Tessellations: Geometry and Art

Contributor: Ashley Nail. Lesson ID: 13100

Tessellations combine art and geometry! These complicated and beautiful patterns are explained through three simple rules. You will study tessellations found in the real world and get to make your own


Geometry, Plane Geometry (2D)

learning style
personality style
Lion, Beaver
Grade Level
Intermediate (3-5)
Lesson Type
Quick Query

Lesson Plan - Get It!

  • Did you know that your bathroom floor can combine geometry and art?

If you thought the tiles on the floors and walls around you were just pretty designs, you might be surprised to learn that these patterns are probably tessellations! These tiles are carefully designed following three rules. If construction workers follow these rules when working on houses, schools, or other buildings in your community, they are creating tessellations!

Imagine you are working with a construction worker to check if the tessellations and tile patterns around you are following the rules.

Let's get started by looking closely at these tile floors.

But before we try to understand tessellations, we need to remember how patterns work!

A pattern is something repeated with a set of rules. You can find patterns with colors, shapes, numbers, and more! For example, this row of shapes has a pattern:

shape pattern

This pattern has three rules.

  1. The first rule in this pattern is that the color of the shapes are green, then blue, then orange. The rule repeats over and over again to make the pattern.
  2. The second rule in this pattern is that the type of shape changes from a circle, to a star, and back to a circle. This rule also repeats over and over again to make a pattern.
  3. The last rule of the pattern is that the sizes of the shapes are small, small, big. These three rules make the pattern we see above.

If we follow these rules, we can guess what the next shape in the pattern will look like! The next shape will be a small green circle!

We see patterns all around us in our lives. For example, look around your house. I bet you can find a floor or a wall with tiles on it similar to this one.

mosaic tile

The tiles cover an entire flat area and are made up of one or more shapes. This is an example of a tessellation.

A tessellation is a type of pattern that covers an entire flat surface with repeating polygons without any gaps or overlapping.

Whew! That seems complicated! But just like our shape pattern from above, we can break this pattern down into three simple rules to follow.

3 Rules of a Tessellation
Rule #1 The shapes must be regular polygons.
Rule #2 The polygons can't overlap or have gaps in the pattern.
Rule #3 Every vertex has to look the same.


Tessellation Rule #1: The shapes must be regular polygons.

A polygon is any shape that is formed by straight lines. The names of polygons tell you how many sides the shape has.

  • Can you think of any polygons?

A regular polygon is when all the sides are equal length. For example, a square is a regular polygon because all four sides are the same length.

Here are examples of other regular polygons labeled by their number of sides:

Each of these polygons is made up of sides that are the same length.

This tile floor is a tessellation made out of all regular hexagons!

mosaic tile hexagons

In order to make a tessellation, we must use regular polygons!

Tessellation Rule #2: The polygons can't overlap or have gaps in the pattern.

Imagine laying tiles on a floor. The floor wouldn't look right or be smooth to walk on if the edges of the tiles overlapped. This is also true if there were gaps between tiles.

For example, you couldn't make a tessellation with just octagons. This 8-sided shape would overlap with each other.

octagons overlapping

However, you could make a tessellation with octagons and squares! No overlapping and no gaps!

tile floor

In order to make a tessellation, we must use one or more regular polygons with no overlapping and no gaps.

Tessellation Rule #3: Every vertex has to look the same.

A vertex is where all the corners of each polygon meet.

For example, look at each corner of our tile floor. Find a corner, or a vertex, and look at all the polygons meeting at this spot.

6.6.6 tessellation

Notice how every vertex you point to is surrounded by three hexagons (six sides). This is named a 6.6.6 tessellation.

Now, look at our example of a tessellation made out of squares and octagons. Let's check to see if every vertex is the same.

4.8.8 tessellation

Each vertex is surrounded by a square (four sides) and two octagons (eight sides). This is named a 4.8.8 tessellation.

In order to make a tessellation, we must follow these three rules for this type of pattern!

Move to the Got It? section to practice using the rules of tessellations!

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