Penrose tiling

A Penrose tiling is a special way of covering a flat surface with shapes so that the pattern never exactly repeats. This is called an aperiodic tiling. Even though the pattern doesn’t repeat, it can still have beautiful symmetries, like mirror images or rotations. Penrose tilings are named after the mathematician and physicist Roger Penrose, who studied them in the 1970s.

There are different kinds of Penrose tilings, but they all use just a few simple shapes that fit together in specific ways. One common version uses two shapes: kites and darts, which are special kinds of quadrilaterals. These shapes can be arranged to create endless, non-repeating patterns.

Penrose tilings have interesting properties. For example, they are self-similar, meaning that if you magnify a small part of the pattern, it looks like the whole pattern again. These tilings also relate to real materials called quasicrystals, which were discovered and recognized with The 2011 Nobel Prize in Chemistry awarded to Dan Shechtman. Penrose tilings helped scientists understand how these unusual materials form.

Background and history

Covering a flat surface with geometric shapes without any overlaps or gaps is called a tiling. Familiar tilings, like using squares on a floor, are called periodic because shifting the pattern by the size of a square shows the same pattern again. These tilings repeat in two directions.

Some tilings, however, do not repeat in any direction and are called aperiodic. Penrose tilings are simple examples of aperiodic tilings using a few special shapes. They were first introduced by Roger Penrose in 1974, using shapes based on pentagons. Later, Penrose and others found ways to use just two shapes to create these non-repeating patterns.

Penrose tilings

Penrose tilings are special ways to cover a flat surface with shapes that don’t repeat in a pattern. There are three main types of Penrose tilings, each using different shapes connected to the pentagon and the golden ratio. To make sure the tiling doesn’t repeat, special rules are used to decide how the shapes fit together.

The first type uses pentagons, stars, and other shapes, with rules about how they can touch. The second type uses shapes called “kites” and “darts,” which are made from smaller triangles. The third type uses two kinds of rhombuses, or diamond shapes, with rules about how their edges match up. These tilings can create very interesting patterns without any part repeating exactly.

Features and constructions

Penrose tilings are special patterns made with shapes that never repeat exactly the same way. They use a special number called the golden ratio, which is about 1.618. This number helps decide the sizes and shapes of the tiles, like kites and darts, making sure they fit together in interesting ways.

These tilings can have fivefold symmetry, meaning they can look the same after being turned by 72 degrees. Even though the whole pattern never repeats, small parts of it can appear many times in different places. This makes Penrose tilings both beautiful and mathematically interesting.

Related tilings and topics

In 1996, a mathematician named Petra Gummelt showed that a special covering, similar to a Penrose tiling, can be made using a single decagonal tile. This works by allowing two types of overlapping regions and using colored patches on the tile to guide how the pieces fit together. This method helps scientists study quasicrystals, which are special materials that have patterns without repeating in a regular way.

There are several other tilings related to Penrose tilings, such as the hexagon-boat-star and Mikulla–Roth tilings. These tilings also have unique patterns and symmetries, making them interesting for mathematicians and scientists to explore.

Art and architecture

People have always liked the way tiles look, and Penrose tilings are especially interesting to look at. Some old patterns used in buildings in North Africa and the Middle East look very similar to Penrose tilings. Artists and builders have used these patterns in many modern buildings and artworks.

Artists have used Penrose tilings in many places, such as universities, buildings, and even streets. For example, the floor of a building at the Indian Institute of Information Technology, Allahabad, has Penrose tiling, and the street Keskuskatu in Helsinki is paved with a Penrose pattern.