Kaleidoscopes

Another thing I learned about from the book about Coxeter was the possibility of a three dimensional kaleidoscope. The principle behind the ordinary two dimensional kaleidoscope is this: the reflection of an object placed between two s mirrors at angle of 180/n to each other forms 2n visible images (including the original object). If you make the object a straight line connecting the midpoints of the mirrors you will see a square (n = 2), a pentagon (n = 2.5) hexagon (n = 3) heptagon (n = 3.5), octagon (n = 4) and so on.

The case of n = 3, where the angle between the mirrors is 60 degrees is shown in the diagram below. The two thicker lines labelled a (coloured red) and b (coloured blue) represent the mirrors. The thinner red and blue lines represent the reflections of the mirrors. For example the blue horizontal line is the reflection of mirror b in mirror a while the red horizontal line is the reflection of mirror a in mirror b. The thin blue diagonal line is the reflection in mirror b of mirror bs reflection in mirror a ... (There is an interesting algebraic structure to these reflections, reflections of reflections and so on. They form what algebraists call a group. But that's a topic for another time perhaps).

The green blob represents an object placed between the mirrors at the centre of straight line drawn between them. The resulting reflections form a hexagon.

The photos below show this setup with two (rather crudely) joined mirrors placed at angle of 60 degrees. In the second photo, the reflection of mirror a in mirror b can be seen more clearly.

In toy kaleidoscopes a third mirror is placed so that the mirrors form an equilateral triangle. What I didn't know is that if you place three mirrors together at just the right angles you can generate three-dimensional images of the platonic solids (the cube, tetrahedron, octahedron, icosahedron and the dodecahedron). What exactly this looks like I have no idea though. I found some detailed instructions on how to make such kaleidoscopes in a nice paper by Roe Goodman ('Alice through looking glass after looking glass', American Mathematical Monthly April, 2004). I've started making a plan and have been practicing cutting glass into the required shapes, so watch this space. Apparently Coxeter used to carry around big mirrors to show this to people and even made a documentary film about them, but I haven't been able to find a copy of that. I've just read here that TVOntario have made a documentary based on Roberts' book, so I wonder if they show any clips from Coxeter's film. Or maybe they reconstructed some of these kaleidoscopes....