Quasiperiodicity and topology transcend dimensions

Abstract

Many of us marvel at the beauty of nature, be it the vivid blossom of the lily, the fractal structures of snowflakes or other intriguing wonders. Their magic lies in the subtle combination of their symmetry and complexity, which come together in ordered patterns. Pythagoras of Samos was the first to express the harmony of nature in terms of mathematics. Since then, physicists have looked for beauty and symmetry not only in the outside world, but also in its mathematical description. Solid materials are composed of numerous atoms arranged in complex structures. Modelling such structures is almost impossible in the absence of order. Fortunately, most solid systems arrange themselves as periodic crystals in which a building block known as the unit cell repeats itself periodically to form a lattice that tiles the whole space. Periodic crystals are the simplest arrangement of atoms, exhibiting both order and symmetry.