This Exploration of Tessellations will guide you through the following:

Exploring Tessellations

Definition ofTessellation

Semi-RegularTessellations

Symmetry inTessellations

RegularTessellations

Create yourown

Tessellation

View artistictessellations

byM.C. Escher

TessellationsAround Us

What is a Tessellation?

A Tessellation is a collection of shapes that fit together to cover a surface without overlapping or leaving gaps.

Tessellations in the World Around Us:

Brick Walls Floor Tiles Checkerboards

Honeycombs Textile Patterns

Art

Can you think of some more?

Are you ready to learn more about Tessellations?

Symmetry inTessellations

CLICK on each topic to learn more…

Once you’ve explored each of the topics above, CLICK HERE to move on.

Regular Tessellations

Semi-RegularTessellations

Regular Tessellations

Regular Tessellations consist of only one type of regular polygon.

Do you remember what a regular polygon is?

A regular polygon is a shape in which all of the sides and angles are equal. Some examples are shown here:

Triangle Square Pentagon Hexagon Octagon

Regular Tessellations

Which regular polygons will fit together without overlapping or leaving gaps to create a Regular Tessellation?

Maybe you can guess which ones will tessellate just by looking at them. But, if you need some help, CLICK on each of the Regular Polygons below to determine which ones will tessellate and which ones won’t:

Triangle OctagonHexagonPentagonSquare

Once you’ve discovered whether each of the regular polygons tessellate or not, CLICK HERE to move on.

Does a Triangle Tessellate?

Regular Tessellations

The shapes fit together without overlapping or leaving gaps, so

the answer is YES.

Does a Square Tessellate?

Regular Tessellations

The shapes fit together without overlapping or leaving gaps, so

the answer is YES.

Does a Pentagon Tessellate?

Regular Tessellations

Gap

The shapes DO NOT fit together because there is a gap. So the

answer is NO.

Does a Hexagon Tessellate?

Regular Tessellations

The shapes fit together without overlapping or leaving gaps, so

the answer is YES.

Hexagon Tessellationin Nature

Does an Octagon Tessellate?

Regular Tessellations

The shapes DO NOT fit together because there are gaps. So the

answer is NO.

Gaps

Regular Tessellations

As it turns out, the only regular polygons that tessellate are:

TRIANGLES

SQUARES

HEXAGONS

Summary of Regular Tessellations:

Regular Tessellations consist of only one type of regular polygon. The only three regular polygons that will tessellate are the triangle, square, and hexagon.

Semi-Regular Tessellations

Semi-Regular Tessellations consist of more than one type of regular polygon. (Remember that a regular polygon is a shape in which all of the sides and angles are equal.)

How will two or more regular polygons fit together without overlapping or leaving gaps to create a Semi-Regular Tessellation? CLICK on each of the combinations below to see examples of these semi-regular tessellations.

Hexagon & Triangle Octagon &

Square

Square & Triangle Hexagon,

Square & Triangle

Once you’ve explored each of the semi-regular tessellations, CLICK HERE to move on.

Semi-Regular Tessellations

Hexagon & Triangle

Can you think of other ways to arrange these hexagons and triangles?

Semi-Regular Tessellations

Octagon & Square

Many floor tiles have these tessellating patterns.

Look familiar?

Semi-Regular Tessellations

Square & Triangle

Semi-Regular Tessellations

Hexagon, Square, & Triangle

Summary of Semi-Regular Tessellations:

Semi-Regular Tessellations consist of more than one type of regular polygon. You can arrange any combination of regular polygons to create a semi-regular tessellation, just as long as there are no overlaps and no gaps.

Semi-Regular Tessellations

What other semi-regular tessellations can you think of?

Translation

Reflection

Glide Reflection

Symmetry in Tessellations

The four types of Symmetry in Tessellations are:

CLICK on the four types of symmetry above to learn more. Once you’ve explored each of them, CLICK HERE to move on.

Rotation

Symmetry in Tessellations

RotationTo rotate an object means to turn it around. Every rotation has a center and an angle. A tessellation possesses rotational symmetry if it can be rotated through some angle and remain unchanged.

Examples of objects with rotational symmetry include automobile wheels, flowers, and kaleidoscope patterns.

CLICK HERE to view someexamples of rotational symmetry.

Back to Symmetry in Tessellations

Rotational Symmetry

Rotational Symmetry

Rotational Symmetry

Back to Rotations

TranslationTo translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance. A tessellation possesses translational symmetry if it can be translated (moved) by some distance and remain unchanged.

A tessellation or pattern with translational symmetry is repeating, like a wallpaper or fabric pattern.

Symmetry in Tessellations

CLICK HERE to view someexamples of translational symmetry.

Back to Symmetry in Tessellations

Translational Symmetry

Back to Translations

ReflectionTo reflect an object means to produce its mirror image. Every reflection has a mirror line. A tessellation possesses reflection symmetry if it can be mirrored about a line and remain unchanged. A reflection of an “R” is a backwards “R”.

Symmetry in Tessellations

CLICK HERE to view someexamples of reflection symmetry.

Back to Symmetry in Tessellations

Reflection Symmetry

Reflection Symmetry

Back to Reflections

Symmetry in Tessellations

Glide ReflectionA glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step. A tessellation possesses glide reflection symmetry if it can be translated by some distance and mirrored about a line and remain unchanged.

CLICK HERE to view someexamples of glide reflection symmetry.

Back to Symmetry in Tessellations

Glide Reflection Symmetry

Glide Reflection Symmetry

Back to Glide Reflections

Symmetry in Tessellations

Summary of Symmetry in Tessellations:

The four types of Symmetry in Tessellations are:

• Rotation

• Translation

• Reflection

• Glide Reflection

Each of these types of symmetry can be found in various tessellations in the world around us, including the artistic tessellations by M.C. Escher.

Exploring Tessellations

We have explored tessellations by learning the definition of Tessellations, and discovering them in the world around us.

Exploring Tessellations

We have also learned about Regular Tessellations, Semi-Regular Tessellations, and the four types of Symmetry in Tessellations.

Create Your Own Tessellation!

Now that you’ve learned all about Tessellations, it’s time to create your own.

You can create your own Tessellation by hand, or by using the computer. It’s your choice!

• CLICK on one of the links below. You will be connected to a website that will give you step-by-step instructions on how to create your own Tessellation.

• BOOKMARK the website so that you can come back to it later.

How to create a Tessellation

by Hand

How to create a Tessellation on the Computer

Once you’ve decided on whether your tessellation will be by hand or on the computer, and you have BOOKMARKED the website, CLICK HERE to move on.

Exploring Tessellations

Before you start creating your own Tessellation, either by hand or on the computer, let’s take one final look at some of the artistic tessellations by M.C. Escher. The following pieces of artwork should help give you Inspiration for your final project.

Good luck!

Resources

• “Totally Tessellated” from ThinkQuest.org

• Tessellations.com

• MathAcademy.com

• CoolMath.com

• MathForum.org

• ScienceU.com

• MathArtFun.com

• MCEscher.com

Click to end