Tessellations

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Tessellations
Lesson 10-4
Images from http//library.thinkquest.org/16661/si
mple.of.regular.polygons/regular.1.html
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A tessellation is a design or pattern in which a
shape is used repeatedly to cover a plane with no
gaps, overlaps, or empty spaces.
Tessellations
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Escher
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Escher
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Escher
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Tessellations

• A regular tessellation is a pattern made with
  only one type of regular polygon.

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The sum of the measures surrounding a point (or
vertex) must be 360.
Tessellations
90?
90?
90?
90?
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Only regular polygons that have an interior angle
which is a factor of 360 will tessellate.
Tessellations
360/120 3
120?
120?
120?
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A tessellation is a design or pattern in which a
shape is use repeatedly to cover a plane with no
gaps, overlaps, or empty spaces.
Tessellations

• 1. A regular tessellation is a pattern made with
  only one type of regular polygon.
• 2. The sum of the measures surrounding a point
  (or vertex) must be 360.
• 3. Only regular polygons that have an interior
  angle which is a factor of 360 will tessellate.
• 4. No regular polygon with more than 6 sides can
  be used in a regular tessellation.

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Can these figures form a regular tessellation?
Yes. This is a regular polygon with a 60
interior angle which is a factor of 360. (360/60
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Yes. This is a regular polygon with a 90
interior angle which is a factor of 360. (360/90
4)
Yes. This is a regular polygon with a 120
interior angle, which is a factor of 360.
(360/120 3)
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Regular triangles 360/60 6
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Regular quadrilaterals 360/90 4
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Regular hexagons 360/120 3
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Can these figures form a regular tessellation?
No. Although this is a regular polygon, it has an
interior angle 135, which is not a factor of
360
No. This is not a regular polygon. It can
tessellate but not in a regular tessellation.
No. This is not a regular polygon. It can
tessellate but not in a regular tessellation.
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Regular octagons 360/135 2.67
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Regular pentagons 360/108 3.33
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Regular heptagons 360/128.57 2.8
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How about these for regular tessellation?

• 1. a 20-sided figure?
• No, because its interior angle is 162, which is
  not a factor of 360.
• (Interior angle measure 180(20 - 2) 162
• 
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• 2. a 10-sided figure?
• No, the interior angle is 144 (not a factor of
  360).
• 3. a 12-sided figure?
• No, the interior angle is 150 (not a factor of
  360).
• Note No regular polygon with more than six
  sides can be used in a regular tessellation.
  

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Semi-regular Tessellations

• If the same combination of regular polygons meet
  at each vertex, it is called a semi-regular
  tessellation.

Notice the regular octagons with interior angles
of 135 and the squares with 90. At each vertex
or point, there is a sum of 135 135 90 360.
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Irregular Tessellations

• Other figures can make tessellations which are
  irregular. The figures used are irregular
  polygons and may be the same or different types.
  
• Here is an irregular tessellation made with kites
  and one with trapezoids.

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Make a special tessellation!

• 1. Begin with a rectangle.
• 2. Cut a piece out of it and stick on another
  side.
• 3. Translate the new figure to create a
  tessellation.

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Or another . . .

• Start with a triangle
• 2. Cut out a piece of it and slide it to another
  side.
• Slide and reflect the figure repeatedly
• to create a tessellation.

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Special Notes on Tessellations

• 1. At each vertex of a tessellation, the sum of
  the measures of the angles must equal 360.
• 2. Any quadrilateral will tessellate.
• 3. Combinations of figures can be used to
  tessellate.
• 4. Only equilateral triangles, squares, and
  regular hexagons can make regular tessellations.

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Images are taken from http//www.tessellations.o
rg http//www.howe-two.com/nctm/tessellations/ex
amples/ http//library.thinkquest.org/16661/simpl
e.of.regular.polygons/regular.1.html