Wallpaper Objectives

1
Wallpaper Objectives

• Be able to classify a wallpaper pattern using
  centers of rotation and lines of reflection and
  glide reflection
• Be able to construct wallpaper patterns given a
  classification
• Be able to analyze cultural tilings using the
  classification

2
Selecting rotations

• Only 2-fold, 3-fold, 4-fold, and 6-fold rotations
  are possible.
• Why 5-fold rotation doesnt exist
• Find a pair, A and B, of 5-fold rotation centers
  which are as close together as possible.
• Rotate B about A and A about B to create another
  pair of 5-fold rotation centers that are even
  closer together.

3
Selecting basic cells

• A basic cell of a wallpaper pattern is one that
  has copies of the motif in all orientations.
• The basic cells tile the plane to form a lattice.

4
Rules for lattices

• If we have reflection or glide reflection lines,
  our lattice must be made of tiles that form two
  sets of parallel lines that are perpendicular.
• Show an example.
• Our lattices should be formed by parallelograms
  (if there is no reflection or glide reflection),
  or rectangles, rhombuses, or squares.

5
Rules for lattices, continued

• If we have at most 2-fold rotation, we use
  rectangles or rhombuses.
• If we have at most 4-fold rotation, we use
  squares.
• If we have at most 3- or 6-fold rotation, we use
  rhombuses that can be used to build hexagons.

6
Generating a wallpaper pattern

• The image on the right shows how to generate a
  cmm pattern (the motif is placed in the shaded
  area the double lines stand for reflection).

7
Generating a wallpaper pattern

• Repeat this process with these generators for p4g
  and pmg.
• When you have created your pattern, make sure you
  get the appropriate classification using the flow
  chart.