Frieze or border Patterns

1
Frieze (or border) Patterns

• We will focus on repeating patterns within a
  strip.
• What is the minimum portion of the strip (or the
  motif) needed to recreate the entire strip?

Incan Strip Patterns
2
Isometries

• What operations can we use on a motif to create a
  strip?
• translation
• glide reflection (across the center line)
• reflection (across the center line or a vertical
  line)
• rotation (about 180 degrees)
• These are called isometries - the rigid motions
  that do not alter lengths, angles, or areas.

3
Examples

• Start with the motif of the letter P.
• Determine what patterns you get if you use
• translation only
• glide reflection only
• reflection across the center line
• reflection across the center line
• Note that translation is always used!

4
All the possibilities

• The remaining three possibilities are
• rotation by 180 degrees only
• reflection across the center line, followed by
  rotation
• reflection across a vertical line, followed by
  rotation
• Note why are these the only 7? Every other
  pairing of isometries results in one of these.
  (Example two glide reflections)

5
Crystallographic notation

• The first letter is always p.
• The second letter is m if there is a vertical
  line of reflection, 1 otherwise.
• The third letter is m if the center line is a
  line of reflection, g if the center line is not a
  line of reflection but is used for glide
  reflection, and 1 otherwise.
• The fourth letter is 2 if there is rotation, 1
  otherwise.