Rotations

Find the shape and symmetries of the two pieces of this cut cube.

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?

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.

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

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?

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

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 one centre of rotation ? Or if you thought that was impossible, could you say why ?

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?