The Geometry of the Universe

One of the most fruitful sources of mathematical intuition is physical space. For not only does physical space provide us with the basic concepts of Euclidean geometry, but it also gives us a pictorial framework for visualizing the very much more general types of space that occur continually throughout mathematics. Moreover, it was the picture of physical space that led to those key ideas of mathematical analysis: continuity and smoothness. Indeed, even the very basic mathematical notion of real number originated from measurement of spatial separation—and of time intervals too, these being, as Albert Einstein’s relativity has told us, geometrical quantities again, whose measurement is essentially bound up with that of space. So it comes as a shock when we also learn from relativity that our now cherished notion of Euclidean geometry does not, after all, describe physical space in the most accurate way. Yet, from these Euclidean beginnings, a more subtle and flexible geometry, known as differential geometry, has grown to maturity. It is in terms of this geometry that Einstein’s theory finds expression. And now, more than sixty years after general relativity was first put forward as a daring original view of the world, the theory stands in excellent agreement with observation. So if we wish to understand how the world is shaped, we must come to terms with this theory.