Geometry: Plane and Fancy
Springer Science & Business Media, 9. jan. 1998 - 162 sider
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.
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Euclid and NonEuclid
12 The Parallel Postulate and its Descendants
13 Proving the Parallel Postulate
Tiling the Plane with Regular Polygons
22 Regular and Semiregular Tessellations
23 Tessellations That Arent and Some Fractals
24 Complex Numbers and the Euclidean Plane
Geometry of the Hyperbolic Plane
42 Graphs and Eulers Theorem
Regular and Semiregular Polyhedra
The Protective Plane and Its Cousin
More Geometry of the Sphere
52 Hamilton Quaternions and Rotating the Sphere
53 Curvature of Polyhedra and the GaussBonnet Theorem
Geometry of Space
62 What Is Curvature?
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