Point substitution processes for decagonal quasiperiodic tilings

• Published in: Acta crystallographica. Section A, Foundations of crystallography Vol. 65; no. 5; pp. 342 - 351
• Main Author: Fujita, Nobuhisa
• Format: Journal Article
• Language: English
• Published: 5 Abbey Square, Chester, Cheshire CH1 2HU, England International Union of Crystallography 01-09-2009; Wiley Subscription Services, Inc
• Subjects: Crystallography; decagonal tilings; Fractals; Geometry; point decorations; Polygons; prototiles; quasiperiodic tilings
• ISSN: 0108-7673; 1600-5724; 2053-2733
• DOI: 10.1107/S0108767309025008

A general construction principle for the inflation rules for decagonal quasiperiodic tilings is proposed. The prototiles are confined to be polygons with unit edges. An inflation rule for a tiling is the combination of expansion and division of the tiles, where the expanded tiles can be divided arbitrarily as long as the set of prototiles is maintained. A certain kind of point decoration process turns out to be useful for the identification of possible division rules. The method is capable of generating a broad range of decagonal tilings, many of which are chiral and have atomic surfaces with fractal boundaries. Two new families of decagonal tilings are presented; one is quaternary and the other ternary. The properties of the ternary tilings with rhombic, pentagonal and hexagonal prototiles are investigated in detail.