Geometry: Plane and Fancy

Εξώφυλλο
Springer Science & Business Media, 9 Ιαν 1998 - 162 σελίδες
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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Περιεχόμενα

Euclid and NonEuclid
1
12 The Parallel Postulate and its Descendants
11
13 Proving the Parallel Postulate
16
Tiling the Plane with Regular Polygons
21
22 Regular and Semiregular Tessellations
26
23 Tessellations That Arent and Some Fractals
35
24 Complex Numbers and the Euclidean Plane
42
Geometry of the Hyperbolic Plane
48
42 Graphs and Eulers Theorem
82
Regular and Semiregular Polyhedra
90
The Protective Plane and Its Cousin
96
More Geometry of the Sphere
105
52 Hamilton Quaternions and Rotating the Sphere
113
53 Curvature of Polyhedra and the GaussBonnet Theorem
121
Geometry of Space
131
62 What Is Curvature?
141

32 Tessellations of the Hyperbolic Plane
57
33 Complex numbers Mobius Transformations and Geometry
64
Geometry of the Sphere
74
63 From Euclid to Einstein
146
References
155
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