Start by investigating reflections of triangles in two mirrors at different angles.  On the right is a very rough short video I made doing this.

How is the angle between the mirrors related to the shape you produce?  What other questions could you ask about these shapes?






Now... why not make a Kaleidoscope?  It's easier than you think - you don't need to use proper mirrors, you can use mirrored acrylic, it's really cheap and easy to cut.  Here is a video of the inside of a Kaleidoscope I made from a Pringles tube!

The mirror strips form an equilateral triangle inside the tube.  Then all you need is a see through chamber at the bottom of the tube to put your bits in.  I used a piece of plastic and the lid of the pringles.  

Then you just rotate the tube and the bits fall about and make these nice patterns!  

All regular polyhedra can be thought of as reflections of a single triangle joining the 3 different axes of symmetry (through the centre of a face, an edge midpoint and a vertex).  

On the right are some images inside some of these kaleidoscopes.  The apparent rays coming out of the shapes are the joins between the mirrors - the axes of symmetry.

Cut 3 mirrors according to the angles between the axes of symmetry of a given polyhedron.  This image gives the angles to cut the mirrors for the cube/octahedron (top) and dodecahedron/icosahedron (bottom).  

Then place a card triangle across the mirrors to give images of 3D polyhedra.  You can calculate the angles for this triangle depending on whether it is part of a:

• triangular face (ie octahedron/icosahedron, in which case it will be a 30-60-90 triangle)
• square face (ie cube: 45-45-90)
• pentagonal face (dodecahedron: 36-54-90)
  

Putting two mirrors parallel to each other gives an infinite number of images.  

The Japanese artist Yayoi Kusama uses this to great effect in her piece Infinity Room - see the pictures on the right.  It is on display at the Tate Modern in London until June 2012 and is well worth going to (and not just for this).

I would like to turn my classroom into an infinity room!

An isometry is a transformation that preserves length.  Reflections, rotations and translations are examples of isometries.  In fact, rotations and translations are just the product of two reflections.  Investigating transformations using two mirrors is really interesting.

If you experiment, you will notice that a rotation is the product of two reflections in intersecting mirrors (the angle or rotation being twice the angle between the mirrors) and a translation is the product of two reflections in parallel mirrors (where the distance translated is twice the distance between the two mirrors).  These are called direct isometries as they preserve direction; writing will appear the same when reflected twice.

If you have an odd number of reflections, you will reverse the direction of the object being reflected.  A reflection in three mirrors is what is called a glide reflection, and can be thought of as the product of a reflection and a translation, like footprints.  These transformations form a group; they tile the whole plane; look at the diagram on the right.

This video from girlsangle demonstrates this in a really fun way!