Aperiodic tiling

Aperiodic tiling

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

The Penrose tilings are a famous example of aperiodic tilings. They are 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 important discovery. They proved that a special tile created by David Smith is an aperiodic monotile. This solved a problem known as the einstein problem.

Aperiodic tilings are important because they help scientists understand quasicrystals. These are special solid materials first discovered in 1982 by Dan Shechtman. Scientists are still learning more about how these materials are built. There are several known ways to create aperiodic tilings, making them interesting to 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 started in 1961 when a logician named Hao Wang looked at 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 time, 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 make patterns called aperiodic tilings. These patterns never repeat themselves in a regular way. One common way is to use special rules for matching the edges of tiles. This makes sure the pattern never repeats. Another way is to fit tiles together in bigger patterns. This creates complex designs that do not repeat.

The Penrose tilings are a famous example made using these methods. They can also be made by projecting shapes from higher dimensions into our 2D space. This makes patterns that never repeat. Many clever ways have been found to build such tilings. This shows how many different ways we can create patterns that never repeat.

Aperiodic tilings in Islamic art

Aperiodic tilings appear in lovely Islamic decorations, like those in the Darb-i Imam shrine in Iran. These patterns might have been created using special tiling methods named girih, which are related to a well-known mathematical design called Penrose tiling.

Physics

Main article: Quasicrystal

Aperiodic tilings were once just ideas in math. But in 1984, a scientist named Dan Shechtman found a special material made from aluminium and manganese. This material had a pattern that was very unusual for crystals. His discovery showed that aperiodic tilings could exist in real materials. Today, scientists study these patterns to learn new ways to build materials.

Confusion regarding terminology

The word "aperiodic" is used in many ways in math, especially when talking about tilings. Sometimes it means the same as "non-periodic." This 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.