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Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics)

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en Limba Engleză Hardback – 28 Sep 2005
In recent years, I have been teaching a junior-senior-level course on the classi­ cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa­ rately. The remainder of the book is an exploration of questions that arise natu­ rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.
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Specificații

ISBN-13: 9780387986500
ISBN-10: 0387986502
Pagini: 528
Dimensiuni: 178 x 254 x 33 mm
Greutate: 1.16 kg
Ediția: 1st Corrected ed. 2000. Corr. 3rd printing 2005
Editura: Springer
Colecția Springer
Seria Undergraduate Texts in Mathematics

Locul publicării: New York, NY, United States

Public țintă

Lower undergraduate

Cuprins

1. Euclid’s Geometry.- 2. Hilbert’s Axioms.- 3. Geometry over Fields.- 4. Segment Arithmetic.- 5. Area.- 6. Construction Problems and Field Extensions.- 7. Non-Euclidean Geometry.- 8. Polyhedra.- Appendix: Brief Euclid.- Notes.- References.- List of Axioms.- Index of Euclid’s Propositions.