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.