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 vi
... prove that this postulate follows from the others and is therefore unnecessary , mathematicians discovered many equivalent formulations . The one that is central to this text is the statement due to Gerolamo Saccheri ( 1733 ) : The sum ...
... prove that this postulate follows from the others and is therefore unnecessary , mathematicians discovered many equivalent formulations . The one that is central to this text is the statement due to Gerolamo Saccheri ( 1733 ) : The sum ...
Σελίδα viii
... Proving the Parallel Postulate Chapter 2 Tiling the Plane with Regular Polygons 2.1 Isometries and Transformation Groups 2.2 Regular and Semiregular Tessellations A 1 1 11 16 21 21 26 2.3 Tessellations That Aren't , and Some Fractals ...
... Proving the Parallel Postulate Chapter 2 Tiling the Plane with Regular Polygons 2.1 Isometries and Transformation Groups 2.2 Regular and Semiregular Tessellations A 1 1 11 16 21 21 26 2.3 Tessellations That Aren't , and Some Fractals ...
Σελίδα 3
... prove it . First we must review what we know about right angles . Euclid's tenth definition says : When a straight line set up on a straight line makes the adjacent angles equal to one another , each of the equal angles is right , and ...
... prove it . First we must review what we know about right angles . Euclid's tenth definition says : When a straight line set up on a straight line makes the adjacent angles equal to one another , each of the equal angles is right , and ...
Σελίδα 4
... prove Postulate 4 ? Since it is so reasonable , why not just assert that it is true ( as Euclid did ) and be done with it ? To answer this question , we must look at the idea of an axiomatic system . We start with a list of undefined ...
... prove Postulate 4 ? Since it is so reasonable , why not just assert that it is true ( as Euclid did ) and be done with it ? To answer this question , we must look at the idea of an axiomatic system . We start with a list of undefined ...
Σελίδα 5
... prove that all right angles are equal , then the fourth postulate would not be independent of the others and we ought to throw it out . Of course , we don't have to throw it out , since it doesn't change our theory . If we could prove ...
... prove that all right angles are equal , then the fourth postulate would not be independent of the others and we ought to throw it out . Of course , we don't have to throw it out , since it doesn't change our theory . If we could prove ...
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