# Tessellation: Definition & Examples

Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

In this lesson you will learn what a tessellation is and how to determine whether or not a pattern is a tessellation. After looking at some examples, you can take a brief quiz to see what you learned.

## What is a Tessellation?

Have you ever looked at a tile pattern or mosaic and wondered how one comes up with something so intricate and creative? Chances are that a geometric concept, such as tessellating, was used in the design.

A tessellation is simply a tiling that has a repeated pattern of one or more shapes. For a pattern to truly be a tessellation, the shapes cannot overlap and can have no spaces between them. The pattern can be created by rotating, translating (sliding), and/or reflecting (mirroring) the shapes.

## Classifying Tessellations

A regular tessellation is made by repeating a regular polygon. Remember that a regular polygon has equal angles and sides. Regular tessellations can be made using an equilateral triangle, a square, or a hexagon.

Some tessellations can be named using a number system. You would first pick a vertex in the pattern; recall that a vertex is a corner of a polygon. It does not matter which vertex you pick. Then identify the polygons around it according to the number of sides each one has. As we look at the examples that follow, we will practice naming them.

The image of a regular tessellation shows a tessellation made of equilateral triangles that are translated horizontally. Let's practice naming it. First, we pick a vertex in the pattern. Next we count to see how many polygons meet at that vertex. There are six. Each polygon is a triangle. Since each triangle has three sides, this a 3.3.3.3.3.3.3 tessellation. Each three represents a triangle that meets at the vertex.

Semi-regular tessellations are made of two or more regular polygons. The image of a semi-regular tessellation is made of hexagons and equilateral triangles. By focusing on the hexagons, we can see the pattern is created by rotating the triangles around the sides of the hexagons. Using our strategy for naming tessellations, we find this is a 3.3.6 tessellation. The threes represent the two triangles and the six represents the hexagon.

## Examples of Tessellations

The patterns of regular and semi-regular tessellations we've seen so far are rather simple. However, tessellations can be very complex. Any figure can be used, not just geometric shapes. Unfortunately, the number system that we used to name our previous examples would not apply to these tessellations. The naming system is used when the tessellation is made of regular polygons, but not for other shapes. For example, if a tessellation has a 10-sided star, there would be no way to determine if the 10 in the name represented a star or a decagon. It would also be impossible to use this system for a shape that does not have a vertex.

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