Tessellation

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles. The tiles fit together with no overlaps and no gaps. In mathematics, tessellation can be used in higher dimensions and different types of geometries. This helps us see how shapes can fit together perfectly.

A periodic tiling has a pattern that repeats. Special types include regular tilings with regular polygonal tiles that are all the same shape. There are also semiregular tilings with more than one shape of regular tiles, arranged the same at every corner. These patterns belong to 17 groups called wallpaper groups. A tiling without a repeating pattern is called "non-periodic". An aperiodic tiling uses a few tile shapes that cannot repeat.

Real tessellations are made from materials like cemented ceramic squares or hexagons. They can be beautiful designs or useful for making strong, water-resistant pavement, floors, or walls. People used tessellations in Ancient Rome and in Islamic art, like in Moroccan architecture and the Alhambra palace. In the twentieth century, M. C. Escher used tessellations in his art, in both regular shapes and curved designs. Tessellations are also used to make beautiful patterns in quilting.

History

Tessellations were used long ago by the Sumerians around 4000 BC to make wall patterns with clay tiles. Later, people used small squared blocks called tesserae to create pretty mosaic tilings with shapes.

In 1619, Johannes Kepler wrote about tessellations in his book Harmonices Mundi. He talked about shapes like those in honeycombs and snowflakes. In 1891, a scientist named Yevgraf Fyodorov found that there are seventeen ways to repeat patterns on a flat surface.

Overview

Tessellation, also called planar tiling, is a way of covering a flat surface with shapes called tiles. There are no gaps or overlaps between the tiles.

Only three regular shapes—an equilateral triangle, a square, and a regular hexagon—can tessellate perfectly all by themselves.

Many other tessellations are possible with different rules. For example, semi-regular tessellations use more than one type of regular polygon. Artists like M. C. Escher created beautiful tessellations that look like animals or natural objects. These can make striking patterns with different colored tiles.

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 been used in buildings to make pretty patterns. Old mosaics used simple shapes. Later, Moorish designs in Islamic architecture used special tiles like Girih and Zellige in famous places such as the Alhambra.

Artists like M. C. Escher got ideas from these patterns. When he visited Spain, Escher made wonderful drawings named "Circle Limit". They show how shapes can fit together in fun ways. You can also see tessellation designs in fabrics, quilts, and origami, where paper is folded into repeating patterns.

In manufacturing

Tessellation helps the manufacturing industry save materials. For example, when making car doors or drink cans from sheet metal, tessellation cuts down on waste.

We can also see tessellation in nature, like the mudcrack-like cracking patterns in thin films. Scientists use micro and nanotechnologies to study how materials arrange themselves.

In nature

Main article: Patterns in nature § Tessellations

The honeycomb is a good example of tessellation 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. They looked at 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. It 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.