Identifying Frieze Patterns Exploration

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Time-40.svg

Objective: Become familiar with identifying border patterns as well as creating them.

Notation

You may use the notation in the text, or use this notation:

The border patterns can be given fairly simple names consisting of 2 (2-letter) symbols. The first symbol is either an 'm' of a '1' depending on if the border pattern has a vertical line of symmetry. The second symbol is an 'm', 'a', '2' or '1'. The correct symbol is determined by a list of questions that need to be asked in order.

Determining the symmetry group can then be accomplished by following this set of questions (we assume that the border patterns run from the left to the right, so that the terms horizontal and vertical are unambiguous):

I. Are there vertical lines of reflection?

Yes -> the symmetry group starts with an M.

No -> the symmetry group starts with a 1.

II. Is there a horizontal line of reflection?

Yes -> the symmetry group ends with an M.

No -> next question

Is there a glide reflection?

Yes -> the symmetry group ends with an G.

No -> next question

Is there 2-fold rotational symmetry?

Yes -> the symmetry group ends with a 2.

No -> the symmetry group ends with a 1.


The correspondance between this system and the one in the text (the IUC notation) is as follows:

  • 1M = p1m1
  • 1G = p1a1
  • 12 = p112
  • 11 = p111
  • MM = pmm2
  • MG = pma2
  • M1 = pm11

Questions

  1. What is the symmetry group for the following border pattern: ... FFFFFFFFFFFFFFFFFFFFFFF...
  2. You can form all 7 border patterns if you start with F. Show the other 6.
  3. What is the symmetry group for the following border pattern: ... BBBBBBBBBBBBBBBBBBBB...
  4. You can form all 7 border patterns if you start with B. Show the other 6.
  5. What is the symmetry group for the following border pattern: ... OOOOOOOOOOOOOOOOOO ...
  6. You can form all 7 border patterns if you start with O. Show the other 6.