Mathematical aspects of quasicrystals

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 R36∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle R_{36^{\circ }}$$\end{document} and R72∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle R_{72^{\circ }}$$\end{document} with equal side lengths. It is crucial that the areas’ ratio is irrational φ=areaR72∘areaR36∘=1+52(goldenratio)(φ2-φ-1=0),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle \varphi = \frac{\textrm{area} \,R_{72^{\circ }}}{\textrm{area}\, R_{36^{\circ }}} = \frac{1+\sqrt{5}}{2}\, (\mathrm{golden\, ratio})\, (\varphi ^2 - \varphi - 1 = 0), \end{aligned}$$\end{document}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).


Introduction
The mathematical theory of quasi-periodic crystals, after the discovery of natural quasicrystals (Bindi et al. 2009), has become an important area of investigation both in theoretical and experimental sciences (see Bindi 2008;Bindi and Stanley 2020;Bindi and Parisi 2023;van Smaalen 2023).
In this note, we present some basic mathematical ideas about tilings in the plane ℝ 2 , starting with the concepts of regular and semi-regular, or periodic and aperiodic, tiles.
In Sect. 3, we examine Penrose tilings, which can be seen as two-dimensional models for quasicrystals. We shall provide details in the case where the prototiles, three rhombi R 7 , R 2 7 and R 3 7 , produce a seven-fold symmetry.
In Sect. 4, we recall the Fibonacci sequence and in Sect. 5, we mention recent works regarding calculus of variations (in particular, the homogenization of Penrose tilings).

Wang tiles
For 0 < < 90 , we denote by R • the planar rhombus with an acute angle of degrees. A tiling (or tessellation) is a covering of a portion S ⊆ ℝ 2 by non-overlapping polygonal (or more generally, compact) shapes called tiles, which may differ in shape, size and orientation. Each tile is the projection of a d-dimensional face of a D-dimensional hypercube A tiling is periodic if it is invariant under two given linearly independent directions (in other words, shifting the pattern without rotating it produces the same tiling).
One long-standing problem has been that of constructing semi-regular tilings from regular ones. Starting from 1960, the problem arose on whether there existed "tilesets" admitting only non-periodic tilings. Such sets are said to be aperiodic.
In 1961, Wang conjectured that every tileset that realizes a tiling of ℝ 2 must also admit a periodic tiling of ℝ 2 . In 1966, his student Berger disproved the conjecture, by finding a set of more than 20,000 square tiles that cover the plane but cannot do it in a periodic way. The prototiles he employed are Wang tiles, namely square tiles with coloured edges. Wang tiles are juxtaposed so that adjacent edges share the same colour. Wang was motivated by his interests in mathematical logic.
The number of tiles needed was drastically reduced in 1996, when an aperiodic tileset of only 13 Wang tiles was discovered. At the same time, it has been shown that it is impossible to obtain aperiodic Wang tilings consisting of 4 tiles or less (1987).
In a 2021 paper published in Advances in Combinatorics Jeandel and Rao (2021) settled the matter for good, they found an aperiodic Wang tiling made of 11 pieces and 4 colours, and showed there cannot exist aperiodic tilings with fewer than 11 Wang tiles.

Penrose tilings
The main discovery regarding aperiodic tilings is due to Penrose, who showed how to tessellate ℝ 2 in a non-periodic fashion using two rhombi ( R 36 • and R 72 • ). The Penrose tiling exhibits a five-fold symmetry, and can be seen as a two-dimensional model of a quasi-crystal.
Since the edge length is 1, we have: and so that where is the golden ratio: an irrational solution to , e i 4 7 , e i 6 7 , e i 8 7 , e i 10 7 , e i 12 7 and their sum is zero: Using elementary formulas, such as cos cos cos = cos( + + ) + cos(− + + ) + cos( − + ) + cos( + − ) 4 A quasi-periodic 7-fold rhombic tiling, whose basic shapes are rhombi with acute angles of 7 , 2 7 and 3 7 radians, enjoys the property that (e.g. R 2 7 ) the prototiles can be arranged to form a seven-pointed star, hence producing 7-fold rotational symmetry.
As in case of pentagonal symmetry, where the length of the long diagonal of R 2 5 equals the golden ratio = 2 cos 5 , here the length of the long diagonal of R 2 7 equals = 2 cos 7 .
Similar to the five-fold case, the ratio of the areas of the 3 prototile rhombi (or their reciprocals) are the irrational solutions to the algebraic equation: namely Actually Quasicrystals consist of long-range ordered, but aperiodic arrangements of atoms. They reveal periodicity only in highdimensional hyperspaces, and no translational invariance. Important for quasicrystals are the so-called rational approximant phases, which are periodic structures close to aperiodic quasicrystalline ones, and which can be described as projections of the quasicrystalline hyper-structure along rational directions in hyperspace with a rational slope approximating the golden ratio.
The method of hyperdimensional constructions and their projection to lower dimensions to obtain quasi-periodic structures are a powerful tool. By suitable rational approximation of irrational quantities, intrinsic in the projection x − 2 cos 2 7 x − 2 cos 4 7 x − 2 cos 6 7 = x 3 + x 2 − 2x − 1.
matrices, periodic structures are related to "perfect" Penrose tilings modulo certain "defects". One such example are "phasons" which, when inserted regularly in Penrose lattices, change them into periodic approximants.

