Tessellation

Shapes can fit together like a puzzle.

This is called a tessellation. It covers a surface with no gaps. The shapes do not overlap either.

Some patterns repeat the same way.

You can use many shapes for this. You can also use just one shape. Triangles, squares, and hexagons work well.

People have used these patterns for a long time.

Ancient Romans made them with small stones. People in Morocco made them for art.

Nature makes them too. Bees make hexagons in honeycombs. These shapes fit perfectly together.

Patterns can also be very beautiful. Artists use them to make quilts and art.

Imagine covering a floor with tiles. You want no gaps between them. You also want no overlaps. This way of covering a surface is called a tessellation.

Some patterns repeat the same way. These are called periodic tilings. You can use just one shape to do this. Only three shapes work perfectly by themselves. These are the equilateral triangle, the square, and the regular hexagon.

You can also use more than one shape. If you use different shapes, it is a semi-regular tiling.

Other patterns do not repeat. These are called non-periodic tilings. A famous kind is the Penrose tiling.

It uses shapes that cannot form a repeating pattern.

People have used these patterns for a long time. The Sumerians used clay tiles for art.

Romans used small blocks to make mosaics. In Spain, the Alhambra palace has beautiful designs.

Even nature uses them. Bees make hexagons in honeycombs.

Imagine covering a floor with beautiful tiles. You want to fit them together perfectly. There should be no gaps between the pieces. There should also be no overlaps where one tile sits on top of another.

This way of covering a surface is called a tessellation. In math, we call the individual shapes tiles. A tessellation can happen on a flat surface called a plane. It can also happen in three-dimensional space. When tiles fill up space, we sometimes call it a honeycomb.

There are different ways these tiles can fit together. A periodic tiling is a pattern that repeats the same way.

You can make a regular tiling using only one kind of shape. Only three shapes can do this perfectly: the equilateral triangle, the square, and the regular hexagon. You can also make semi-regular tilings. These use more than one type of shape. However, every corner must look exactly the same.

Some patterns are much more mysterious. A non-periodic tiling does not have a repeating pattern.

There is a special kind called an aperiodic tiling. These use a small set of shapes that can never form a repeating pattern. Mathematicians have found that periodic patterns on a flat surface fall into seventeen different groups. These are known as wallpaper groups.

People have loved these patterns for thousands of years. Around 4000 BC, the Sumerians used clay tiles for wall decorations.

In ancient Rome, people made mosaics using small square blocks called tesserae.

The Alhambra palace in Spain has very famous geometric designs.

In 1619, Johannes Kepler studied these shapes. He wrote about how honeycombs and snowflakes use hexagons. In the twentieth century, the artist M. C. Escher made famous art using these patterns.

You can see tessellations all around you every day. They are not just for math books or art galleries. They can be used in quilts to make pretty designs.

Builders use them for sturdy floors and walls that keep water out. Even bees are master mathematicians. They build hexagonal cells in honeycombs to store their honey.

Nature uses these shapes to stay strong and organized.

A tessellation, or tiling, is the process of covering a surface using geometric shapes called tiles. To be a true tessellation, the tiles must fit together perfectly. This means there can be no gaps between the shapes and no overlaps where one tile sits on top of another.

In mathematics, this concept is studied as planar tiling when it occurs on a flat surface, known as a Euclidean plane. Mathematicians can also extend these ideas to higher dimensions. When tiles fill a three-dimensional space, the pattern is often called a honeycomb.

To understand how these patterns work, we must look at the relationship between tiles. An edge is the line where two bordering tiles meet. A vertex is the specific point where three or more tiles intersect.

Some patterns are edge-to-edge, meaning the sides of the polygons match up perfectly. In these cases, a tile cannot share a partial side with its neighbor. A common example of a non-edge-to-edge tiling is a brick wall, where the long side of one brick meets the corners of two others.

Tessellations are categorized by how much they repeat. A periodic tiling follows a pattern that repeats regularly. These patterns can be organized into seventeen distinct mathematical categories called wallpaper groups.

In contrast, a non-periodic tiling lacks a repeating pattern. There is a special subset called aperiodic tilings. These use a specific set of shapes, known as prototiles, that are mathematically incapable of forming a repeating pattern.

The Penrose tiling is a famous example that uses two different quadrilateral shapes to create this effect.

We can further classify tilings by the types of polygons they use. A regular tessellation is highly symmetric and uses only one type of regular polygon. Only three shapes can form a regular tessellation: the equilateral triangle, the square, and the regular hexagon. A semi-regular, or Archimedean, tessellation uses more than one type of regular polygon. However, it must be isogonal, meaning every vertex looks exactly the same. There are eight known types of semi-regular tessellations. For example, a tiling might have a vertex configuration of 4.8.8, meaning one square and two octagons meet at every corner.

History shows that humans have used these geometric principles for millennia. Around 4000 BC, the Sumerians created wall decorations using clay tiles.

During classical antiquity, Romans used small square blocks called tesserae to create intricate mosaics.

Islamic art also features sophisticated geometric tilings, such as the colorful zellige patterns found in the Alhambra palace in Spain.

In 1619, Johannes Kepler conducted an early documented study of these structures. He explored the hexagonal patterns found in snowflakes and honeycombs. Later, in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling belongs to one of the seventeen isometry groups. This work is considered the unofficial start of the mathematical study of tessellations.

The significance of tessellations extends from art to the natural world. In nature, bees create hexagonal cells in honeycombs to organize their hives.

In human construction, tilings provide durable and water-resistant surfaces for floors and walls. Artists like M. C. Escher used tessellations for incredible visual effects. He often used irregular, interlocking shapes like animals to create complex patterns in both Euclidean and hyperbolic geometry. Even in domestic life, tessellations appear in the decorative patterns of quilts.

Mathematics continues to find new complexities in these patterns. The Swiss geometer Ludwig Schläfli pioneered the study of higher-dimensional versions of these shapes, called polytopes. He created the Schläfli symbol notation to describe them compactly. For instance, a tiling of regular hexagons can be written as {6,3} to show it has six-sided polygons meeting in groups of three. While many rules exist, such as the Conway criterion for periodic tiling, no general rule has been found to determine if any given shape can tile a plane. This leaves many interesting problems for mathematicians to solve.