Semiregular Tiling

1
Semiregular Tiling

• Our goal find all the semiregular tilings
• Our strategy first, find all the ways you can
  surround a point with regular polygons
• Review Rules 1 and 2

2
Rules 3-5

• Rule 3 You cannot use four different types of
  polygons.
• Rule 4 You cannot have k.n.m if k is odd and m
  and n are different.
• Rule 5 You cannot have 3.k.n.m unless km.

3
Possible Configurations

• We already know three configurations 6.6.6,
  4.4.4.4, and 6.6.6.6.6.6.
• The book gives us 3.7.42.
• You will be figuring out the rest in problems 7-9.

4
Creating the tilings

• Given a configuration, we then determine how many
  tilings we can create from it.
• Examples
• 3.7.42 produces no tilings
• One configuration you will find is 3.3.3.4.4. It
  produces two tilings.
• There are 8 semiregular tilings overall - we have
  found 3 of them.

5
Irregular Tilings

• Note that we can easily form irregular tilings
  with rectangles and parallelograms.
• Our goal is to find ways to take regular or
  simple irregular tilings and produce more complex
  ones.

6
Parallel Translation

• Start with a square, rectangular, parallelogram,
  or hexagonal grid.
• Cut a shape along one side and translate it to
  the opposite side.
• This process can be repeated for other pairs of
  parallel sides.

7
Escher Example
8
Glide Reflection

• Start with a tiling where the basic shape has a
  pair of parallel sides.
• Cut out a shape on one side, flip it over and
  attach to the other side.

9
M. C. Escher Example