Tessellations

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Tessellations
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Why Tessellations?
State of NJ - Core Curriculum Standards Mathematic
s - 4. 07 Geometric Properties Standard
State of NJ - Core Curriculum Standards Mathematic
s - 4. 11 Patterns Relationships
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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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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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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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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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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 ...