Tessellations

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Tessellations
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Tessellation

• A tessellation or a tiling is a way to cover a
  floor with shapes so that there is no overlapping
  or gaps.

• Remember the last jigsaw puzzle piece you put
  together? Well, that was a tessellation. The
  shapes were just really weird.

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Examples

• Brick walls are tessellations. The rectangular
  face of each brick is a tile on the wall.

• Chess and checkers are played on a tiling. Each
  colored square on the board is a tile, and the
  board is an example of a periodic tiling.

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Examples

• Mother nature is a great producer of tilings. The
  honeycomb of a beehive is a periodic tiling by
  hexagons.

• Each piece of dried mud in a mudflat is a tile.
  This tiling doesn't have a regular, repeating
  pattern. Every tile has a different shape. In
  contrast, in our other examples there was just
  one shape.

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Alhambra

• The Alhambra, a Moor palace in Granada, Spain, is
  one of todays finest examples of the
  mathematical art of 13th century Islamic artists.

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Tesselmania

• Motivated by what he experienced at Alhambra,
  Maurits Cornelis Escher created many tilings.

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Regular tiling

• To talk about the differences and similarities of
  tilings it comes in handy to know some of the
  terminology and rules.

• Well start with the simplest type of tiling,
  called a regular tiling.It has three rules

1. The tessellation must cover a plane with no gaps
   or overlaps.
2. The tiles must be copies of one regular polygon.
3. Each vertex must join another vertex.

• Can we tessellate using these game rules? Lets
  see.

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Regular tiling

• Tessellations with squares, the regular
  quadrilateral, can obviously tile a plane.

• Note what happens at each vertex. The interior
  angle of each square is 90º. If we sum the angles
  around a vertex, we get 90º 90º 90º 90º
  360º.

• How many squares to make 1 complete rotation?

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Regular tiling

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

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Triangles

• Triangles?

• 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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Semiregular tiling

• A semiregular tiling has the same game rules
  except that now we can use more than one type of
  regular polygon.

• To name a tessellation, work your way around one
  vertex counting the number of sides of the
  polygons that form the vertex.

• Go around the vertex such that the smallest
  possible numbers appear first.

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Semiregular tiling

• Here is another example made from three triangles
  and two squares

• There are only 8 semiregular tessellations, and
  weve now seen two of them the 4.6.12 and the
  3.3.4.3.4

• Your in-class construction will help you find the
  remaining 6 semiregular tessellations.

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Demiregular tiling

• The 3 regular tessellations (by equilateral
  triangles, by squares, and by regular hexagons)
  and the 8 semiregular tessellations you just
  found are called 1-uniform tilings because all
  the vertices are identical.

• If the arrangement at each vertex in a
  tessellation of regular polygons is not the same,
  then the tessellation is called a demiregular
  tessellation.

• If there are two different types of vertices, the
  tiling is called 2-uniform. If there are three
  different types of vertices, the tiling is called
  3-uniform.

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Examples

• There are 20 different 2-uniform tessellations of
  regular polygons.

3.4.6.4 / 4.6.12
3.3.3.3.3.3 / 3.3.3.4.4 / 3.3.4.3.4
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Summary

• Regular Tessellation
• Only one regular polygon used to tile
• Semiregular Tessellation
• Uses more than one regular polygon
• Has the same pattern of polygons AT EVERY VERTEX
• Demiregular Tessellation
• Uses more than one regular polygon
• Has DIFFERENT patterns of polygons used at
  vertices
• Must name all different patterns.

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Name the Tessellation
Regular? SemiRegular? DemiRegular?
SemiRegular 4.6.12
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Name the Tessellation
Regular? SemiRegular? DemiRegular?
Demiregular 3.12.12/3.4.3.12
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Name the Tessellation
Regular? SemiRegular? DemiRegular?
Demiregular 3.3.3.3.3.3/3.3.4.12
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Name the Tessellation
Regular? SemiRegular? DemiRegular?
DemiRegular 3.6.3.6/3.3.6.6
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Name the Tessellation
Regular? SemiRegular? DemiRegular?
SemiRegular 3.3.4.3.4