Geometry: Plane and Fancy
Springer 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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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 transformation move origin orthogonal pair of antipodal parallel postulate pattern pentagons perpendicular polyhedra polyhedral surface polyhedron possible Problem projective plane proof Proposition prove quaternions real number rectangle regions regular polygons right angles rotation Schlegel diagram semiregular tilings shortest path side smaller snowflake space spherical square straight line Suppose symmetry tangent tessellation translation triangle ABC unit circle unit sphere vector vertex vertices