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GEOMETRY: Plane and Fancy offers students a fascinating tour through parts of geometry they are unlikely to see in the rest of their studies while, at the same time, anchoring their excursions to the well known parallel postulate of Euclid. The author shows how alternatives to Euclid's fifth postulate lead to interesting and different patterns and symmetries. In the process of examining geometric objects, the author incorporates the algebra of complex (and hypercomplex) numbers, some graph theory, and some topology. Nevertheless, the book has only mild prerequisites. Readers are assumed to have had a course in Euclidean geometry (including some analytic geometry and some algebra) at the high school level. While many concepts introduced are advanced, the mathematical techniques are not. Singer's lively exposition and off-beat approach will greatly appeal both to students and mathematicians. Interesting problems are nicely scattered throughout the text. The contents of the book can be covered in a one-semester course, perhaps as a sequel to a Euclidean geometry course.
1 Euclid and Non-Euclid.- 1.1 The Postulates: What They Are and Why.- 1.2 The Parallel Postulate and Its Descendants.- 1.3 Proving the Parallel Postulate.- 2 Tiling the Plane with Regular Polygons.- 2.1 Isometries and Transformation Groups.- 2.2 Regular and Semiregular Tessellations.- 2.3 Tessellations That Aren’t, and Some Fractals.- 2.4 Complex Numbers and the Euclidean Plane.- 3 Geometry of the Hyperbolic Plane.- 3.1 The Poincaré disc and Isometries of the Hyperbolic Plane.- 3.2 Tessellations of the Hyperbolic Plane.- 3.3 Complex numbers, Möbius Transformations, and Geometry.- 4 Geometry of the Sphere.- 4.1 Spherical Geometry as Non-Euclidean Geometry.- 4.2 Graphs and Euler’s Theorem.- 4.3 Tiling the Sphere: Regular and Semiregular Polyhedra.- 4.4 Lines and Points: The Projective Plane and Its Cousin.- 5 More Geometry of the Sphere.- 5.1 Convex Polyhedra are Rigid: Cauchy’s Theorem.- 5.2 Hamilton, Quaternions, and Rotating the Sphere.- 5.3 Curvature of Polyhedra and the Gauss-Bonnet Theorem.- 6 Geometry of Space.- 6.1 A Hint of Riemannian Geometry.- 6.2 What Is Curvature?.- 6.3 From Euclid to Einstein.- References.