Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Khun Phaen Escapes to Freedom

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

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.

Friezes

Stage: 3

Published January 2001,February 2011.

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?

The tyres:

The patterns that you find on car tyres are invariably based on one of the SEVEN (only) frieze patterns. Frieze patterns are those patterns that do not spread all over the plane (like a tessellation) but are confined to a strip. They are generated by a limited number of transformations - a translation, half turn, glide and reflection.

The seven frieze patterns are replicated below.

Can you describe what is different about each of these patterns?

Starting with a basic shape of your own can you generate patterns based on the seven friezes above?

How about looking at some of the tyre treads and see if you can match them to these patterns.

You may also like to design your tyre treads on a more complex motive than the b's, d's, p's and q's used above.