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

Age 11 to 14 Challenge Level:

Why do this problem?

This problem follows on from Mirror, Mirror... but it works equally well on its own. It could be used as extension work for students learning to draw reflections accurately.

Alternatively, this problem and the two related problems Mirror, Mirror... and Who Is the Fairest of Them All? could form a unit of work on combined transformations. All three problems ask students to consider the effect of combining two transformations, and then challenge them to describe the single transformations that produce the same results.

Possible approach

These printable resources may be useful: ...on the Wall,
Isometric Paper,
Square Paper.

As an extension task, all that is needed is to provide the problem verbally or as a worksheet to pairs of students who could then make sense of it together. When they have established the combined transformation for one specific example, a teacher intervention may be appropriate, to move the focus to the general case -asking the key questions below.

With a full class, encourage different students to start with different flag positions, and different choices of x or y axis as the first mirror line. The teacher intervention above could become a full class discussion. In theory, all students will have the same combined transformation, which should be a perfect moment for a comment on evidence versus proof.

Key questions

What if the flag was in a different place?

What if you reflected in the other line first?

What if the lines were 45 degrees apart?

What if the lines were 30 or 60 degrees ? (use isometric paper )

What, precisely, does the final position of the flag depend on?

Can you prove it?

Possible extension

Ask students to summarise their findings in exactly 20 words (!) - then ask what other transformations might be combined in a similar manner.

Ask (suitably experienced) students to create a dynamic geometry file that demonstrates their findings.

Possible support

Try Mirror, Mirror... before attempting this problem.

Ensure students use square paper and encourage them to draw their reflections accurately.

Compound transformations. Creating and manipulating expressions and formulae. Rotations. Curious. Generalising. Games. Reflections. Mathematical reasoning & proof. Interactivities. Visualising.

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