Author:
Sbordone Carlo,Guida Margherita
Abstract
AbstractA 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 $$\displaystyle R_{36^{\circ }}$$
R
36
∘
and $$\displaystyle R_{72^{\circ }}$$
R
72
∘
with equal side lengths. It is crucial that the areas’ ratio is irrational $$\begin{aligned} \displaystyle \varphi = \frac{\textrm{area} \,R_{72^{\circ }}}{\textrm{area}\, R_{36^{\circ }}} = \frac{1+\sqrt{5}}{2}\, (\mathrm{golden\, ratio})\, (\varphi ^2 - \varphi - 1 = 0), \end{aligned}$$
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).
Funder
Università degli Studi di Napoli Federico II
Publisher
Springer Science and Business Media LLC
Subject
General Earth and Planetary Sciences,General Agricultural and Biological Sciences,General Environmental Science