Exploring Tessellations
This Exploration of Tessellations will
guide you through the following:
Definition of
Tessellation
Regular
Tessellations
Symmetry in
Tessellations
Tessellations
Around Us
Semi-Regular
Tessellations
View artistic
tessellations
by
M.C. Escher
Create your
own
Tessellation
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
Honeycombs
Floor Tiles
Textile
Patterns
Can you think of some more?
Checkerboards
Art
Are you ready to learn more about
Tessellations?
CLICK on each topic to
learn more…
Regular
Tessellations
Semi-Regular
Tessellations
Once you’ve explored each of the topics
above, CLICK HERE to move on.
Symmetry in
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
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
Square
Pentagon
Hexagon
Octagon
Once you’ve discovered whether each of the regular
polygons tessellate or not, CLICK HERE to move on.
Regular Tessellations
Does a Triangle Tessellate?
The shapes fit together without
overlapping or leaving gaps, so
the answer is YES.
Regular Tessellations
Does a Square Tessellate?
The shapes fit together without
overlapping or leaving gaps, so
the answer is YES.
Regular Tessellations
Does a Pentagon Tessellate?
Gap
The shapes DO NOT fit together
because there is a gap. So the
answer is NO.
Regular Tessellations
Does a Hexagon Tessellate?
The shapes fit together without
overlapping or leaving gaps, so
the answer is YES.
Hexagon Tessellation
in Nature
Regular Tessellations
Does an Octagon Tessellate?
Gaps
The shapes DO NOT fit together
because there are gaps. So the
answer is NO.
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
Look familiar?
Many floor tiles have these
tessellating patterns.
Semi-Regular Tessellations
Square & Triangle
Semi-Regular Tessellations
Hexagon, Square, & Triangle
Semi-Regular Tessellations
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?
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 you’ve explored each of them, CLICK HERE to move on.
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 some
examples of rotational symmetry.
Back to Symmetry in Tessellations
Rotational Symmetry
Rotational Symmetry
Rotational Symmetry
Back to Rotations
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 some
examples of translational symmetry.
Back to Symmetry in Tessellations
Translational Symmetry
Back to Translations
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 some
examples of reflection symmetry.
Back to Symmetry in Tessellations
Reflection Symmetry
Reflection Symmetry
Back to Reflections
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 some
examples 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, SemiRegular 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