Penrose tiling

A Penrose tiling is a special way to cover a flat surface with shapes so the pattern never repeats exactly. 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 special 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 make a small part of the pattern bigger, 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 overlaps or gaps is called a tiling. Familiar tilings, like using squares on a floor, repeat their pattern. We call these periodic tilings.

Some tilings do not repeat at all. These are called aperiodic tilings. Penrose tilings are simple examples of aperiodic tilings. They use 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. Special rules help the shapes fit together so the pattern never repeats.

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,” 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 where no part looks exactly the same as another.

Features and constructions

Penrose tilings are special patterns made with shapes that never repeat exactly. 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 found a new way to cover a surface with tiles. She used one special tile that looks like a ten-sided shape. By letting pieces overlap in two ways and using colors to show how they fit, this method helps scientists study quasicrystals. Quasicrystals are special materials with patterns that don’t repeat in a regular way.

There are other tilings related to Penrose tilings, like the hexagon-boat-star and Mikulla–Roth tilings. These also have special patterns and symmetries, which make them interesting for mathematicians and scientists.

Art and architecture

People like the way tiles look, and Penrose tilings are very interesting. Some old patterns in buildings in North Africa and the Middle East look like 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 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.