Fibonacci chain
This is a basic representation of an aperiodic system in ℝ . Let us recall the Fibonacci numbers (F k ) k∈ℕ : where each number is the sum of the two integers preceding it, according to the inductive definition

It is well known that
The Fibonacci tiling is obtained by projection from a 2D periodic structure, and represents an aperiodic covering of ℝ generated by two segments L (long) and S (short), whose The segments are juxtaposed as follows: as the length of the chain tends to ∞ , the ratio between the number of L edges and S edges contained in the chain tends to the golden ratio (Fig. 3). The chain does not have a translational symmetry, but has a clear "local" order. For example, there are no consecutive short segments SS, and no triples LLL. The projection acts from a 2d periodic lattice onto a straight line with slope , a physical line.
To find a rational approximant, one defines an irrational "strip" which contains the lattice points that will be projected onto the physical line, thus filling the Fibonacci sequence.
The difference between the Fibonacci algorithm and the approximant method becomes clearer once we look at their respective recursive formulas: and

Homogenization of Penrose tilings
The homogenization theory deals with the description of the macroscopic properties of media with fine microstructure and has important applications to the study of the properties of composite materials. Originally ( ∼ 1970) it was introduced in case of periodic structures, when the period is "small" with respect to the size of the region in which the system is to be studied.
The main problem consisting in going back from the microscopic quantities to the macroscopic ones, which are measurable.
The problem of homogenization of integral energies goes back to E. Sanchez Palencia, I. Babuska, E. De Giorgi-S. Spagnolo, F. Murat-L. Tartar, P. Marcellini for which we quote the reference books (Bensoussan et al. 1978) andDal Maso (1993).
Functionals of the form ( Ω ⊂ ℝ 2 ) for u ∈ W 1,2 (Ω;ℝ m ) , a Sobolev space of L 2 functions with gradient in L 2 , are considered and some natural limit as → 0 is described as with explicit expression for f H due to periodicity. In other words, we are performing the asymptotic analysis of fast-oscillating integral functionals depending on a smallscale parameter , as goes to zero.
Let us concentrate here to the special case where [a ij (x)] is a bounded symmetric matrix, such that (for a > 0) a n d we s u p p o s e a ij ( If Ω ⊂ ℝ 2 is bounded, there exists, for any ∈ L 2 (Ω) and > 0 a unique solution u to the minimum problem such that u = 0 are on the boundary Ω . (Dirichlet problem). It can be proved that there exists a constant symmetric positive-definite matrix [ ij ] such that u converges in L 2 as → 0 to the solution u of the minimum problem and u = 0 on Ω.
The interesting fact is that the matrix ij is different from Recently (Braides et al. 2009), a homogenization theorem was proved for energies which follow the geometry of an a-periodic Penrose tiling. The authors consider mixtures of two linear conducting materials with different dieletric constants depending on the type of tiles. The coefficient a = a(y) only takes two values , ∈ ℝ depending on the tile type y (see Braides et al. 2009). One can prove a homogenization-type theorem for energies that abide by the geometry of an aperiodic Penrose tiling. The result is obtained by showing that the alreadyexisting general homogenization theorems can be applied. The method applies to general quasicrystalline geometries.
Funding Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
Data availability Data are available.

Conflict of interest
The authors declare no conflict of interest.
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