Geometry of Surfaces [[electronic resource] /] / by John Stillwell

• Record Nr.: UNINA9910480073903321
• Autore: Stillwell John
• Pubbl/distr/stampa: New York, NY : , : Springer New York : , : Imprint : Springer, , 1992
• ISBN: 1-4612-0929-3
• Edizione: [1st ed. 1992.]
• Descrizione fisica: 1 online resource (XI, 236 p.)
• Collana: Universitext, , 0172-5939
• Disciplina: 516.3/62
• Soggetti: Geometry
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: "With 165 Figures."
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: 1. The Euclidean Plane -- 1.1 Approaches to Euclidean Geometry -- 1.2 Isometries -- 1.3 Rotations and Reflections -- 1.4 The Three Reflections Theorem -- 1.5 Orientation-Reversing Isometries -- 1.6 Distinctive Features of Euclidean Geometry -- 1.7 Discussion -- 2. Euclidean Surfaces -- 2.1 Euclid on Manifolds -- 2.2 The Cylinder -- 2.3 The Twisted Cylinder -- 2.4 The Torus and the Klein Bottle -- 2.5 Quotient Surfaces -- 2.6 A Nondiscontinuous Group -- 2.7 Euclidean Surfaces -- 2.8 Covering a Surface by the Plane -- 2.9 The Covering Isometry Group -- 2.10 Discussion -- 3. The Sphere -- 3.1 The Sphere S2 in ?3 -- 3.2 Rotations -- 3.3 Stereographic Projection -- 3.4 Inversion and the Complex Coordinate on the Sphere -- 3.5 Reflections and Rotations as Complex Functions -- 3.6 The Antipodal Map and the Elliptic Plane -- 3.7 Remarks on Groups, Spheres and Projective Spaces -- 3.8 The Area of a Triangle -- 3.9 The Regular Polyhedra -- 3.10 Discussion -- 4. The Hyperbolic Plane -- 4.1 Negative Curvature and the Half-Plane -- 4.2 The Half-Plane Model and the Conformai Disc Model -- 4.3 The Three Reflections Theorem -- 4.4 Isometries as Complex Functions -- 4.5 Geometric Description of Isometries -- 4.6 Classification of Isometries -- 4.7 The Area of a Triangle -- 4.8 The Projective Disc Model -- 4.9 Hyperbolic Space -- 4.10 Discussion -- 5. Hyperbolic Surfaces -- 5.1 Hyperbolic Surfaces and the Killing-Hopf Theorem -- 5.2 The Pseudosphere -- 5.3 The Punctured Sphere -- 5.4 Dense Lines on the Punctured Sphere -- 5.5 General Construction of Hyperbolic Surfaces from Polygons -- 5.6 Geometric Realization of Compact Surfaces -- 5.7 Completeness of Compact Geometric Surfaces ; -- 5.8 Compact Hyperbolic Surfaces -- 5.9 Discussion -- 6. Paths and Geodesies -- 6.1 Topological Classification of Surfaces -- 6.2 Geometric Classification of Surfaces -- 6.3 Paths and Homotopy -- 6.4 Lifting Paths and Lifting Homotopies -- 6.5 The Fundamental Group -- 6.6 Generators and Relations for the Fundamental Group -- 6.7 Fundamental Group and Genus -- 6.8 Closed Geodesic Paths -- 6.9 Classification of Closed Geodesic Paths -- 6.10 Discussion -- 7. Planar and Spherical Tessellations -- 7.1 Symmetric Tessellations -- 7.2 Conditions for a Polygon to Be a Fundamental Region -- 7.3 The Triangle Tessellations -- 7.4 Poincaré’s Theorem for Compact Polygons -- 7.5 Discussion -- 8. Tessellations of Compact Surfaces -- 8.1 Orbifolds and Desingularizations -- 8.2 Prom Desingularization to Symmetric Tessellation -- 8.3 Desingularizations as (Branched) Coverings -- 8.4 Some Methods of Desingularization -- 8.5 Reduction to a Permutation Problem -- 8.6 Solution of the Permutation Problem -- 8.7 Discussion -- References.
• Sommario/riassunto: Geometry used to be the basis of a mathematical education; today it is not even a standard undergraduate topic. Much as I deplore this situation, I welcome the opportunity to make a fresh start. Classical geometry is no longer an adequate basis for mathematics or physics-both of which are becoming increasingly geometric-and geometry can no longer be divorced from algebra, topology, and analysis. Students need a geometry of greater scope, and the fact that there is no room for geometry in the curriculum un­ til the third or fourth year at least allows us to assume some mathematical background. What geometry should be taught? I believe that the geometry of surfaces of constant curvature is an ideal choice, for the following reasons: 1. It is basically simple and traditional. We are not forgetting euclidean geometry but extending it enough to be interesting and useful. The extensions offer the simplest possible introduction to fundamentals of modem geometry: curvature, group actions, and covering spaces. 2. The prerequisites are modest and standard. A little linear algebra (mostly 2 x 2 matrices), calculus as far as hyperbolic functions, ba­ sic group theory (subgroups and cosets), and basic topology (open, closed, and compact sets).

Kodaly in the second grade classroom : developing the creative brain in the 21st century / / Micheál Houlahan, Philip Tacka

• Record Nr.: UNINA9910788397003321
• Autore: Houlahan Mícheál
• Pubbl/distr/stampa: New York, New York : , : Oxford University Press, , 2015; ©2015
• ISBN: 0-19-027302-X; 0-19-024849-1
• Descrizione fisica: 1 online resource (297 p.)
• Collana: Kodaly Today Handbook Series
• Disciplina: 372.87049
• Soggetti: School music - Instruction and study; Second grade (Education) - Curricula - United States
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.; "Lesson Plans Included"--Cover.
• Nota di contenuto: Evaluating a LessonUnit Plans; Unit 1: Grade 1 Review; Unit 2: Teaching do; Unit 3: Teaching Half Note; Unit 4: Teaching re; Unit 5: Teaching Four Sixteenth Notes; Unit 6: Teaching do Pentatonic Scale; Unit 7: Teaching Quadruple Meter; 6 Assessment and Evaluation; Grade 2 Assessments; Notes; Index
• Sommario/riassunto: Since the mid-twentieth century, Zoltan Kodaly's child-developmental philosophy for teaching music has had significant positive impact on music education around the world, and is now at the core of music teaching in the United States and other English speaking countries. The Kodaly Today handbook series is the first comprehensive system to update and apply the Kodaly concepts to teaching music in elementary school classrooms. Kodaly in the Second Grade Classroom provides teachers with a step-by-step road map for developing children's performance, creative movement, and literacy skills in an or