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

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?

Can you place the blocks so that you see the relection in the picture?

What is the missing symbol? Can you decode this in a similar way?

These images are taken from the Topkapi Palace in Istanbul, Turkey. Can you work out the basic unit that makes up each pattern? Can you continue the pattern? Can you see any similarities and. . . .

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

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

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices. . . .

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

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

Follow these instructions to make a five-pointed snowflake from a square of paper.

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

Someone at the top of a hill sends a message in semaphore to a friend in the valley. A person in the valley behind also sees the same message. What is it?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

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

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.

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

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

Use the blue spot to help you move the yellow spot from one star to the other. How are the trails of the blue and yellow spots related?

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 information on these cards to draw the shape that is being described.

This interactivity allows you to sort letters of the alphabet into two groups according to different properties.

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

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

Plex lets you specify a mapping between points and their images. Then you can draw and see the transformed image.

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

This activity investigates how you might make squares and pentominoes from Polydron.

Can you make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?

Can you deduce the pattern that has been used to lay out these bottle tops?

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

An article for students and teachers on symmetry and square dancing. What do the symmetries of the square have to do with a dos-e-dos or a swing? Find out more?

When dice land edge-up, we usually roll again. But what if we didn't...?

Here is a chance to create some Celtic knots and explore the mathematics behind them.

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Use the interactivity to create some steady rhythms. How could you create a rhythm which sounds the same forwards as it does backwards?

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Can all but one square of an 8 by 8 Chessboard be covered by Trominoes?

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Create a symmetrical fabric design based on a flower motif - and realise it in Logo.