Tessellations

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Tessellations
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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
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In nature and life, they appear often
Honeycombs...
Mud flats...
In games, like checkers..
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Tessellations in Nature
snake skin
spider web
armadillo armor
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Tessellations are all around us

• Look for tessellations in walls, patios and
  pavements.

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Tessellations are all around us
Sometimes an unusual shape will tessellate
Common shapes can be arranged in unusual ways
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Tessellations are all around us
Sometimes 2 or more different shapes will
tessellate.
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Tessellations Around the House

• Tessellations can be found in quilts, floor
  tiling, and wallpaper.

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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.
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more science.
This image suggests the relationship between
tessellations, symmetry, and x-ray crystallograpy.
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and even more science.
Hydrogen Peroxide, among other chemicals, has
this tessellating, crystal structure.
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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?

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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.
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Regular tiling

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

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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!.

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Pentagons

• Will pentagons work?

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

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Hexagons

• Hexagons?

• The interior angle is 120º, and 120º 120º
  120º 360º.

• How many hexagons to make 1 complete rotation?

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Heptagons

• Heptagons? Octagons?

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Regular tiling

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

• That was an easy game. Lets make it a bit more
  rewarding.

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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.
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This is NOT a regular polygon tessellation, becaus
e...
vertex
The plane is not filled at the vertex, because
there is space left over.
space
A regular polygon tessellation, can be changed
using alterations to the sides of the polygon.
These alterations are called transformations.
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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.
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Translations - a slide
this side
the alteration
moves here
Get practice template and try creating a
translation.
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Lets try together.. a translation or slide.
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Reflections - mirror images
the alteration
flips here
Get practice template and try creating a
reflection.
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Lets try together.. a reflection, flip or
mirror image
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Rotations - turns
the alteration
here
rotates around this vertex
Get practice template and try creating a
rotation...
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Lets try together.. a rotation or turn
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Tessellations by M.C. Escher
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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.
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Make Your Own Escher Tessellations 1
Use this as a template to create your
tessellation.
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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.
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Make Your Own Escher Tessellations 2
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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.
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Make Your Own Escher Tessellations 3
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Professional Tessellation Designs
Sun and Moon M.C. Escher
Fish 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?

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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.

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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.

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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
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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.
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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.int
  ro.html - site for construction of
  tessellations. http//forum.swarthmore.edu/sum95/s
  uzanne/links.html - great list of tessellation
  links

• Math. Com - List of good tessellation links
• http//test.math.com/students/wonders/tessellation
  s.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 Universitys Tilings Around Us Site.
  http//www.ScienceU.com/geometry/articles/tiling/t
  ilings.html

• Other links from Forum.
• http//forum.swarthmore.edu/library/topics/transfo
  rm_g/

The end ...
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Paper Folding Movie
Click to play the movie.