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.
But What is Geometry?
The Greek words that form the word geometry, mean "earth" and "measure". Though decorative geometric patterns that are still in use today can be found on almost the oldest man-made objects that can be found in museums, geometry itself was originally a very practical study, needed for dividing land fairly†, for construction, and for almost any manufacturing process, and for that other most ancient of sciences: astronomy.
†In fact, we may have the tax-man and the flooding of the Nile to thank for geometry. Herodotus, writing in the 5th century BC says:
The king moreover (so they say) divided the country among all the Egyptians by giving each an equal square parcel of land, and made this his source of revenue, appointing the payment of a yearly tax. And any man who was robbed by the river of a part of his land would come Sesotris and declare what had befallen him; then the king would send men to look into it and measure the space by which the land was diminished, so that thereafter it should pay in proportion to the tax originally imposed. From this, to my thinking the Greeks learned the art of geometry.
The ancient Greeks, with their love for argument and intellectual speculation, moved geometry from the realm of the practical to that of the theoretical and ideal. The study of geometry was the first flowering in humanity of the rule of pure reason. Greek philosophers revered geometry. It was through geometry that we learned that reason was both more useful, and more powerful, than superstition and magic. Without Greek geometry the scientific method might not have emerged for many hundreds or thousands of years. When Europe emerged from the dark ages into the renaissance, geometry, and mathematics generally, were among the most important disciplines that were readopted from Islamic scholars.
But for pure mathematicians today, geometry is on the whole a very different subject from what is was until the early nineteenth century. Until then the theorems of geometry were perceived as being absolutely true: only a handful of individuals ever considered the possibility that Euclid's geometry might not accurately describe the universe in which we live, or even that a universe with a different geometry might even be logically possible. Johann Bolyai, one of the two mathematicians usually credited with the discovery that other geometries were possible, said, with reason, "I have created a new universe from nothing". Nowadays mathematicians would, if they could, be happy to invent some entirely new geometry, with previously undreamt of laws. Provided the theory seemed to be consistent it would be an equally valid geometry as the geometry of the universe we actually live in, and equally worthy of study.
For engineers and craftsmen (and especially for the new breed of crasftsman who creates computer images and animations), geometry remains a practical subject. Map makers and surveyors, computer programmers, chemical engineers designing a complex network of pipes, need answers to practical geometrical problems that pure mathematicians regard as being unworthy of notice. Because of this, in some ways the practioners of these professions might be closer in spirit to the ancients, than are "real" mathematicians.
A great difference in the geometry that I will be presenting in this book from that of the Greeks, is that modern geometry is most often presented based on the idea of transforming one object into another, whereas Greek geometry was mainly concerned with finding methods of constructing (usually drawing) particular curves and shapes: for Greeks the concept of transformation (and motion), did not sit comfortably with their patterns of reasoning and was used with reluctance. In modern geometries the main focus of study is the identification of properties that are unchanged by whatever transformations are permitted in the geometry under consideration, and these - "the" geometric properties, are different in different geometries. For example, there are very interesting geometries in which straight lines and circles become interchangeable, so that a circle and a straight line are in some sense "the same". In such a geometry a symmetry exists between such apparently, in our eyes, different objects, and it is this that makes the idea of symmetry so fundamental.