The original version of this story appeared in Quanta Magazine.
Mathematics started with numbers—clear, concrete, intuitive. Over the last two centuries, however, it has become a far more abstract enterprise. One of the first major steps down this road was taken in the late 18th and early 19th centuries. It involved a field called group theory, and it changed math—theoretical and applied—as we know it.
Groups generalize essential properties of the whole numbers. They have transformed geometry, algebra, and analysis, the mathematical study of smoothly changing functions. They’re used to encrypt messages and study the shapes of viruses. Physicists