Section 1.7 Overview
¶After clarifying the distinction between geometries and models in Section 1.1, we introduced a simple geometry in Section 1.2, then explored several finite models, with unique parallel lines (Section 1.3), mutiple parallel lines (Section 1.4), and no parallel lines (Section 1.5). We also explored unexpected properties of the geometry of the earth (Section 1.6).
The remaining chapters provide a more systematic account of several “almost”-Euclidean geometries. We first review some of the properties of Euclidean geometry in Chapter 2, then consider a model of city streets in Chapter 3. A more axiomatic treatment of “Euclidean geometry without the parallel postulate” is given in Chapter 4, followed by models for the standard non-Euclidean geometries: the Poincaré disk model of hyperbolic geometry (Chapter 5), the Klein disk model of single-elliptic geometry (Chapter 6), and the sphere model of double-elliptic geometry (Chapter 7). This discussion of non-Euclidean models culminates in a geometric discussion of area in both hyperbolic and elliptic geometry (Chapter 8). The remaining chapters provide brief introductions to projective geometry (Chapter 9), tilings (Chapter 10), and special relativity (Chapter 11).