内容

目次

Volume 1 • Lengths—The Pythagorean theorem • Consequences of the Pythagorean theorem • Areas • Areas by slicing and scaling • Areas by cut and paste • Areas by counting • Unsolvable problems in Euclidean geometry • Does every set have a size? • Bibliography Volume 2 • The fundamental theorem of algebra • The Brouwer fixed point theorem • Tools • Lebesgue covering dimension • Fat curves and Peano curves • The arc, the simple closed curve, and the Cantor set • Algebraic topology • Characterization of the 2-sphere • 2-manifolds • Arcs in $\mathbb{S}^2$ are tame • R. L. Moore's decomposition theorem • The open mapping theorem • Triangulation of 2-manifolds • Structure and classification of 2-manifolds • The torus • Orientation and Euler characteristic • The Riemann-Hurwitz theorem • Bibliography Volume 3 • A graphical introduction to hyperbolic geometry • Hyperbolic geometry • Gravity as curvature • Curvature by polyhedral approximation • Curvature as a length derivative • Theorema egregium • Curvature appendix • Bibliography