Geometry: Plane and Fancy

Forside
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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Innhold

Euclid and NonEuclid
3
12 The Parallel Postulate and its Descendants
13
13 Proving the Parallel Postulate
18
Tiling the Plane with Regular Polygons
23
22 Regular and Semiregular Tessellations
28
23 Tessellations That Arent and Some Fractals
37
24 Complex Numbers and the Euclidean Plane
44
Geometry of the Hyperbolic Plane
50
42 Graphs and Eulers Theorem
84
Regular and Semiregular Polyhedra
92
The Protective Plane and Its Cousin
98
More Geometry of the Sphere
107
52 Hamilton Quaternions and Rotating the Sphere
115
53 Curvature of Polyhedra and the GaussBonnet Theorem
123
Geometry of Space
133
62 What Is Curvature?
143

32 Tessellations of the Hyperbolic Plane
59
33 Complex numbers Mobius Transformations and Geometry
66
Geometry of the Sphere
76
63 From Euclid to Einstein
148
References
157
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