A non‐quadratic irrationality associated to an enneagonal quasiperiodic tiling of the plane

Franco, B. J. O.; Silva, F. W. O. da; Inácio, E. C.

doi:10.1002/pssb.2221950102

Cited by 3 publications

(2 citation statements)

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“…In this section we present the heptagonal, enneagonal, tetradecagonal and octadecagonal patterns, which are not encountered in Moroccan geometric art. Franco et al (1996) proposed enneagonal tiling of the plane using geometric shapes. Seven-and ninefold quasiperiodic tilings very similar to ours were obtained by Whittaker & Whittaker (1988), who used the method of projection of N-dimensional space.…”

Section: New Quasiperiodic Patternsmentioning

confidence: 99%

“…Makovicky (1992Makovicky ( , 2004Makovicky ( , 2007Makovicky ( , 2008 2011), Makovicky & Fenoll Hach-Alí (1997), Makovicky et al (1998), Castera & Jolis (1991) and Castera (1996Castera ( , 2003 studied the octagonal and decagonal patterns, Rigby (2005), Lu & Steinhardt (2007), Saltzman (2008) and Al Ajlouni (2012) were interested in the decagonal patterns, Makovicky & Makovicky (2011) studied the dodecagonal structure, and Bonner & Pelletier (2012) constructed a sevenfold pattern. In another context, Whittaker & Whittaker (1988) proposed heptagonal and enneagonal tiling, and Franco et al (1996) also proposed enneagonal quasiperiodic tiling.…”

Section: Introductionmentioning

confidence: 99%

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Moroccan ornamental quasiperiodic patterns constructed by the multigrid method

2014

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The similarity between the structure of Islamic decorative patterns and quasicrystals has aroused the interest of several crystallographers. Many of these patterns have been analysed by different approaches, including various kinds of ornamental quasiperiodic patterns encountered in Morocco and the Alhambra (Andalusia), as well as those in the eastern Islamic world. In the present work, the interest is in the quasiperiodic patterns found in several Moroccan historical buildings constructed in the 14th century. First, the zellige panels (fine mosaics) decorating the Madrasas (schools) Attarine and Bou Inania in Fez are described in terms of Penrose tiling, to confirm that both panels have a quasiperiodic structure. The multigrid method developed by De Bruijn [Proc. K. Ned. Akad. Wet. Ser. A Math. Sci. (1981), 43, 39–66] and reformulated by Gratias [Tangente (2002), 85, 34–36] to obtain a quasiperiodic paving is then used to construct known quasiperiodic patterns from periodic patterns extracted from the Madrasas Bou Inania and Ben Youssef (Marrakech). Finally, a method of construction of heptagonal, enneagonal, tetradecagonal and octadecagonal quasiperiodic patterns, not encountered in Moroccan ornamental art, is proposed. They are built from tilings (skeletons) generated by the multigrid method and decorated by motifs obtained by craftsmen.

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Section: New Quasiperiodic Patternsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Moroccan ornamental quasiperiodic patterns constructed by the multigrid method

2014

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Two-Dimensional Graphics

Trott

2004

The Mathematica GuideBook for Graphics

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Cyclotomic Aperiodic Substitution Tilings

Pautze¹

2017

Symmetry

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Abstract:The class of Cyclotomic Aperiodic Substitution Tilings (CASTs) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry D n or D 2n , i.e., the tiling contains an infinite number of patches of any size with dihedral symmetry D n or D 2n only by iteration of substitution rules on a single tile.

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A non‐quadratic irrationality associated to an enneagonal quasiperiodic tiling of the plane

Abstract: A enneagonal tiling of the plane is proposed. A self-sirnilar pattern is obtained by using eight basic shapes. This pattern presents rotational symmetry and no translational invariance. ' ) C.P. 702, CEP 30.161-970. Belo Horizontc, Brazil. ') CEP 36.570.000, ViGosa (MG). Brazil.

Cited by 3 publications

References 14 publications

Moroccan ornamental quasiperiodic patterns constructed by the multigrid method

Moroccan ornamental quasiperiodic patterns constructed by the multigrid method

Two-Dimensional Graphics

Cyclotomic Aperiodic Substitution Tilings

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