morphisms’ in my shoe. My tactic for dimensions, picked on about halfway through The Poincaré Conjecture (so don’t blame O’Shea if it’s nonsense), is to think of them not in terms of how many different directions you can go in, but how many terms you’d require to describe an object if you wanted someone to steal it for you. You’d need to tell the thief not only its exact location in the owner’s room (that’s three dimensions for starters), but also what type of object it was, the size of it, its colours, what it was made of — and there’s a seven-dimensional occasional table already. In short, view the extra space dimensions as qualities and, though it’s not really the same thing, my heart plumps up again.
Henri Poincaré (‘a monster of mathematics’) was born in Nancy in 1854, ambi-dextrous, short-sighted and clumsy as a goose. His sister described him as ‘being buried in books, but never seeming to work’. The great Prussian professor Felix Klein tried to trump the young genius on some points of geometry, worked himself into a complete breakdown and destroyed his intellectual lead in Europe. ‘It was a dirty street fight with concealed switchblades,’ says O’Shea, ‘and Poincaré was adept at twisting the knife.’
O’Shea is not trying to explain Poincaré’s conjecture in any detail, or even many of the basic concepts essential to it (although his endnotes, glossaries and other appendices go on for nearly 70 pages). Rather, he wants to give us a feel for the buzz of discovery, to make us quail at the breadth of the problem’s importance. A proper mathematical discussion is riddled with algebra, complex numbers and incantations from topology. To know about those, you have also to know about ‘kissing circles’, the ‘soul conjecture’ and the ‘primitive roots of unity’. No wonder the Church has lost its voice: mathematics has stolen its language.
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