... combinatorics, a sort of glorified dice-throwing.

Pi is not merely the ubiquitous factor in high school geometry problems it is stitched across the whole tapestry of mathematics, not just geometry's little corner of it. Pi occupies a key place in trigonometry too. It is intimately related to e, and to imaginary numbers. Pi even shows up in the mathematics of probability

Sometimes in studying Ramanujan's work, [George Andrews] said at another time, I have wondered how much Ramanujan could have done if he had had MACSYMA or SCRATCHPAD or some other symbolic algebra package.