1 - The language of symmetry

Summary

You boil it in sawdust, you salt it in glue

You condense it with locusts and tape

Still keeping one principal object in view–

To preserve its symmetrical shape.

The Hunting of the Snark, Lewis Carroll

Symmetry, to a mathematician, encompasses much more than it does in everyday usage. One of the pioneers of this grander view was the distinguished and influential German mathematician Felix Klein. In 1872, on the occasion of his appointment to a chair at the University of Erlangen at the remarkably early age of 23, Klein proposed to the mathematical world that it should radically extend its received view of symmetry, to encompass things which had never been thought of as symmetrical before. Our quotation from Lewis Carroll, alias Charles Dodgson, mathematician and Fellow of Christ Church College, Oxford, was written only four years later. Perhaps Dodgson had heard about Klein's ideas and had them in mind as he composed his nonsensical verse.

In his historic short article, the young Klein synthesized over fifty years of mathematical development in a new and profoundly influential way. It is today difficult to fully appreciate the significance of what he said, because his lecture crystallized one of those paradigm shifts which, after they have happened, seem so obvious that it hard to imagine how anyone could ever have thought otherwise.