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DETERMINING QUASICRYSTAL STRUCTURES ON SUBSTITUTION Shigeki Akiyama a
 

Summary: DETERMINING QUASICRYSTAL STRUCTURES ON SUBSTITUTION
TILINGS
Shigeki Akiyama a
and Jeong-Yup Lee b
a: Department of Mathematics, Faculty of Science, Niigata University, 8050 Ikarashi-2,
Nishi-ku, Niigata, Japan (zip: 950-2181); akiyama@math.sc.niigata-u.ac.jp
b: KIAS 207-43, Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130-722, Korea;
jylee@kias.re.kr
Abstract. Quasicrystals are characterized by the diffraction patterns which consist of pure bright
peaks. Substitution tilings are commonly used to obtain geometrical models for quasicrystals. We
consider certain substitution tilings and show how to determine a quasicrystalline structure for the
substitution tilings computationally. In order to do this, it is important to have the Meyer property
on the substitution tilings. We use the recent result of Lee-Solomyak in [18] which determines the
Meyer property on the substitution tilings from the expansion maps.
Keywords: Quasicrystals, Pure point diffraction, Self-affine tilings, Overlap coincidence, Meyer
sets, Algorithm.
1. Introduction
For the study of geometric structures of quasicrystals, substitutions are commonly used to create
mathematical models. Many known examples such as the Penrose tiling and the Robinson tiling
can be created by this method. Mathematically quasicrystals are characterized as the structures of

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics