Illusion

A security camera, taking pictures each half a second, films a cyclist going by. In the film, the cyclist appears to go forward while the wheels appear to go backwards. Why?

Weighty Problem

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it is facing the other way round.

Rolling Triangle

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

Attractive Rotations

Stage: 3 Challenge Level:

Take a look at the image below. How do you think it was created?
Did you notice any symmetry in the image?

Here is a diagram which shows how we created the image. We started with a triangle (shaded) and then used the coordinate grid to help us to rotate it through multiples of $90^{\circ}$ around the point $(0,0)$.

Create some images of your own by rotating a shape through multiples of $90^{\circ}$.
You might like to start with a triangle as we did, or you might want to use other shapes.

How can you use a coordinate grid to help you to rotate each point around $(0,0)$?
What is the relationship between the coordinates of the points as they rotate through multiples of $90^{\circ}$?

Here are some more ideas to explore:

Can you use an isometric grid to rotate a shape through multiples of $60^{\circ}$?

Try creating some images based on other rotations, such as $30^{\circ}$ or $72^{\circ}$ or... (you will need to use a protractor for these). What do you notice about the rotational symmetry of your images?

What is the rotational symmetry of your final image if you rotate through multiples of $80^{\circ}$ or $135^{\circ}$? Can you explain why?

Here is the kind of image you could try to create:
Send us pictures of your rotation patterns along with your interesting mathematical discoveries.