Corner Mirror

 Look at the mirrors from different angles. Can you always see yourself? Close one eye and look at the corner, where is your eye is the mirror? See what happens when you move your head around. Can you use the corner mirror and pieces to recreate the chain of 12 rings shown in the image?
Maths

The corner mirror is also called a corner reflector. It is three flat mirrors, all at right-angles, like the inside of a cube. When light enters the cube, it hits all three mirrors, and is reflected back towards the source. So light from your eye will return directly to your eye, so you are looking directly at yourself from any angle.

The direction of a ray in three dimensions can be written as three coordinates (x, y, z). When the ray hits a mirror one component reverses direction, while the other two stay the same. After hitting all three mirrors, all three components are reversed, so the direction the ray is travelling is (-x, -y, -z).
The 12 Ring problem can be puzzling until you realise the rings do not need to lie flat. Each mirror doubles the number of objects so, to make complete rings, reflect the half-ring in one mirror and reflect the quarter-rings in two mirrors. These rings are then reflected in the other mirrors to make eight copies of each piece.

History
People
Applications

Corner reflectors consisting of three mirrors are used to make safety reflectors for cars, bicycles, and signs. They have also been used to bounce laser beams back to the earth from the surface of the moon.

Corner reflectors are also placed on runways, lifeboats and buoys to make these objects shows up particularly strongly on radar screens. If we place eight corner reflectors back-to-back, like an octahedron, radio waves will be reflected back from any direction.

Maths at Home

Bounce a tennis ball at any wall or floor in the corner of a room. It should return to you after bouncing off the three surfaces.

Here are some instructions to make your own corner mirror.