# Tessellations What is a tessellation If many copies of a shape can be used to cover a surface without leaving any gaps between them then we say that the shape will tessellate T

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```					Tessellations
What is a tessellation?
If many copies of a shape can be used to cover a
surface, without leaving any gaps between them, then
we say that the shape will tessellate.
The pattern that is formed is called a tessellation.

Simply put: A tessellation is like
a puzzle that repeats a
particular pattern.

For example:
In nature and life, they appear often:

Honeycombs...

Mud flats...

In games, like checkers..
Tessellations in Nature

snake skin

spider web
Tessellations are all around us:

Look for tessellations in walls, patios and
pavements.
Tessellations are all around us:

Common shapes can be       Sometimes an unusual
arranged in unusual ways   shape will tessellate
Tessellations are all around us:

Sometimes 2 or more different shapes will tessellate.
Tessellations
Around the House
• Tessellations
can be found
in quilts,
floor tiling,
and
wallpaper.
Study of Science
X-ray Crystallography

This picture depicts the
transformation of eight
points in an array to make
a very small crystal lattice,
which tessellates.
more science….

This image suggests the
relationship between
tessellations, symmetry,
and x-ray crystallograpy.
and even more
science….

Hydrogen Peroxide,
among other
chemicals, has this
tessellating, crystal
structure.
Math in Tessellations
• Many different geometric shapes can be used
to create tessellations.

• Which of the shapes below would fit with
itself to make a good tessellation design?
The mathematics of the
Regular Tessellation...

• A regular polygon tessellation is constructed from
regular polygons.

• Regular polygons have equal sides and equal angles.

• The regular polygons must fill the plane at each
vertex, with repeating patterns and no overlapping
pieces.

Note: this pentagon does not
fit into the vertex… therefore
it is not a regular tessellation.
Regular tiling

• Which other regular
polygons do you
think can tile the
plane?
Triangles

• Triangle?
• Yep!
• How many
triangles to
make 1 complete
rotation?

• The interior angle of every equilateral triangle is 60º. If we
sum the angles around a vertex, we get 60º + 60º + 60º +
60º + 60º + 60º = 360º again!.
Pentagons

• Will pentagons work?

• The interior angle of a pentagon is 108º, and 108º + 108º
+ 108º = 324º.
Hexagons

• Hexagons?
• The interior angle is 120º, and 120º + 120º +
120º = 360º.
• How many hexagons to make 1 complete
rotation?
Heptagons

• Heptagons?
Octagons?
• Not without getting overlaps.
In fact, all polygons with more than six sides will
overlap.
Regular tiling

• So, the only regular polygons that tessellate
the plane
are triangles, squares and hexagons.

• That was an easy game. Let’s make it a bit
more rewarding.
A famous mathematician, Kepler studied
tessellations and noted the regular
tessellations or (tilings) of the plane.

There are only three regular tessellations;

one of triangles

one of squares

and one of hexagons.
This is NOT a regular polygon tessellation,
because...

vertex

The plane is not filled
at the vertex, because
space              there is space left over.

A regular polygon tessellation, can be changed using
“alterations” to the sides of the polygon. These
alterations are called transformations.
Three Common Transformations

1. Translation, which is a slide of one side of the
polygon.

2. Reflection, which is a flip or mirror image of one
side of the polygon.

3. Rotation, which is a turn of a side around one
vertex of the polygon.
Translations - a slide
this side

the alteration           moves here

Get practice template and try creating a translation….
Let’s try together….. a translation or slide….
Reflections - mirror images

the alteration

flips
here

Get practice template and try creating a reflection….
Let’s try together….. a reflection, flip or mirror image
Rotations - turns

the alteration

here
rotates around
this vertex

Get practice template and try creating a rotation...
Let’s try together….. a rotation or turn
Tessellations by M.C. Escher
Tessellations 1
shape that will tessellate
e.g. a rectangle.

Step 2: Remove a shape or
shapes from one side of the
rectangle and fix them to the
opposite side.
Tessellations 1
Use this as a template to create your tessellation.
Tessellations 2
from three of the sides and fix them to the opposite
sides like this.
Tessellations 2
Tessellations 3
and fix them to the opposite sides like this.
Tessellations 3
Professional Tessellation
Designs

Sun and Moon                          Fish
M.C. Escher                           Robert Ingalls

• Many artists have created master works of art
using the simple rules of tessellations.
• M.C. Escher and Robert Ingalls are among many
tessellation artists.
• Can you you spot the repeating shape?
You Can Be a
Tessellation Artist

You need to first create a tessellating template.
• Step 1: using grid paper cut out a perfect square.
• Step 2: with a pencil make a line on two adjacent
sides of the square. Be sure not to draw through a
corner.
• Step 3: cut out the pieces and attach them with
tape to the opposite sides from where they came.
 You have now created your tessellation template.
Seeing a Figure
• Looking at your tessellation template, study the
sides and the shape to decide what sort of figure
you might see.
• The person who designed this form saw an
elephant and an elf.
• Are there any other different forms that you
might see?
• When looking for the shape be sure not to see too
much detail as it can make the art work too
crowded.
Putting It All Together
template and trace it
on to a piece of paper.   Elephants

• When tracing you can      Translation
symmetry
use different types of
symmetry to change
• Be sure that your
whole page is filled                    Elves and
Elephants
with a repeating
pattern.                                Combinations of
symmetry
creatively.
Extras     ...
Is this a regular tessellation?

What kind of tessellation is this?

Tessellation internet activity

Cool site for interactive Javascript tessellations.

Constructing tessellations - Claris/Appleworks,
Windows Paint and more.
• Totally Tessellated - ThinkQuest winner - great site, instruction, information.

• Tessellations Tutorials - Math Forum site
http://forum.swarthmore.edu/sum95/suzanne/tess.intro.html - site for construction of tessellations.

• Math. Com - List of good tessellation links
http://test.math.com/students/wonders/tessellations.html

• World of Escher site - commerical site with gallery of Dutch artist,Escher who was
famous for his tessellation art.
http://WorldOfEscher.com/gallery/

• Science University’s Tilings Around Us Site.
http://www.ScienceU.com/geometry/articles/tiling/tilings.html

http://forum.swarthmore.edu/library/topics/transform_g/

The end ...
Paper Folding Movie

Click to play the movie.

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