Tessellations

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Tessellations
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What Is a Tessellation?

• A shape or tile that repeats to fill a surface
  without any gaps or overlaps.
• The name comes from the word tessella, the small
  square tile used in ancient Roman mosaics.

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What is a tessellation?
Formal definition A careful juxtaposition of
elements into a coherent pattern sometimes called
a mosaic or tiling.
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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Patterns

• Very basic tessellations are simply a repeating
  pattern.
• The rule is that you have to make sure that the
  shapes fit together leaving no open space on the
  page.

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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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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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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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Reflections - mirror images
the alteration
flips here
Get practice template and try creating a
reflection.
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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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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 ...