# Identifying Frieze Patterns Exploration

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

## Materials

- Printed copy of the Identifying Frieze Patterns Exploration.

## Exploration

### Crystallographic Notation

The crystallographic notation for the frieze patterns is made up of four letters/numbers. The label always starts with a P, and the rest of the label is determined by the symmetries.

P | M or 1 | M, a or 1 | 2 or 1 |

The first symbol is always a P | Vertical mirror line gives and M, otherwise we have 1 |
If the axis is a mirror line we get M, if there is a glide reflection we get a, otherwise we have a 1 |
Two fold rotation gives a 2, otherwise we get a 1 |

### Alternative Notation

The border patterns can be given fairly simple names consisting of 2 symbols. The first symbol is either 'M' or '1', depending on if the border pattern has a vertical line of symmetry. The second symbol is 'M', 'G', '2' or '1', depending on what other symmetries are present. (This notational system is derived from these meanings: M designates a mirror symmetry, G a glide-reflection symmetry, 2 a 2-fold rotation symmetry, and 1 the absence of a symmetry.)

Determining the symmetry group can then be accomplished by following a 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). The order of the questions is important!

M or 1 | M, G, 2 or 1 |

Vertical mirror line gives and M, otherwise we have 1 |
Is the axis is a mirror line? - M; Is there is a glide reflection? - G; Is there a 2-fold rotation? - 2; otherwise we get a 1 |

The correspondence 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

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