This Exploration of Tessellations will guide you through the following:

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Exploring Tessellations
This Exploration of Tessellations will guide you
through the following
Definition ofTessellation
Symmetry inTessellations
RegularTessellations
Semi-RegularTessellations
TessellationsAround Us
Create yourownTessellation
View artistictessellationsbyM.C. Escher
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What is a Tessellation?
A Tessellation is a collection of shapes that fit
together to cover a surface without overlapping
or leaving gaps.
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Tessellations in the World Around Us
Brick Walls
Floor Tiles
Checkerboards
Honeycombs
Textile Patterns
Art
Can you think of some more?
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Are you ready to learn more about Tessellations?
CLICK on each topic to learn more
Regular Tessellations
Semi-RegularTessellations
Symmetry inTessellations
Once youve explored each of the topics above,
CLICK HERE to move on.
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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
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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 wont
Triangle
Octagon
Hexagon
Pentagon
Square
Once youve discovered whether each of the
regular polygons tessellate or not, CLICK HERE to
move on.
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Regular Tessellations
Does a Triangle Tessellate?
The shapes fit together without overlapping or
leaving gaps, so the answer is YES.
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Regular Tessellations
Does a Square Tessellate?
The shapes fit together without overlapping or
leaving gaps, so the answer is YES.
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Regular Tessellations
Does a Pentagon Tessellate?
Gap
The shapes DO NOT fit together because there is a
gap. So the answer is NO.
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Regular Tessellations
Does a Hexagon Tessellate?
The shapes fit together without overlapping or
leaving gaps, so the answer is YES.
Hexagon Tessellationin Nature
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Regular Tessellations
Does an Octagon Tessellate?
Gaps
The shapes DO NOT fit together because there are
gaps. So the answer is NO.
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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.
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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
Square Triangle
Hexagon, Square Triangle
Octagon Square
Once youve explored each of the semi-regular
tessellations, CLICK HERE to move on.
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Semi-Regular Tessellations
Hexagon Triangle
Can you think of other ways to arrange these
hexagons and triangles?
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Semi-Regular Tessellations
Octagon Square
Look familiar?
Many floor tiles have these tessellating patterns.
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Semi-Regular Tessellations
Square Triangle
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Semi-Regular Tessellations
Hexagon, Square, Triangle
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Semi-Regular Tessellations
Summary of Semi-Regular Tessellations Semi-Regula
r 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?
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Symmetry in Tessellations
The four types of Symmetry in Tessellations are
Rotation
Translation
Reflection
Glide Reflection
CLICK on the four types of symmetry above to
learn more. Once youve explored each of them,
CLICK HERE to move on.
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Symmetry in Tessellations
Rotation To 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
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Rotational Symmetry
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Rotational Symmetry
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Rotational Symmetry
Back to Rotations
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Symmetry in Tessellations
Translation To 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.
Back to Symmetry in Tessellations
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Translational Symmetry
Back to Translations
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Symmetry in Tessellations
Reflection To 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.
Back to Symmetry in Tessellations
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Reflection Symmetry
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Reflection Symmetry
Back to Reflections
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Symmetry in Tessellations
Glide Reflection A 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
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Glide Reflection Symmetry
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Glide Reflection Symmetry
Back to Glide Reflections
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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.

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Exploring Tessellations
We have explored tessellations by learning the
definition of Tessellations, and discovering them
in the world around us.
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Exploring Tessellations
We have also learned about Regular Tessellations,
Semi-Regular Tessellations, and the four types of
Symmetry in Tessellations.
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Create Your Own Tessellation!

• Now that youve learned all about Tessellations,
  its time to create your own.
• You can create your own Tessellation by hand, or
  by using the computer. Its 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 youve decided on whether your tessellation
will be by hand or on the computer, and you have
BOOKMARKED the website, CLICK HERE to move on.
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Exploring Tessellations
Before you start creating your own Tessellation,
either by hand or on the computer, lets 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!
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Resources

• Totally Tessellated from ThinkQuest.org
• Tessellations.com
• MathAcademy.com
• CoolMath.com
• MathForum.org
• ScienceU.com
• MathArtFun.com
• MCEscher.com

Click to end