Aperiodic tilings are a type of tiling where the tiles can fit together in a specific, non-repeating way to cover a surface. Aperiodic tilings were first introduced by mathematician Roger Penrose in the 1970s and have since been studied extensively in the field of mathematics.
One of the most famous aperiodic tilings is the Penrose tiling, named after its creator. These tilings are formed using two different types of tiles: fat rhombuses and thin rhombuses. The tiles are arranged in a way that allows them to fit together without forming a repeating pattern, creating a highly intricate and complex tiling.
Aperiodic tilings have a number of interesting properties that make them unique. For example, they have a property known as quasiperiodicity, which means that although they do not have a repeating pattern, they do exhibit some local order. This property has led to the study of aperiodic tilings in a number of different fields, including physics and chemistry.
In addition to the Penrose tiling, there are many other types of aperiodic tilings that have been discovered and studied. Some of these include the Ammann-Beenker tiling, the Octagorean tiling, and the Chaim Goodman-Strauss tiling.
Aperiodic tilings have a number of practical applications, including in the design of quasicrystals, which are materials that exhibit quasiperiodic properties. They have also been used in the design of decorative patterns and in the creation of computer algorithms.
Overall, aperiodic tilings are a fascinating and important area of study in mathematics and have a number of interesting properties and applications.
See an update as of March 2023 regarding a new aperiodic tiling discovered by David Smith.