What do Mathematicians do?
G. H. Hardy, who was one of England's finest twentieth century mathematicians, wrote a famous essay, A Mathematician's Apology, on what he thought being a mathematician was all about. (Incidentally, the back of my copy of this little book mentions that "Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it is like to be a creative artist'). This is part of what he says at the start of section 10 of his essay.
Hardy also said: "I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world". However, history has proved him wrong, because his work in Number Theory underlies modern forms of encryption that are used for forming secure Internet connections). A mathematician, like a painter or a poet is a maker of patterns. If his patterns are more permanent than theirs it is because they are made with ideas. A painter makes patterns with shapes and colours, a poet with words. ... A mathematician, on the other hand, has no material to work with but his ideas, and so his patterns are likely to last longer, since ideas wear less with time than words.
For Hardy, mathematics is about the distillation of knowledge into a pure realm of ideas. Many modern mathematical ideas are extremely abstract, and apply equally in very different models or applications. This abstraction, carried too far "piling subtlety of generalisation upon subtlety of generalisation" [Hardy, quoting Alfred North Whitehead], leads to ideas that are "insipid". For Hardy "And so in mathematics; a property common to too many objects can hardly be exciting, and mathematical ideas become dim unless they also have plenty of individuality. ... [quoting Whitehead again]'it is the large generalization, limited by a happy particularity, that is the fruitful conception.'"
But just what are these patterns of ideas anyway? For Hardy, and probably for almost all professional mathematicians of any time in the last two hundred years, they are elegant proofs of beautiful and "deep" theorems. The mathematician will almost always strive to leave the details of the particular behind, first discerning, and then proving, some previously unknown mathematical "truth" lying beneath.
My two penn'orth
Here I part company with "real" mathematicians. In this book, there will be many patterns of shapes and colours, and rather fewer of ideas. Hardy would not have approved. In another place he writes, "There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain." He also pours scorn on those who seek to improve the standing mathematics by pointing out its usefulness to scientific advancement and technological progress.
Like the ancient Egyptians, who knew that a rope twelve cubits long knotted at regular one cubit intervals could be used to draw a right-angle accurately in the ground by making it into a 3-4-5 triangle, a mathematical "fact", whether or not I know how to prove it, will for me, provided I can use it to create interesting patterns, be enough in itself.
Another thing I have in common with the ancient Egyptians, and with the Greeks, is that the mathematics in this book will be almost entirely geometry.