Penrose Tiling

A Penrose tiling is a way of covering a plane completely with non-overlapping polygons that never repeats in a regular, periodic pattern no matter how far it extends. The mathematician and physicist Roger Penrose introduced the first such tiling in a 1974 paper, later simplifying it to versions built from just two prototile shapes, either two rhombi or a pair called kites and darts. Despite having no translational symmetry, a Penrose tiling can show reflection symmetry and fivefold rotational symmetry, and its patterns are self-similar across scales. The tilings turned out to describe real physical quasicrystals, materials whose atoms are arranged in an ordered but non-repeating structure, a discovery that earned Dan Shechtman the 2011 Nobel Prize in Chemistry.