Imagine that you're standing on the Equator and you draw a straight line going north-south, perpendicular to the Equator. OK? Then you draw another straight line next to it, perpendicular to the Equator again. You have a pair of straight parallel lines, right? No, they meet at the poles. Madness! Don't worry, even Euclid, the father of geometry, couldn't cope with parallel lines meeting. They meet because they are bending through another dimension. If you think that's weird, check out relativity. It's mental.
I love geometry, but it's a passion that's hard to share. Even my mother, a very patient listener, who was fascinated when I explained how parallel lines meet, had had enough of it by then. For me, the neat shapes in geometry textbooks are pure intellectual crack cocaine. Circles, ellipses, cones, cyclides, spirals, and a host of heavenly abstract mathematical forms exist forever in any conceivable universe.
I have a hunch that this universe is hyperspherical. I often think about it, but don't meet many other people who do. I interviewed Patrick Moore once, and asked him was what shape he thought the universe was. He said he didn't know, quite dismissively, adding in a quieter voice that he had asked Einstein the same question and that he didn't know either. My money's on a flat hypersphere, and it's worth a flutter at Ladbrokes.
I was invited by the Victoria and Albert Museum to have a look at some of Leonardo da Vinci's notebooks. There are 21 in existence. Bill Gates bought one for $34m, but the cash value has probably rocketed since The Da Vinci Code. The V&A has three volumes known as the Foster Codices. A troupe of security guards, a couple of professors and the rustle of tracing paper accompanied the unveiling of the tiny leatherbound tablets from their cryogenic vault.
I was allowed to touch them, but didn't want to. Our host was a scholar of the works, he knew them intimately, loved them, and his hands trembled as he turned the pages. Gates's $34m suddenly seemed a nominal sum for one of these notebooks. It was like looking right into the man's brain.
What makes the notebooks such a hit is that they are so heavily illustrated. I imagine Newton's notebooks are fascinating to a mathematician, and Einstein's to a physicist, but if you don't understand the maths, there's only so much you can wonder at. The Foster Codices, however, are peppered with beguiling, enigmatic drawings. There's something for everybody: sketches of horses, hats, perpetual-motion engines, diggers, arches, trusses, pulleys.
For me, the most satisfying aspect was that it was clear that Leonardo was a geometry fanatic, a curves man. I've never seen so many shapes. The idea of genius didn't exist until the 18th century, but it's hard to think of any other way of describing these brilliant cerebral excursions. The man had a go at everything, some of it, like the perpetual-motion stuff, he got wrong. And maybe that's what makes these priceless treasures so inspirational.