This Exploration of Tessellations will guide you through the following: Exploring Tessellations Definition of Tessellation Semi-Regular Tessellations.

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?


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


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

Does a Triangle Tessellate?


The shapes fit together without overlapping or leaving gaps, so

the answer is YES.

Does a Square Tessellate?



the answer is YES.

Does a Pentagon Tessellate?


Gap

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

answer is NO.

Does a Hexagon Tessellate?



the answer is YES.

Hexagon Tessellationin Nature

Does an Octagon Tessellate?


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

answer is NO.

Gaps

Figures that Tessellate

• Find the measure of an angle of a regular polygon using the following formula

• If is a factor of 360, then the n-gon will tessellate


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.

Are you ready to learn more about Tessellations?




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


Hexagon & Triangle

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


Octagon & Square

Many floor tiles have these tessellating patterns.

Look familiar?


Square & Triangle


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.


What other semi-regular tessellations can you think of?

Translation

Reflection

Glide Reflection

Symmetry in Tessellations

The four types of Symmetry in Tessellations are:

Rotation


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.


CLICK HERE to view someexamples of translational symmetry.


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”.


CLICK HERE to view someexamples of reflection symmetry.


Reflection Symmetry

Reflection Symmetry

Back to Reflections


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.


Glide Reflection Symmetry

Glide Reflection Symmetry

Back to Glide Reflections


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.


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


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!

* He was born Maurits Cornelis Escher in 1898, in Leeuwarden, Holland.

M.C. Escher developed the tessellating shape as an art form

*Escher was a graphic artist, who specialized in woodcuts and lithographs.

* His father wanted him to be an architect, but bad grades in school and a love of drawing and design led him to a career in the graphic arts.

His interest began in 1936, when he traveled to Spain and saw the tile patterns used in the Alhambra.

Escher saw tile patterns that gave him ideas for his art work

Alhambra Palace

* The Alhambra is a walled city and fortress in Granada, Spain. It was built during the last Islamic Dynasty (1238-1492).

* The palace is lavishly decorated with stone and wood carvings and tile patterns on most of the ceilings, walls, and floors.

The Alhambra Palace is afamous example ofMoorish architecture.It may be the most wellknown Muslim construction.

Islamic art does not usuallyuse representations of living beings, but usesgeometric patterns,especially symmetric(repeating) patterns.

By “distorting” the basic shapes he changed them into animals, birds, andother figures.The effect can beboth startling and beautiful.

Escher Horses

Lets make a simple tessellating shape

Begin with a simple geometric shape - the square

Change the shape of one side

Copy this line on the opposite side

Rotate the line and repeat it on the remaining edges

Erase the original shape

Add lines to the inside of the shapes to turn them into

pictures.

Add color to enhance your picture.

By repeating your shape you create a tessellated picture

This Exploration of Tessellations will guide you through the following: Exploring Tessellations Definition of Tessellation Semi-Regular Tessellations.

Documents

regular tessellations

regular polygons

escher tessellations

exploration of tessellations

type of regular polygon

square triangle slide

nature slide

hexagon tessellation