The gateway to an empire

In mathematics it is new ways of looking at old things that seem to be the most prolific sources of far-reaching discoveries. A particular fact may have been known for centuries, and it may have been sterile or of only minor interest all that time, when suddenly some original mind glimpses it from a new angle and perceives the gateway to an empire. What the first flash of intuition sees may take years or even centuries to open up and explore completely, but once a start in the right direction is made, discovery and development go forward at an ever-increasing speed…

Regularities and repetitions in patterns suggest at once to a modern mathematician the abstract groups behind the patterns; and the various transformations of one problem, not necessarily mathematical, into another again spell “group” and raise the question what, if anything, in the problems remains the same, or invariant, under all these transformations…When faced with a new problem, mathematicians frequently try to restate it so that it is equivalent to one whose solution is already known.

—Eric Temple Bell, Mathematics: Queen and Servant of Science