Two, Three, Four, and More

A beautifully illustrated exploration of the remarkable world of mathematics in low dimensionsHighly valued results in mathematics are often general: proofs in all dimensions, solutions for all degrees of polynomials. Yet great richness can be found in particular cases. John Stillwell presents a visual panorama of the multitudes that live in dimensions 2, 3, and 4—and a few higher dimensions—through the interplay of geometry, topology, and algebra.We tend to think of the dimensions 2, 3, and 4 as the dimensions of our physical experience in space and time, but they are also of exceptional mathematical interest. Together with the degrees 2, 3, and 4 of equations, they cover some of the most remarkable mathematical objects and pose problems ranging from easy, to hard, to unsolvable. Each chapter of this engaging book focuses on a dimension or degree, starting at 2 dimensions and ascending to 4 dimensions and then beyond to objects such as the Leech lattice in 24 dimensions. The gradual increase in dimension reveals what is gained and what is lost with a bit more space to work in. Along the way, Stillwell brings the mathematics to life through rich examples, engaging historical context, and vivid anecdotes about the mathematicians who uncovered surprising solutions—or revealed why some problems can’t be solved.With hundreds of illustrations, Two, Three, Four, and More is an illuminating mathematical tour by one of the discipline’s leading expositors, revealing how these intriguing special cases underpin some of the deepest ideas in mathematics.