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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
armadillo armor
Tessellations are all around us:
Look for tessellations in walls, patios and
pavements.
Tessellations are all around us:
Common shapes can be
arranged in unusual ways
Sometimes an unusual
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
space
The plane is not filled
at the vertex, because
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
Make Your Own Escher
Tessellations 1
Step 1: Start with a simple
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.
Make Your Own Escher
Tessellations 1
Use this as a template to create your tessellation.
Make Your Own Escher
Tessellations 2
You could start with a hexagon and cut semi-circles
from three of the sides and fix them to the opposite
sides like this.
Make Your Own Escher
Tessellations 2
Make Your Own Escher
Tessellations 3
Start with a square and cut segments from two sides
and fix them to the opposite sides like this.
Make Your Own Escher
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
• Take your tessellation
template and trace it
on to a piece of paper.
• When tracing you can
use different types of
symmetry to change
your picture.
• Be sure that your
whole page is filled
with a repeating
pattern.
• Colour in your picture
creatively.
Elephants
Translation
symmetry
Elves and
Elephants
Combinations of
symmetry
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,
HyperCard, HyperStudio, Geometry Sketchpad,
Windows Paint and more.
Links
• Totally Tessellated - ThinkQuest winner - great site, instruction, information.
http://library.advanced.org/16661/
• Tessellations Tutorials - Math Forum site
http://forum.swarthmore.edu/sum95/suzanne/tess.intro.html - site for construction of tessellations.
http://forum.swarthmore.edu/sum95/suzanne/links.html - great list of tessellation links
• 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
• Other links from Forum.
http://forum.swarthmore.edu/library/topics/transform_g/
The end ...
Paper Folding Movie
Click to play the movie.