Aperiodic tiling

In the mathematics of tessellations, a non-periodic tiling is a special kind of pattern made by repeating shapes that does not repeat itself in a regular, repeating way. This means you can cover a surface with these shapes, but there is no exact pattern that repeats over and over again. An aperiodic set of prototiles is a group of basic shapes that can only create these non-repeating patterns.

The Penrose tilings are a famous example of aperiodic tilings, named after the mathematician who first described them. In March 2023, researchers David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss made an exciting discovery. They proved that a special tile created by David Smith is an aperiodic monotile, solving a problem known as the einstein problem. This problem asked whether a single shape could create such non-repeating patterns.

Aperiodic tilings are important because they help scientists understand quasicrystals, special solid materials first discovered in 1982 by Dan Shechtman, who later won a Nobel Prize in 2011. Even though we know these materials exist, scientists are still learning more about their detailed structure. There are several known ways to build aperiodic tilings, making them a fascinating area of study in both mathematics and science.

Definition and illustration

Imagine covering a flat surface completely with shapes so that there are no gaps or overlaps. This is called a tiling. A periodic tiling looks the same if you shift it by certain distances — like repeating patterns on graph paper.

An aperiodic tiling is special because it never looks the same after any shift. One famous example is the Penrose tiling, which uses specific shapes to create beautiful, never-repeating patterns. These tilings are interesting because they break the usual rules of regular, repeating designs.

History

The idea of aperiodic tilings began in 1961 when a logician named Hao Wang studied how tiles can cover a flat surface. Later, in 1964, Robert Berger found a special group of tiles that could only cover the surface in a non-repeating way. Over the years, scientists found smaller and simpler groups of tiles that could do this. By 2023, they even discovered a single tile shape that could create a non-repeating pattern all by itself.

Today, many mathematicians and scientists study these special tilings to learn more about patterns in nature and math.

Constructions

There are several ways to create patterns called aperiodic tilings, which do not repeat themselves in any regular pattern. One common method uses special rules to match the edges of tiles, ensuring the pattern never repeats. Another method uses a process where tiles fit together in larger patterns, creating complex, non-repeating designs.

The Penrose tilings are a famous example made using these methods. They can also be created by projecting shapes from higher dimensions into our regular 2D space, which results in patterns that never repeat. Many clever techniques have been developed to build such tilings, showing just how many different ways we can create patterns that never settle into a repeating cycle.

Aperiodic tilings in Islamic art

Aperiodic tilings can be found in beautiful Islamic decorations, such as those in the Darb-i Imam shrine in Iran. These patterns may have been made using special tiling methods called girih that are similar to a famous mathematical design known as Penrose tiling.

Physics

Main article: Quasicrystal

Aperiodic tilings were once thought to be only mathematical ideas. But in 1984, a physicist named Dan Shechtman discovered a special kind of material made from aluminium and manganese that showed a unique pattern. This pattern had fivefold symmetry, which was very unusual for crystals. This discovery showed that aperiodic tilings could exist in real materials. Today, scientists study these patterns to understand how materials can be built in new ways.

Confusion regarding terminology

The word "aperiodic" has been used in many different ways in math, especially when talking about tilings. Sometimes it means the same as "non-periodic," which just means a tiling that doesn’t repeat itself in a pattern. Other times, it refers to tilings made from a special set of tiles that can only create non-repeating patterns.

The word "tiling" can also be confusing. For example, with Penrose tiling, there are infinitely many ways to arrange the tiles, even though they look similar up close. This shows how tricky these terms can be, so experts try to use them carefully.