Geometry: Plane and FancySpringer 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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Σελίδα vi
... Euclid's ax- iomatic system is based on five postulates , of which four are reasonably intuitive . The fifth postulate , however , is quite another story . In the pro- cess of attempting to prove that this postulate follows from the ...
... Euclid's ax- iomatic system is based on five postulates , of which four are reasonably intuitive . The fifth postulate , however , is quite another story . In the pro- cess of attempting to prove that this postulate follows from the ...
Σελίδα vii
... claims they make . I also expect them to challenge me if they are not convinced about claims I make . This is the ideal environment for mathematical rigor . Contents Preface Chapter 1 Euclid and Non - Euclid 1.1 Preface vii Preface.
... claims they make . I also expect them to challenge me if they are not convinced about claims I make . This is the ideal environment for mathematical rigor . Contents Preface Chapter 1 Euclid and Non - Euclid 1.1 Preface vii Preface.
Σελίδα ix
David A. Singer. Contents Preface Chapter 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 Chapter 2 Tiling the Plane with Regular ...
David A. Singer. Contents Preface Chapter 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 Chapter 2 Tiling the Plane with Regular ...
Σελίδα x
... Euclid to Einstein 146 References 155 1 Euclid and Non - Euclid CHAPTER 1.1 The Postulates X Contents.
... Euclid to Einstein 146 References 155 1 Euclid and Non - Euclid CHAPTER 1.1 The Postulates X Contents.
Σελίδα 1
David A. Singer. 1 Euclid and Non - Euclid CHAPTER 1.1 The Postulates : What They Are and Why At the beginning of Euclid's monumental thirteen volume text The Ele- ments [ 18 ] , there is a list of 23 Definitions , five Postulates , and ...
David A. Singer. 1 Euclid and Non - Euclid CHAPTER 1.1 The Postulates : What They Are and Why At the beginning of Euclid's monumental thirteen volume text The Ele- ments [ 18 ] , there is a list of 23 Definitions , five Postulates , and ...
Περιεχόμενα
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 |
157 | |
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A₁ Algebra angle sum antipodal points assume assumption axioms Calculus called Chapter closed curve color commutative law complex numbers compute congruent conjugate Möbius transformation construct convex coordinate corresponding cube defect defined described disc edge elliptic geometry equal equation Euclid Euclidean geometry Euler's theorem exactly example exterior angles fact fifth postulate flip formula geodesic graph h-lines hexagon hyperbolic geometry hyperbolic plane ideal point intersect inverse isometry Koch snowflake lemma length line segment Mathematics Möbius band Möbius transformation move non-Euclidean origin orthogonal pair of antipodal parallel postulate pattern pentagons perpendicular polyhedra polyhedral surface polyhedron possible Problem projective plane proof Proposition prove quaternions radius real number regular and semiregular regular polygons right angles rotation Schlegel diagram semiregular tilings shortest path side smaller snowflake space spherical straight line Suppose symmetry tessellation tile the plane translation unit circle vector vertex vertices