These have been likened to the yin and yang of mathematics: geometry is space, algebra is time; geometry is like painting, algebra is like music; and so on. Less fancifully, geometry is about form, algebra is about structure—in particular, the structure that lurks within equations. And as Descartes showed with his invention of “Cartesian coordinates,” equations can describe forms: x2 + y2 = 1, for example, describes a circle of radius 1. So algebra and geometry turn out to be intimately related, exchanging what André Weil called “subtle caresses.”
