The Foundations of Geometry and the Non-Euclidean Plane

Forside
Springer Science & Business Media, 6. des. 2012 - 512 sider
This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref erence if necessary.

Inni boken

Hva folk mener - Skriv en omtale

Vi har ikke funnet noen omtaler på noen av de vanlige stedene.

Innhold

MAPPINGS
10
THE REAL NUMBERS
20
AXIOM SYSTEMS
34
PART ONE ABSOLUTE GEOMETRY
48
INCIDENCE AXIOM AND RULER POSTULATE
65
BETWEENNESS
73
SEGMENTS RAYS AND CONVEX SETS
84
ANGLES AND TRIANGLES
95
CIRCLES
226
ABSOLUTE GEOMETRY AND SACCHERI
239
SACCHERIS THREE HYPOTHESES
255
EUCLIDS PARALLEL POSTULATE
269
BIANGLES
292
EXCURSIONS
317
PART TWO NONEUCLIDEAN GEOMETRY
332
BRUSHES AND CYCLES
347

THE GOLDEN AGE OF GREEK MATHEMATICS
111
EUCLIDS ELEMENTS Optional
121
PASCHS POSTULATE AND PLANE
131
CROSSBAR AND QUADRILATERALS
144
MEASURING ANGLES AND THE PROTRACTOR
155
ALTERNATIVE AXIOM SYSTEMS Optional
172
MIRRORS
182
CONGRUENCE AND THE PENULTIMATE
192
PERPENDICULARS AND INEQUALITIES
204
REFLECTIONS
216
ROTATIONS TRANSLATIONS
360
THE CLASSIFICATION OF ISOMETRIES
371
SYMMETRY
386
HOROCIRCLES
402
THE FUNDAMENTAL FORMULA
421
CATEGORICALNESS AND AREA
444
QUADRATURE OF THE CIRCLE
464
e Hints and Answers
494
Notation Index
503
Opphavsrett

Andre utgaver - Vis alle

Vanlige uttrykk og setninger

Bibliografisk informasjon