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 v
... assumptions about and principles of geometry . That being said , I should mention that the word " curvature ” does not even appear until the end of the fifth chapter of the book . Before then , it is hidden within the idea of the sum of ...
... assumptions about and principles of geometry . That being said , I should mention that the word " curvature ” does not even appear until the end of the fifth chapter of the book . Before then , it is hidden within the idea of the sum of ...
Σελίδα vi
... assumption that the angle sum is al- ways 180 ° . We consider the process of tiling the plane with regular poly- gons . Section 2.1 sets up the machinery of isometries and transformation groups . In Section 2.2 we find all regular and ...
... assumption that the angle sum is al- ways 180 ° . We consider the process of tiling the plane with regular poly- gons . Section 2.1 sets up the machinery of isometries and transformation groups . In Section 2.2 we find all regular and ...
Σελίδα 2
... assumptions made specifically about geometry , which are to be taken as true without proof . The Axioms are assumptions about mathematical truth in general , not specific to geometry . Although Euclid gives the Postulates first , let us ...
... assumptions made specifically about geometry , which are to be taken as true without proof . The Axioms are assumptions about mathematical truth in general , not specific to geometry . Although Euclid gives the Postulates first , let us ...
Σελίδα 4
... assumption should not contradict other assumptions . If our axioms were inconsistent , we could use logical arguments to deduce nonsense . 2. They should be complete . There must be enough assumptions so that we are able to determine ...
... assumption should not contradict other assumptions . If our axioms were inconsistent , we could use logical arguments to deduce nonsense . 2. They should be complete . There must be enough assumptions so that we are able to determine ...
Σελίδα 6
... assumption that through a point not on a given line it is possible to find more than one line not meeting the given line . We will explore this assumption in Chapter 3 . It was not until 1854 that a different geometry appeared ...
... assumption that through a point not on a given line it is possible to find more than one line not meeting the given line . We will explore this assumption in Chapter 3 . It was not until 1854 that a different geometry appeared ...
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 |
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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