The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction. The first proof of this...
In an old Indian parable, six blind men each touch a different part of an elephant. They disagree about what the elephant must look like: Is it smooth or rough? Is it like a snake (so thinks the man touching the trunk) or a fan (as the man touching the ear proposes)? If the blind men had combined their insights, they might have been able to give a correct account of the nature of the elephant.
The central objects of study in topology are spaces called manifolds, which look flat when you zoom in on them. The surface of a sphere, for instance, is a two-dimensional manifold. Topologists understand such two-dimensional manifolds very well. And they have developed tools that let them make sense of three-dimensional manifolds and those with five or more dimensions. But in four dimensions...
Repetition doesn’t always have to be humdrum. In mathematics, it is a powerful force, capable of generating bewildering complexity. Even after decades of study, mathematicians find themselves unable to answer questions about the repeated execution of very simple rules — the most basic “dynamical systems.” But in trying to do so, they have uncovered deep connections between those rules and other...
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