Alex's Adventures in Numberland - Alex Bellos [169]
When hyperbolic space was first conceived it appeared to go against any sense of reality, yet it has become accepted as equally ‘real’ as flat or spherical surfaces. Every surface has its own geometry, and we need to choose the one that best applies, or, as Henri Poincaré once said: ‘One geomtry cannot be more true than another; it can only be more convenient.’ Euclidean geometry, for example, is the most appropriate for schoolchildren armed with rulers, compasses and flat pieces of paper, while spherical geometry is the most appropriate for airline pilots navigating flight paths.
Physicists are also interested in which geometry is most appropriate for their purposes. Riemann’s ideas about the curvature of surfaces provided Einstein with the equipment to make one of his greatest breakthroughs. Newtonian physics assumed that space was Euclidean, or flat. Einstein’s theory of general relativity, however, stated that the geometry of space-time (3-D space plus time considered as the fourth dimension) was not flat but curved. In 1919 a British scientific expedition in Sobral, a town in the northeast of Brazil, took images of the stars behind the sun during a solar eclipse and found that they had shifted slightly from their real positions. This was explained by Einstein’s theory that the light from the stars was curving a round the sun before it reached Earth. While the light appeared to bend around the sun when seen in three-dimensional space, which is the only way we can see things, it was actually following a straight line according to the curved geometry of space-time. The fact that Einstein’s theory correctly predicted the position of the stars vindicated his general theory of relativity and it is what made him a global celebrity. The London Times headline blazoned: ‘Revolution in Science, New Theory of the Universe, Newtonian Ideas Overthrown’.
Einstein was concerned with space-time, which he showed to be curved. What about the curvature of our universe without considering time as a dimension? In order to see which geometry best fits the behaviour of our three spatial dimensions on a large scale, we need to see how lines and shapes behave over extremely large distances. Scientists are hoping to discover this from the data that is being gathered by the Planck satellite, launched in May 2009, which is measuring cosmic background radiation – the so-called ‘afterglow’ of the Big Bang – to a higher resolution and sensitivity than ever before. Considered opinion is that the universe is either flat or spherical, although it is still possible that the universe might be hyperbolic. It is wonderfully ironic to think that a geometry originally thought to be nonsensical might actually reflect the way things really are.
At around the same time that mathematicians were exploring the counter-intuitive realm of non-Euclidean space, one man was turning upside-down our understanding of another mathematical notion: infinity. Georg Cantor was a lecturer at Halle University in Germany, where he developed a trail-blazing theory of numbers in which infinity could have more than one size. Cantor’s ideas were so unorthodox that they initially provoked ridicule from many of his peers. Henri Poincaré, for example, described his work as ‘a malady, a perverse illness from which some day mathematics would be cured’, while Leopold Kronecker, Cantor’s former teacher and professor of maths at Berlin University, dismissed him as a ‘charlatan’ and a ‘corruptor of youth’.
This war of words probably contributed to Cantor’s nervous breakdown in 1884, aged 39, the first of many mental-health episodes and hospitalizations. In his book on Cantor, Everything and More, David Foster Wallace writes: ‘The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist,