Date: Fri, 14 Aug 1998 11:02:06 -0600 (CST)
Subject: Frieze patterns proof

Name: Brian
Level: All

Question:
I am doing a research project on frieze patterns (also called band patterns or border patterns). I know that there are exactly seven different types of frieze patterns, but I have been unable to find a proof for this. If you could tell me this proof, or tell me some internet sites on which I can find material on this, I would be very grateful.

Sincerely,
Brian Bairstow

Hi Brian

Proof outlines can be found in lots of recent books dealing with geometry. For example, Coxeter, INTRODUCTION TO GEOMETRY (1969) section 3.7 (pp. 47-49).
But you can prove it yourself faster than looking up the proof.

There are only 5 types of isometries that map a horizontal strip to itself:

1. a translation
2. a half-turn (i.e. rotation through 180 degrees)
3. a reflection in a vertical mirror
4. a reflection in a horizontal mirror
5. a glide reflection.
Each of the five generates a frieze group. By putting pairs of them together, you will find only two further possibilities.

I hope this helps.
Chris

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