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Effect of rotational symmetry on colored lattices

Effect of rotational symmetry on colored lattices A colored lattice Lc has a geometrical lattice L. A subgroup lattice L′ of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L′ in L is given by Δ, the determinant of the matrix (tjk) that converts L into L′. This is the order of the factor group {L/L′}, and is also the number n of colors present. The crystal systems—i.e., the combinations of rotational symmetry axes—of Lc, L, and L′ are all the same, but L and L′ may have different centerings; this results in 39 combinations of centerings, called here the 39 symmetry types of colored lattices. These types are tabulated here, together with the special forms taken by (tjk) and the formulas for Δ. In only 7 of the 39 types can the number of colors be arbitrary; in most types certain numbers of colors are impossible. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Proceedings of the National Academy of Sciences PNAS

Effect of rotational symmetry on colored lattices

Proceedings of the National Academy of Sciences , Volume 75 (12): 5751 – Dec 1, 1978

Effect of rotational symmetry on colored lattices

Proceedings of the National Academy of Sciences , Volume 75 (12): 5751 – Dec 1, 1978

Abstract

A colored lattice Lc has a geometrical lattice L. A subgroup lattice L′ of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L′ in L is given by Δ, the determinant of the matrix (tjk) that converts L into L′. This is the order of the factor group {L/L′}, and is also the number n of colors present. The crystal systems—i.e., the combinations of rotational symmetry axes—of Lc, L, and L′ are all the same, but L and L′ may have different centerings; this results in 39 combinations of centerings, called here the 39 symmetry types of colored lattices. These types are tabulated here, together with the special forms taken by (tjk) and the formulas for Δ. In only 7 of the 39 types can the number of colors be arbitrary; in most types certain numbers of colors are impossible.

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Publisher
PNAS
Copyright
Copyright ©2009 by the National Academy of Sciences
ISSN
0027-8424
eISSN
1091-6490
Publisher site
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Abstract

A colored lattice Lc has a geometrical lattice L. A subgroup lattice L′ of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L′ in L is given by Δ, the determinant of the matrix (tjk) that converts L into L′. This is the order of the factor group {L/L′}, and is also the number n of colors present. The crystal systems—i.e., the combinations of rotational symmetry axes—of Lc, L, and L′ are all the same, but L and L′ may have different centerings; this results in 39 combinations of centerings, called here the 39 symmetry types of colored lattices. These types are tabulated here, together with the special forms taken by (tjk) and the formulas for Δ. In only 7 of the 39 types can the number of colors be arbitrary; in most types certain numbers of colors are impossible.

Journal

Proceedings of the National Academy of SciencesPNAS

Published: Dec 1, 1978

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