### Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

### The Frieze Tree

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

### Friezes

Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

# ...on the Wall

##### Stage: 3 Challenge Level:

This problem follows on from Mirror, Mirror...

You might find it helpful to copy this diagram onto squared paper .

Reflect the flag in one of the lines. Reflect the resulting image in the other line.
Can you describe the single transformation you would need to get from the first flag to the last flag?

Does it matter in which line you reflect first?

Try this with the flag in other positions.

Now try it with lines that meet at $45^{\circ}$ and at $60^{\circ}$ (you might find it helpful to use isometric paper for the $60^{\circ}$ case).

Again, try it with the flag in different positions

Can you predict what single transformation you would need to get from the first flag to the last flag if the lines meet at $\theta^{\circ}$?
Can you prove your answer?

If you have enjoyed this problem, you may like to have a go at Who is the fairest of them all? .