| Abstrakt: |
A two-dimensional model of a quasi-crystal is the Penrose tiling (1974), which is an aperiodic "disjoint" covering of the plane generated by two rhombi R 36 ∘ and R 72 ∘ with equal side lengths. It is crucial that the areas' ratio is irrational φ = area R 72 ∘ area R 36 ∘ = 1 + 5 2 (golden ratio) (φ 2 - φ - 1 = 0) , which in turn reveals a local five-fold symmetry, forbidden for crystals. Recent advances on "Wang tiles", that is square tiles that cover the plane but cannot do it in a periodic fashion, are due to Jeandel and Rao (An aperiodic set of 11 Wang tiles, Advances in Combinatorics, pp 1–37, 2021), giving a definitive answer to the problem raised by Hao Wang in 1961. Other recent applications to variational problems in Homogenization are also mentioned (Braides et al. in C R Acad Sci Paris 347(11–12):697–700, 2009). [ABSTRACT FROM AUTHOR] |
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