Two-Dimensional Spaces, Volumes 1, 2, and 3

James W. Cannon, Brigham Young University, Provo, UT.

• Volume 1Lengths—The Pythagorean theoremConsequences of the Pythagorean theoremAreasAreas by slicing and scalingAreas by cut and pasteAreas by countingUnsolvable problems in Euclidean geometryDoes every set have a size?BibliographyVolume 2The fundamental theorem of algebraThe Brouwer fixed point theoremToolsLebesgue covering dimensionFat curves and Peano curvesThe arc, the simple closed curve, and the Cantor setAlgebraic topologyCharacterization of the 2-sphere2-manifoldsArcs in $\mathbb{S}^2$ are tameR. L. Moore's decomposition theoremThe open mapping theoremTriangulation of 2-manifoldsStructure and classification of 2-manifoldsThe torusOrientation and Euler characteristicThe Riemann-Hurwitz theoremBibliographyVolume 3A graphical introduction to hyperbolic geometryHyperbolic geometryGravity as curvatureCurvature by polyhedral approximationCurvature as a length derivativeTheorema egregiumCurvature appendixBibliography

The books contain an incredible range of ideas, both in terms of topic and in terms of sophistication. The author has great taste in topics and provides very valuable insights."" - Richard Evan Schwartz, Brown University