Geometry: Plane and FancySpringer Science & Business Media, 6 Δεκ 2012 - 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 vii
... parallel postulate . It has been my practice , in teaching the course for which this book forms the basis , to begin by presenting this proof , preceded by the warning that it is not correct . I believe that the best way to understand ...
... parallel postulate . It has been my practice , in teaching the course for which this book forms the basis , to begin by presenting this proof , preceded by the warning that it is not correct . I believe that the best way to understand ...
Σελίδα viii
... 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 Polygons 2.1 Isometries and Transformation Groups 2.2 Regular and Semiregular ...
... 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 Polygons 2.1 Isometries and Transformation Groups 2.2 Regular and Semiregular ...
Σελίδα 1
... Postulates , and five Common Notions . This book focuses on just one of these , the fifth postulate , com- monly known as the " Parallel Postulate . " Before we can do that , though , it will be necessary to get some idea of what these ...
... Postulates , and five Common Notions . This book focuses on just one of these , the fifth postulate , com- monly known as the " Parallel Postulate . " Before we can do that , though , it will be necessary to get some idea of what these ...
Σελίδα 6
... parallel postulate from the other postulates , but also that we cannot disprove it either . Euclid was absolutely right in making it a postulate , since otherwise the question of whether such lines meet or not could not be resolved . It ...
... parallel postulate from the other postulates , but also that we cannot disprove it either . Euclid was absolutely right in making it a postulate , since otherwise the question of whether such lines meet or not could not be resolved . It ...
Σελίδα 11
... Parallel Postulate and its Descendants In this section we will look at alternative formulations of the parallel postulate . Originally , many of these ... Parallel Postulate and its Descendants 11 The Parallel Postulate and Its Descendants.
... Parallel Postulate and its Descendants In this section we will look at alternative formulations of the parallel postulate . Originally , many of these ... Parallel Postulate and its Descendants 11 The Parallel Postulate and Its Descendants.
Περιεχόμενα
1 | |
Tiling the Plane with Regular Polygons | 21 |
Geometry of the Hyperbolic Plane | 48 |
Geometry of the Sphere | 74 |
More Geometry of the Sphere | 105 |
Geometry of Space | 131 |
References | 155 |
Άλλες εκδόσεις - Προβολή όλων
Συχνά εμφανιζόμενοι όροι και φράσεις
A₁ Algebra angle sum antipodal points assume assumption axioms called Chapter closed curve color commutative law complex numbers compute congruent conjugate Möbius transformation construct convex corresponding cube defect defined described disc divide edge elliptic geometry equal equation Euclid Euclidean geometry Euler's theorem exactly example exterior angles fact fifth postulate flip formula geodesic geometric object 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 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 rectangle regular polygons right angles rotation Schlegel diagram semiregular tilings shortest path side smaller snowflake space spherical straight line Suppose symmetry tangent tessellation translation triangle ABC unit circle vector vertex vertices