This problem explores the shapes and symmetries in some national flags.

Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Use the interactivity to find out how many quarter turns the man must rotate through to look like each of the pictures.

Explore the effect of reflecting in two intersecting mirror lines.

Why not challenge a friend to play this transformation game?

Use the clues about the symmetrical properties of these letters to place them on the grid.

Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?

A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

How many different symmetrical shapes can you make by shading triangles or squares?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

A triangle ABC resting on a horizontal line is "rolled" along the line. Describe the paths of each of the vertices and the relationships between them and the original triangle.

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.

This article for teachers suggests ideas for activities built around 10 and 2010.

Put your visualisation skills to the test by seeing which of these molecules can be rotated onto each other.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

What is the same and what is different about these tiling patterns and how do they contribute to the floor as a whole?

Watch this "Notes on a Triangle" film. Can you recreate parts of the film using cut-out triangles?

Look carefully at the video of a tangle and explain what's happening.

A brief video looking at how you can sometimes use symmetry to distinguish knots. Can you use this idea to investigate the differences between the granny knot and the reef knot?

Can you picture where this letter "F" will be on the grid if you flip it in these different ways?

This problem provides training in visualisation and representation of 3D shapes. You will need to imagine rotating cubes, squashing cubes and even superimposing cubes!

The first part of an investigation into how to represent numbers using geometric transformations that ultimately leads us to discover numbers not on the number line.

What is the relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

Can you work out what kind of rotation produced this pattern of pegs in our pegboard?

Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?

How many different transformations can you find made up from combinations of R, S and their inverses? Can you be sure that you have found them all?

Does changing the order of transformations always/sometimes/never produce the same transformation?

See the effects of some combined transformations on a shape. Can you describe what the individual transformations do?

I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only. . . .

A design is repeated endlessly along a line - rather like a stream of paper coming off a roll. Make a strip that matches itself after rotation, or after reflection

Points off a rolling wheel make traces. What makes those traces have symmetry?

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

What mathematical words can be used to describe this floor covering? How many different shapes can you see inside this photograph?

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

This article describes a practical approach to enhance the teaching and learning of coordinates.

This article describes the scope for practical exploration of tessellations both in and out of the classroom. It seems a golden opportunity to link art with maths, allowing the creative side of your. . . .

Investigate the transfomations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0. -1 and +1.

Sort the frieze patterns into seven pairs according to the way in which the motif is repeated.

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

What happens to these capital letters when they are rotated through one half turn, or flipped sideways and from top to bottom?

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

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

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