# Geometry

Mathematics is the Queen of Sciences, but Geometry is the soul of mathematics.

Geometry may be the aspect of mathematics more fundamental to human experience than any other. Even in cultures in which the idea of counting and number remained at a rudimentary level, craftsmen have ornamented the objects they create with intricate geometric patterns. Humans are almost never purely practical. The urge to doodle and to adorn is, except for the occasional minimalist architect, almost impossible to resist.

Educated during the 1970's I learned little geometry at school, and none at all at university so far as I can recall. For two thousand years the study of Euclid's Elements had been a key part of education throughout Europe and Arabia. According to many writers it was only surpassed by the Bible and the Koran in terms of sales, influence, and the number of editions of it and commentaries upon it.^{†})†When looking to buy a copy recently I could not find it any bookshops - which may seem more surprising when I tell you that I was looking in Cambridge. It is hard to imagine anywhere more likely to stock it. Fortunately the Elements are available on the internet, both to order, and online.
But, because pictures can mislead, and give rise to unconscious assumptions that render proofs invalid or incomplete, the use of diagrams or pictures became frowned upon, and perhaps partly because of this, geometry became unfashionable (even though in the mid-nineteenth century it had probably as profound an effect on artistic thought, and philosophy, as anything has had since).

Here is a little story, quoted from the preface to Tristam Needham's excellent book "Visual Complex Analysis" (don't worry if you haven't the faintest idea what Complex Anlysis might be).

Imagine a society in which the citizens are encouraged, indeed compelled up to a certain age, to read (and sometimes to write) musical scores. All quite admirable. However, this society has a very curious - few remember how it all started - and disturbing law:

Music must never be listened to or performed!.Though its importance is universally acknowledged, for some reason music is not widely appreciated in this society. To be sure, professors still excitedly pore over the great works of Bach, Wagner, and the rest, and they do their utmost to communicate to their students the beautiful meaning of what they find there, but still they become tongue-tied when brashly asked the question, "What's the point of this?!"

This was the fate of mathematics, and perhaps especially of geometry, from the mid-nineteenth until the late twentieth centuries: Visual intuition, and visual aesthetic enjoyment, was anethema. Even language was suspect. Abstract symbolism, formalism, the law.

Fortunately things have been changing. The availability of cheap and powerful computers means that images, of which even the most brilliant mathematician of the past could have only an inkling, can be reproduced on modern computers in a matter of seconds, and give an insight, if not into the meaning of some mathematical theory, at least into the beauty which inspires those who study it in depth. Educators have realised that in a culture dominated by the instant, and by slick visuals, that mathematicians need to "get with the program", if their discipline is not to become the preserve of an even tinier minority than it is already.