Tessellation

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries. This idea helps us understand how shapes fit together perfectly to cover an area.

A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. Patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern.

Real physical tessellations are made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor, or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the Moroccan architecture and decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting.

History

Tessellations were used a long time ago by the Sumerians around 4000 BC to decorate walls with patterns made from clay tiles. Later, in ancient times, people used small squared blocks called tesserae to create beautiful mosaic tilings with geometric designs.

In 1619, Johannes Kepler wrote about regular and semiregular tessellations in his book Harmonices Mundi, describing shapes like those found in honeycombs and snowflakes. Many years later, in 1891, a scientist named Yevgraf Fyodorov discovered that there are seventeen different ways to repeat patterns across a flat surface.

Overview

Tessellation, also called planar tiling, is a way of covering a flat surface using shapes called tiles without leaving any gaps or overlapping. Common rules include no gaps between tiles and no tile corners lying on another tile’s edge. Only three regular shapes—an equilateral triangle, a square, and a regular hexagon—can tessellate perfectly on their own to fill the entire plane.

Many other tessellations are possible with different rules. For example, semi-regular tessellations use more than one type of regular polygon but still have the same arrangement at every corner. Artists like M. C. Escher created beautiful tessellations using irregular shapes that look like animals or natural objects. These can create striking patterns when different colored tiles are used.

In mathematics

Further information: Euclidean tilings by convex regular polygons, Uniform tiling, and List of Euclidean uniform tilings

Mathematicians have special words for talking about tilings. An edge is where two tiles meet, often a straight line. A vertex is where three or more tiles meet. In an isogonal or vertex-transitive tiling, every vertex looks the same. The fundamental region is a shape like a rectangle that repeats to make the whole tiling. For example, a tiling of squares has four squares meeting at each vertex.

Tiles can share full sides or just parts of sides. An edge-to-edge tiling is one where tiles share whole sides. The "brick wall" pattern is not edge-to-edge because bricks share long sides with two others. A monohedral tiling uses just one shape of tile. One famous monohedral tiling is the Voderberg tiling, discovered in 1936, which uses a special non-regular shape called an enneagon. Another is the Hirschhorn tiling, using irregular pentagons.

There are three perfectly regular tilings made of equilateral triangles, squares, or regular hexagons. These are very symmetric and use just one type of regular shape. Semi-regular tilings use more than one type of regular polygon, like squares and octagons together. There are eight such semi-regular patterns.

Main article: Wallpaper group

Tilings that repeat in two directions can be grouped into 17 wallpaper groups. Some believe all 17 are shown in the beautiful tiles of the Alhambra palace in Granada, Spain. These patterns help us understand how designs repeat.

Main articles: Aperiodic tiling and List of aperiodic sets of tiles

Some tilings never repeat exactly, called aperiodic tilings. The famous Penrose tilings use two different quadrilaterals to make patterns that never repeat. These help scientists study structures in nature that also lack repeating patterns, called quasicrystals. Another interesting tiling is made with Wang tiles, squares with colored edges that must match up. These can only make non-repeating patterns.

An einstein tile is a single shape that forces a non-repeating pattern. The first such tile, called a "hat", was discovered in 2023.

Further information: Four colour theorem

Sometimes the color of a tile is part of the design. The four colour theorem says that any map can be colored with just four colors so that no two neighboring areas share the same color. This works for tilings too.

See also: Conway Criterion

Besides regular shapes, many other polygons can make tilings. Any triangle or quadrilateral can tile the plane. Even shapes with more sides, like pentagons and hexagons, can work. Some special shapes made of many sides can also tile perfectly.

Voronoi or Dirichlet tilings are made by starting with a set of points. Each tile is the area closest to one point. These tiles are always convex polygons. The opposite process, Delaunay triangulation, makes triangles between points and is useful for computer simulations.

Main article: Honeycomb (geometry)

Tessellations can also fill three dimensions, like stacking boxes to fill a room. The cube is one shape that can do this perfectly. These three-dimensional patterns are called honeycombs. In nature, some crystals form patterns like this.

Tessellations can even work in curved spaces, not just flat ones. In hyperbolic geometry, which is more curved than normal space, special patterns of regular shapes can tile the entire area.

In art

Further information: Mathematics and art

Tessellations have long been used in architecture to create beautiful decorative patterns. Ancient mosaics often featured simple geometric shapes, while later cultures, like the Moorish designs in Islamic architecture, used special tiles such as Girih and Zellige in famous buildings like the Alhambra.

Artists like M. C. Escher were inspired by these patterns. During his visit to Spain, Escher created amazing drawings called "Circle Limit", showing how shapes can fit together in interesting ways. Tessellation designs are also found in textiles, quilts, and even origami, where paper is folded into repeating patterns.

In manufacturing

Tessellation is used in the manufacturing industry to help save materials. For example, when making shapes like car doors or drink cans from sheet metal, tessellation helps reduce waste.

We can also see tessellation in nature, such as the mudcrack-like cracking patterns that appear in thin films. Scientists study these patterns using micro and nanotechnologies to understand how materials organize themselves.

In nature

Main article: Patterns in nature § Tessellations

The honeycomb is a great example of tessellation found in nature, with its hexagonal cells. In plants, a "tessellate" pattern can appear as a checkered design on flower petals, tree bark, or fruit. Some flowers, like the fritillary, and certain species of Colchicum, show this pattern.

Cracks in materials can also create tessellations. These patterns, known as Gilbert tessellations, help explain formations like mudcracks and crystals. For example, basaltic lava flows often form hexagonal columns due to cooling and cracking, such as at the Giant's Causeway in Northern Ireland. Another interesting example is tessellated pavement, seen at Eaglehawk Neck on the Tasman Peninsula of Tasmania.

Natural patterns also appear in foams. Scientists study how to pack foam cells closely together. In 1887, Lord Kelvin suggested a way using a special shape, while in 1993, two researchers proposed an even better way to reduce the space needed.

In puzzles and recreational mathematics

Main articles: Tiling puzzle and recreational mathematics

Tessellations are fun to use in puzzles and games! They inspire many types of tiling puzzles, like traditional jigsaw puzzles and the classic tangram. Modern puzzles often use shapes made from triangles and squares, called polyiamonds and polyominoes. Famous puzzle creators like Henry Dudeney and Martin Gardner used tessellations in fun math challenges. For example, they explored shapes that can be broken into smaller copies of themselves, called "rep-tiles". One exciting puzzle is “squaring the square,” where you must fill a square with smaller squares of different sizes.

Examples

Some beautiful ways to cover a flat surface with shapes are called tessellations. One example is the Triangular tiling, which uses identical triangles to fill the plane — this is one of the three simple, or regular, tilings. Another interesting pattern is the Snub hexagonal tiling, which is a mix of different shapes and is called a semiregular tiling.

There are many more creative tilings! The Floret pentagonal tiling uses a special kind of five-sided shape. The Voderberg tiling makes a spiral pattern with nine-sided figures. These patterns show how shapes can fit together in amazing ways to cover a surface completely.