Cyclotomic Aperiodic Substitution Tilings

Abstract

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.

An integer representation for periodic tilings of the plane by regular polygons

“…Shutov and Maleev [28] refined the Baake construction of Penrose tilings using 5th roots of the unity. Pautze [29] used 2nth roots of the unity to represent substitution aperiodic tilings with dihedral symmetry.…”

Darb-e Imam Tessellations: A Mistake of 250 Years

“…This 1453-dating of the Safavid tessellations has been persistent: see, e.g., Lu and Steinhardt (2007), Zheludev and Ozbay (2007), Saltzman (2008), Dewar (2009), Ferrero et al (2009), Juhel (2012), Todesco (2012), Al Ajlouni (2012Ajlouni ( , 2013, Au-Yang and Perk (2013), Tennant (2014), Topper (2014), Ebrahimi and Aliabadi (2015), Williams and Ostwald (2015), Aube (2016), Bier (2011Bier ( , 2012Bier ( , 2017, Cromwell (2015Cromwell ( , 2016, Caste ´ra (2016), Makovicky (2007Makovicky ( , 2016, Bonner (2017), Lissek et al (2017), Pautze (2017), Wichmann and Wade (2017). 9 In addition, it seems that Bonner's (2003) claim that four designs (his Figs.…”