Abstract
Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.
Publisher
International Union of Crystallography (IUCr)
Subject
Inorganic Chemistry,Physical and Theoretical Chemistry,Condensed Matter Physics,General Materials Science,Biochemistry,Structural Biology
Reference18 articles.
1. A radial analogue of Poisson’s summation formula with applications to powder diffraction and pinwheel patterns
2. Baake, M. & Grimm, U. (2013). Aperiodic Order: A Mathematical Invitation, Vol. 1. Cambridge University Press.
3. Bellissard, J. (1992). From Number Theory To Physics, edited by M. Waldschmidt, P. Moussa, J.-M. Luck & C. Itzykson, pp. 538-630. Berlin: Springer-Verlag.
4. Bellissard, J., Hermann, D. J. L. & Zarrouti, M. (2000). Directions in Mathematical Quasicrystals (CRM Monograph Series), Vol. 13, edited by M. Baake & R. V. Moody, pp. 207-258. Providence, RI: American Mathematical Socitey.