Determining quasicrystal structures on substitution tilings

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 and Solomyak, 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

Acknowledgements

This research was supported by the Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0011150) and the Japanese Society for the Promotion of Science (JSPS), Grant in aid 21540010.

Notes

Note

1. It is possible to take a smaller R. See 12 for details.