Mathematics Project

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Mathematics Project
Escher

• The master of tessellation

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Biography of Escher
Maurits Cornelis Escher was born on June 17, 1898
in Leeuwarden,Netherlands. His works involve
repeated patterns and regular divisions of the
plane, impossible constructions, and infinite
space.
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• Though his early work tended toward realistic
  portrayals of the landscape and architecture
  observed during his travels, it reflected a
  greater fascination for structural constructs
  than for the landscape itself.
• Through his wondrous creations, Escher was able
  to lay a symbolic bridge between the realms of
  art and science.

Tessellations
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What Is a Tessellation? 
The plane can be covered with a pattern in such a
way as to leave no region uncovered.No matter the
pattern how it is looked like,but it must cover
the whole plane without any gaps.Then it will be
formed a tessellation.
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Tessellation of Regular Polygon

• If the interior angle of a polygon can exactly
  divide 360o(whole circle), it can form a
  tessellation.
• The interior angles of equilateral triangle,
  square, regular hexagon are 60o, 90o and 120o
  respectively. They can exactly divide 360o. So
  than they can form a tessellation.

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They can form tessellation.
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a tessellation of triangles
a tessellation of squares
a tessellation of hexagons
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Other polygons
BUT
Some gaps here.
Overlap!
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How to create a tessellation?
Step One

• Start with a square. Select your favorite color.

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Step Two

• Start at the top corner, wiggle around as you are
  going down and exit the lower corner of the
  square. A section of the square will be selected.
  

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Step Three

• Cut a shape form one side of a tile, tape it on
  the opposite side directly across from the cut
  out.

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Step Four

• Repeat the process but this time go from top to
  bottom or bottom to top.

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Step Five

• Cut it out and replace it to the
  bottom.(translating it to the opposite side.)
• Look at the figure. What does it look like to
  you?

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Step Six

• Use the pencil or paint brush (with contrasting
  colors selected) to decorate the figure. Do not
  add anything to the outer edges of the silhouette.

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Step Seven

• Copy the figure and place beside it,then change
  the color of the second figure that you have been
  formed already.

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Step Eight

• Continue sliding and positioning first one figure
  and then the second until you have one row
  completed.

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Step Nine

• "copy" by sliding one figure to the side. Lasso
  the other figure, fit it into position up against
  the darker figure.
• If you cant do it, continue this procedure
  until
• you have created a tessellation.

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Sky Water
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Day and Night
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Lizards
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Fishes
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Fish and Boats
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SUN AND MOON
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butterfly
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Example 1 Tessellation by using squares and
translation
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Example 2 Tessellation by using equilateral
triangles and rotation
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Example 2 Tessellation by using squares and
reflection
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Wa Ying College F.2 Mathematics Assessment 2009
2010 Tessellation

• Tasks
• Students have to study the power point
  presentation
• refer to web page http//www.waying.edu.hk/ma
  thematics
• 2 a) Students have to create 1 tessellation
  picture.
• b) Students have to explain how they alter and
  transform the geometric figures of the
  tessellation .

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• Content
• The project should contain the following parts
• (a) Cover (with title, name, class and
  class number)
• (b) A table showing the transformation of
  the geometric figures of the tessellation
  .

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(c) 4 paper models of the new figure for
tessellation
Parts (b) and (c) should be handed in on 25th
November, 2009.
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(d) A full paged tessellation drawing with
colour.
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Marks Allocation

• This is an individual assignment.
• The tessellation should be presented on a new A4
  size paper. The final tessellated patterns in
  should fill up the whole page.
• The report should be in English.

Date of Submission 9th December, 2009
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Requirements

• The total marks of this assignment is 10.
• Marks are allocated as follow
• Tessellation pictures 4 marks
• Explanation (in table form) 5 marks
• Presentation (eg cover, binding and tidiness)
  .. 1 mark