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....

Introduction to Geometry

I was very excited to get my copy of Donald Coxeter's Introduction to Geometry through the post, all the way from Germany (a very good condition second hand copy via AbeBooks.com). This was a Christmas present from my mum and dad (thanks mum and dad!) Here are some pictures:

Full of very interesting stuff on groups and symmetries, kaleidoscopes, tilings of the plane, the platonic solids, affine, projective and hyperbolic geometry ....

I first learned about Donald Coxeter from a biography called King of Infinite Space: The Man Who Saved Geometry by Siobahn Roberts (a book I found while browsing in my local public library). He was a very interesting and somewhat eccentric man by all accounts, who almost single handedly revitalised the study of geometry in mathematics in the 20th century. This was at a time when the fashion in mathematics was for abstraction, formal rigour, set-theoretic reductions and a complete absence of pictures and diagrams -- a philosophy of mathematics most strongly expressed at the time by the Bourbaki collective (a group of mainly French mathematicians who wrote under the pseudonym 'General Bourbaki').

Here is a nice quote about Coxeter from Benoit Mandelbrot:

He was viewed as a throwback... He was a bit marginal ... I remember feeling the strength of his style. The enjoyment Coxeter always had handling shapes, models, and letting models help him dream, is something I find very attractive and very important -- the spirit of loving shapes and the role of the eye and the hand, that what I dound so marvelous in Coxeter.

Most people are not strong enough to have a well-defined personal style ... The should bend according to fashion or circumstance and he clearly did not bend. He kept with his classical tradition of geometry, which had been totally flattened -- pulverized would be even closer -- by Bourbaki. to learn mathematics without pictures is criminal, a ridiculous enterprise.

(Roberts, p. 127)

One of the things I learned about from Roberts' book was Coxeter's pop-up dodecahedron (I've since found this in other sources, so I'm not sure whether Coxeter actually inveneted it). I've already made quite a few of these. Here is a picture of one and a little film clip:

Coxeter apparently used to make a joke of it in his classes, looking vaguely around and asking 'Now where did I put my dodacehedron?' then opening up a book so that it leapt out. The instructions for making them in Roberts' book are not as clear as they could be, but it's quite simple. First cut out two copies of this network of pentagons:

You need quite stiff card for this, or the dodacehedron will crumple. An alternative is to print the networks or ordinary paper and then glue them on to cardboard, which is what I did. Here is a PDF of the network. Once you've cut out two copies of the network, score lightly along the edges of the pentagon at the centre. Then places one network on top of the other, like this:

where the network underneath (shown in blue) is rotated 180 degrees with respect to the one on top (shown in red). Your elastic band needs to have a diameter slightly less than the diameter of the network. Keeping the two networks flat, place the elastic band on top of the network as shown here:

Then, still keeping the networks flat (press down with your hand or put a book on top to keep everything in place), thread the elastic band alternatively over and under the points of the pentagons, like this:

All you have to do then is gently let go and the dodecahedron should pop up into place.