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# What Is a Tessellation in Math? Definition, Types & Real-World Examples

Jul 25, 2024

Have you ever looked at a honeycomb and noticed how the shapes fit together perfectly?

If so, you’ve already encountered tessellations.

In this guide, we’ll explain what tessellation is, explore the types, show you real-life examples, and share some fun facts too.

## What Is a Tessellation?

A tessellation  is a pattern of geometric shapes that fit together perfectly on a plane without any gaps or overlaps and can repeat in all directions infinitely.

Tessellations can be composed of one or more simple polygons, which are two-dimensional shapes with any number of straight sides. Examples of polygons include triangles, squares, rectangles, hexagons, and others.

Here’s an example of tessellating polygons:

We can see how this repeating pattern of octagons and squares fits together neatly and predictably, without leaving any gaps between them.

Now let’s see an example of a pattern that isn’t an example of tessellation.

Notice the gaps between these pentagons?

As we said, tessellations are patterns that can repeat in all directions infinitely. This composition cannot maintain its pattern infinitely, therefore, it is not a tessellation.

Curved shapes such as circles and ovals can tesselate but not on their own.

When circles are arranged side by side, they create curved gaps between them.

To create a tessellating pattern with circles, you would need another curved shape that seamlessly fits into these gaps such as crescents or curved triangles.

Take a look at this example:

Notice the shapes between circles?

They allow the circles to tessellate.

### What Is the Difference Between Tessellations and Tiles?

While both tiles and tessellations can be made up of polygons without leaving any gaps or overlaps, there’s a significant difference between them.

As we’ve said earlier, tessellations must have discernable repeating patterns. Tiles, on the other hand, may or may not include them.

To be sure we’ve got the difference right, let's check out this pair of examples.

In example #1, we see hexagons and rhombuses of equal size that fit together to cover the plane and form a pattern that is repeatable and predictable. So, example #1 shows a tessellation:

In example #2, we see rectangles filling up the plane without leaving any gaps or overlaps but also without a repeatable pattern. So, example 2 shows a tiling, not a tessellation:

Example #2

## Types of Tessellations

There are three different types of tessellations: regular, semi-regular, and irregular tessellations.

Let’s explore each of them.

### Regular Tessellations

Regular tessellations are patterns made of regular polygons—shapes with sides of equal length and angles of equal size—that fit together perfectly without any gaps or overlaps.

Only three regular polygons—triangles, squares, and hexagons—can form tessellations by themselves.

### Semi-Regular Tessellations

Semi-regular tessellations are made from two or more types of regular polygons such as triangles, squares, hexagons, and octagons.

The image below shows a combination of octagons and squares.

### Irregular Tessellations

Irregular tessellations are composed of shapes that aren't regular polygons, but they still fit together without leaving any gaps or overlaps.

With irregular tessellations, there's a limitless number of figures you can create.

Check out our video on the difference between tiles and tessellations, how tessellation works, and more.

## Tessellating Patterns in Nature

You've probably seen tessellations countless times without realizing it.

Here are a few examples of patterns found in nature that tessellate:

• Honeycomb: Beehives are made up of hexagonal cells that tessellate perfectly, allowing bees to maximize the efficiency of their storage and nesting space.
• Pineapple Skin: If you take a closer look at the skin of a pineapple, you'll notice a tessellating pattern of hexagons which isn't just decorative—it makes the fruit strong, helping it handle environmental challenges as it grows.
• Turtle Shells: The patterns on a turtle's shell are often made up of tessellating shapes due to the arrangement of scutes, the hard plates covering it. These patterns protect the turtle, help it grow, provide camouflage, and make it easier to identify different species.
• Fish Scales: The scales on a fish's body often overlap in a tessellating pattern. The pattern of fish scales has several important functions: it protects from harm and predators, helps it swim smoothly, and keeps its body temperature stable.

Tessellating patterns on a turtle's shell.

Throughout history, people have created fascinating designs inspired by tessellations. These can be found in various kinds of art and architecture. Here are some of the most notable examples:

• Mosaics: Mosaics are colorful artworks made by arranging small pieces of glass or stone. They decorate walls, floors, and ceilings in places like temples and churches.
• Islamic Geometric Patterns: Islamic art features beautiful designs made of shapes like squares and triangles, often seen on the floors, walls, and ceilings of mosques and palaces.
• M.C. Escher's Artwork: M.C. Escher's art plays with shapes that fit together perfectly, like birds and fish. His drawings challenge how we see things, making us think about space and reality.
• Floor Tiles and Pavements: Many floor tiles and pavements are designed with tessellations. Hexagonal, square, or triangular tiles are arranged in repeating patterns to cover surfaces efficiently and create beautiful designs.

Tiles with tessellating patterns.

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