Tessellation

Tessellation of a flat surface refers to the repeated placement of shapes with no overlaps and no gaps. These shapes are also called tiles. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

In math

The three regular polygons that can tesselate.

Using only regular pentagons to cover a flat plane like this leaves star-shaped holes everywhere.

Only three regular polygons can cover a perfectly level surface: triangles, squares, and hexagons. Other shapes, like pentagons, will need help from other shapes like rhombi.

Roger Penrose in front of Penrose tiles in a bathroom in Philadelphia.

Penrose tilings are special cases where the pattern never repeats, no matter how much it continues.

In real life

Since ancient times, people have used tessellations to decorate buildings.

The artist M.C. Escher was brilliant at painting pictures involving tessellation. He was inspired by the walls and floors in Muslim buildings.

Tilings like this one inspired MC Escher.

Tessellations appear a lot in nature. One of the best examples is the honeycomb. Honeycombs are built with hexagonal holes to hold honey and bees. If you can build a square that's big enough to hold 225 bees, you can bend the edges into a hexagon and hold 260 bees. If you want to have a lot of area with only a certain amount of perimeter, the best shape is a circle, and the hexagon is the closest thing that the bees can have.

A detail from M.C. Escher's painting, Metamorphosis, displayed in a museum in Brazil.

• 

  All of the buildings in this picture of Belfast in Ireland are rectangular to fit together, with some spaces for alleyways.

• 

  This bee can hold more honey with the hexagon walls of its honeycomb than with square walls.

• 

  The James Webb telescope has 18 big hexagonal mirrors.

• 

  The yolks of these eggs fit together like hexagons due to the way that circles tesselate.

• 

  The same thing seems to happen with these bubbles.

In puzzles

Many puzzles like Tangram are built around tessellation.

• 

  This square is made of seven pieces...

• 

  but with a little imagination, it can be changed into a number of shapes like this one.

• 

  This game is called Tetris, and the goal is to fit plenty of shapes together.

• 

  This rectangle is made of 12 pentominoes, which are in turn made of 9 smaller pentominoes, so all together, the rectangle has 108 small pentominoes.

.

Tessellation Media

• 

  A temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC), showing a tessellation pattern in coloured tiles

• 

  Roman rhombille mosaic

• 

  A rhombitrihexagonal tiling: tiled floor in the Archeological Museum of Seville, Spain, using square, triangle, and hexagon prototiles

• 

  The elaborate and colourful zellige tessellations of glazed tiles at the Alhambra in Spain that attracted the attention of M. C. Escher

• 

  An example of a non-edge‑to‑edge tiling: the 15th convex monohedral pentagonal tiling, discovered in 2015

• 

  A Pythagorean tiling is not an edge‑to‑edge tiling.

• 

  This tessellated, monohedral street pavement uses curved shapes instead of polygons. It belongs to wallpaper group p3.

• 

  A Penrose tiling, with several symmetries, but no periodic repetitions

• 

  A set of 13 Wang tiles that tile the plane only aperiodically

• 

  Random Truchet tiling