Irregular Tiling

1
Irregular Tiling

• Our goal find ways to generate irregular tilings
  from regular or simple irregular tilings
• Methods so far
• parallel translation
• glide reflection

2
Midpoint Rotation

• Start with any tiling using quadrilaterals or
  triangles. (Note we can tile a plane with any of
  these.)
• Pick a side and find its midpoint.
• Given any shape on one side of the midpoint,
  rotate it through 180 degrees to form the other
  side.

3
Escher Example
4
Side Rotation

• Start with any regular tiling.
• Cut out a shape on one side, rotate it to an
  adjacent side using the endpoint.

5
Escher Example
6
Practice

• Using the triangular grid paper
• create an irregular tiling using midpoint
  rotation on one side of an equilateral triangle
• create an irregular tiling using side rotation on
  a hexgon
• Given the patterns, try to identify which method
  from today was used to create it.

7
Conway Criterion

• A loop without crossing and punctures will tile
  the plane if you can divide it into 6 arcs with
  endpoints A, B, C, D, E, and F where
• the arc AB is the parallel translate of ED
• arcs BC, CD, EF, and FA have rotational symmetry
  about their midpoints

8
Example

• Explain why the below shape fits the Conway
  Criterion.
• Check out more examples at http//www.mpls.k12.mn.
  us/rrumppe/ConTess.html