This book is a comprehensive look at geometry – its history, discoverers, and applications - including how it works, and why it works. Along the way are imaginative twists and opposing viewpoints.
For example, the father of geometry, Euclid, as is often recounted in high school geometry lessons, said that two parallel lines never meet. Along came Bernard Riemann to suggest that, of course parallel lines never meet—parallel lines do not even exist. These two geniuses of geometry did not come to blows over opposing ideas in their theorems, postulates, and proofs. In fact they lived two thousand years apart (Euclid around 300 BCE and Riemann in the 1800s), but had Euclid been employed in the same math department as Riemann, the geometry of Riemann certainly would have fascinated Euclid. Euclid’s geometry took place on a plane. Riemann’s occurred on an ellipse. No doubt both would have appreciated differential geometry, which studies the geometries of curves, spaces, and manifolds. For instance, the geometry of manifolds underlies string theory, which is the attempt to merge quantum mechanics with Albert Einstein’s general theory of relativity. In this theory, subatomic particles are modeled as tiny one-dimensional “stringlike” entities rather than the more conventional approach in which they are modeled as zero-dimensional point particles. Perhaps the blend of geometries of the past and future will help prove current scientific hypotheses on string theory.
Readers will also benefit from the book’s historical perspective.
Chapter 1: History of Geometry
Chapter 2: Branches of Geometry
Chapter 3: Biographies
Appendix of Geometric Terms and Concepts 238