Colorings Associated with Periodic and Non-Periodic Crystals
title_short On Bravais Colorings Associated with Periodic and Non-Periodic Crystals
title_full On Bravais Colorings Associated with Periodic and Non-Periodic Crystals
title_fullStr On Bravais Colorings Associated with Periodic and Non-Periodic Crystals
title_full_unstemmed On Bravais Colorings Associated with Periodic and Non-Periodic Crystals
title_sort on bravais colorings associated with periodic and non-periodic crystals
publisher Archīum Ateneo
publishDate 2011
url https://archium.ateneo.edu/mathematics-faculty-pubs/99
http://philjournalsci.dost.gov.ph/33-vol-140-no-1-june-2011/438-on-bravais-colorings-associated-with-periodic-and-non-periodic-crystals
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