Semi-perfect colourings of hyperbolic tilings

Peñas, Ma. Louise Antonette N De Las; Felix, Rene P.; Gozo, Beaunonie R.; Laigo, Glenn R.

doi:10.1080/14786435.2010.524901

Cited by 4 publications

(2 citation statements)

References 4 publications

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“…Succeeding papers (De Las Peñ as et al, 2006, 2011Gozo, 2010) reduced the restrictions on the patterns being colored. These papers considered patterns whose tiles did not form a single G-orbit.…”

Section: Introductionmentioning

confidence: 96%

“…Such colorings are called perfect colorings. On the other hand, a method for identifying the color groups arising from index-2 subgroups of symmetry groups was outlined in De Las Peñ as et al (2011). This method was then applied to tilings of the hyperbolic plane to obtain various colorings, some of which are perfect, and others with associated color groups of index 2 in the symmetry group (called semi-perfect colorings; Felix & Loquias, 2008).…”

Section: Introductionmentioning

confidence: 99%

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Color groups arising from index-nsubgroups of symmetry groups

Felix

Junio

2015

Acta Cryst Sect A

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One of the main goals in the study of color symmetry is to classify colorings of symmetrical objects through their color groups. The term color group is taken to mean the subgroup of the symmetry group of the uncolored symmetrical object which induces a permutation of colors in the coloring. This work looks for methods of determining the color group of a colored symmetric object. It begins with an index n subgroup H of the symmetry group G of the uncolored object. It then considers H-invariant colorings of the object, so that the color group H(*) will be a subgroup of G containing H. In other words, H ≤ H(*) ≤ G. It proceeds to give necessary and sufficient conditions for the equality of H(*) and G. If H(*) ≠ G and n is prime, then H(*) = H. On the other hand, if H(*) ≠ G and n is not prime, methods are discussed to determine whether H(*) is G, H or some intermediate subgroup between H and G.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Color groups arising from index-nsubgroups of symmetry groups

Felix

Junio

2015

Acta Cryst Sect A

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Symmetry groups of single-wall nanotubes

et al. 2013

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This work investigates the symmetry properties of single-wall carbon nanotubes and their structural analogs, which are nanotubes consisting of different kinds of atoms. The symmetry group of a nanotube is studied by looking at symmetries and color fixing symmetries associated with a coloring of the tiling by hexagons in the Euclidean plane which, when rolled, gives rise to a geometric model of the nanotube. The approach is also applied to nanotubes with non-hexagonal symmetry arising from other isogonal tilings of the plane.

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Symmetry groups associated with tilings on a flat torus

et al. 2015

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This work investigates symmetry and color symmetry properties of Kepler, Heesch and Laves tilings embedded on a flat torus and their geometric realizations as tilings on a round torus in Euclidean 3-space. The symmetry group of the tiling on the round torus is determined by analyzing relevant symmetries of the planar tiling that are transformed to axial symmetries of the three-dimensional tiling. The focus on studying tilings on a round torus is motivated by applications in the geometric modeling of nanotori and the determination of their symmetry groups.

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Semi-perfect colourings of hyperbolic tilings

Cited by 4 publications

References 4 publications

Color groups arising from index-nsubgroups of symmetry groups

Color groups arising from index-nsubgroups of symmetry groups

Symmetry groups of single-wall nanotubes

Symmetry groups associated with tilings on a flat torus

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