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What is a Tessellation?

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Did you know that most kitchen floor tiles are tessellations? Read this lesson and you'll find out why and how you can create and design your own tessellation using many different shapes.

Definition

By definition, a tessellation is tiling that uses shapes to cover a surface with no gaps or overlaps. Picture a kitchen floor with tiles and you are looking at a tessellation.

Kitchen floor tessellation.
tessellations

This particular kitchen floor tessellation is made up of all squares. But, tessellations aren't limited to just squares. They can be any shape or any combination of shapes. And the shapes don't have to follow a particular pattern. You can have a random tessellation of random shapes if you wanted.

And, tessellations don't always have to be flat. They can also be three-dimensional. Like when you take some building bricks and you make a wall or other solid structure with no gaps.

A 3-D tessellation.
tessellations

Identifying Tessellations

What identifies a particular design as a tessellation is that it follow these two rules:

  • 1. It can be repeated over and over again
  • 2. There must be no gaps and no overlaps

As long as the design follows these two rules, then it is a tessellation.

Let's look at some more examples of tessellations now.

Single Shapes

If you limit your tessellation to a single shape, you get a pattern like your kitchen tiled floor. You can use any shape that will allow this such as triangles, squares, rectangles, and hexagons.

Triangle tessellation.
tessellations

Square tessellation.
tessellations

Hexagon tessellation.
tessellations

This last tessellation using just hexagons is actually a pattern that you can find in the real world. Honey bees store honey in the hexagonal holes they make as part of their beeswax.

Once you create your tessellation, you can use various colors to make your pattern. As you can see in the above, you can use two colors or more. Even if you used just two colors and one shape, you can still create some very interesting tessellations such as this one that looks like a sad face looking at a happy face.

Tessellation using only small black and white triangles arranged in different ways.
tessellations

This particular tessellation was made using only small black and white triangles arranged in various ways.

All of the above are tessellations because there are no gaps nor overlaps. Only a few shapes can be used by themselves to create tessellations. Some shapes require another shape be used with them to create a tessellation.

Two Shapes

When you add another shape, you can create more detailed patterns. Your second shape will depend on what you choose for your first shape as your first shape will create the spaces for your second shape. For example, if you use octagons for your first shape, your second shape will be the squares in between the octagons.

A tessellation with hexagons and squares.
tessellations

Your octagons with squares is a tessellation because there are no gaps between the shapes and no overlaps.

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