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Tessellations

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Identifying Tessellations
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Dylan came storming in the door after a busy day at school. He slammed his books down on the kitchen table.

“What is the matter?” his Mom asked sitting down at the table.

“Well, I made this great geodesic dome. It is finished and doing great, but Mrs. Patterson wants me to investigate other shapes that you could use to make a dome. I don’t want to do it. I feel like my project is finished,” Dylan explained.

“Maybe Mrs. Patterson just wanted to give you an added challenge.”

“Maybe, but what other shapes can be used to form a dome? The triangle makes the most sense,” Dylan said.

“Yes, but to figure this out, you need to know what other shapes tessellate,” Mom explained.

“What does it mean to tessellate? And how can I figure that out?”

Pay attention to this Concept and you will know how to answer these questions by the end of it.

Guidance

We can use translations and reflections to make patterns with geometric figures called tessellations.

A tessellation is a pattern in which geometric figures repeat without any gaps between them.

In other words, the repeated figures fit perfectly together. They form a pattern that can stretch in every direction on the coordinate plane.

Take a look at the tessellations below.

This tessellation could go on and on.

We can create tessellations by moving a single geometric figure. We can perform translations such as translations and rotations to move the figure so that the original and the new figure fit together.

How do we know that a figure will tessellate?

If the figure is the same on all sides, it will fit together when it is repeated. Figures that tessellate tend to be regular polygons. Regular polygons have straight sides that are all congruent. When we rotate or slide a regular polygon, the side of the original figure and the side of its translation will match. Not all geometric figures can tessellate, however. When we translate or rotate them, their sides do not fit together.

Remember this rule and you will know whether a figure will tessellate or not! Think about whether or not there will be gaps in the pattern as you move a figure.

Sure. To make a tessellation, as we have said, we can translate some figures and rotate others.

Take a look at this situation.

Create a tessellation by repeating the following figure.

First, trace the figure on a piece of stiff paper and then cut it out. This will let you perform translations easily so you can see how best to repeat the figure to make a tessellation.

This figure is exactly the same on all sides, so we do not need to rotate it to make the pieces fit together. Instead, let’s try translating it. Trace the figure. Then slide the cutout so that one edge of it lines up perfectly with one edge of the figure you drew. Trace the cutout again. Now line the cutout up with another side of the original figure and trace it. As you add figures to the pattern, the hexagons will start making themselves!

Check to make sure that there are no gaps in your pattern. All of the edges should fit perfectly together. You should be able to go on sliding and tracing the hexagon forever in all directions. You have made a tessellation!

Do the following figures tessellate? Why or why not? 

Example A

Solution: Yes, because it is a regular polygon with sides all the same length.

Example B

Solution: No, because it is a circle and the sides are not line segments.

Example C

Solution: Yes, because it is made up of two figures that tessellate.

Now let's go back to the dilemma at the beginning of the Concept.

First, let’s answer the question about tessellations. What does it mean to tessellate?

To tessellate means that congruent figures are put together to create a pattern where there aren’t any gaps or spaces in the pattern. Figures can be put side by side and/or upside down to create the pattern. The pattern is called a tessellation.

How do you determine which figures will tessellate and which ones won’t?

Regular polygons will tessellate as long as one of their interior angles is divisible by 360^\circ . One interior angle of a regular pentagon is \frac{180(5-2)}{5}=\frac{540}{5}=108^\circ . Because 108 is not a factor of 360, a regular pentagon will not tessellate. Try it out to prove it to yourself! A regular hexagon, on the other hand, does tessellate. One interior angle of a regular hexagon is \frac{180(6-2)}{6}=\frac{720}{6}=120^\circ . Because 120 is a factor of 360, a regular hexagon will tessellate.

Vocabulary

Tessellation
a pattern made by using different transformations of geometric figures. A figure will tessellate if it is a regular geometric figure and if the sides all fit together perfectly with no gaps.

Guided Practice

Here is one for you to try on your own.

Draw a tessellation of equilateral triangles.

Solution

In an equilateral triangle each angle is 60^\circ . Therefore, six triangles will perfectly fit around each point.

Video Review

Explore More

Directions: Will the following figures tessellate?

  1. A regular pentagon
  2. A regular octagon
  3. A square
  4. A rectangle
  5. An equilateral triangle
  6. A parallelogram
  7. A circle
  8. A cylinder
  9. A cube
  10. A cone
  11. A sphere
  12. A rectangular prism
  13. A right triangle
  14. A regular heptagon
  15. A regular decagon

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