The Foundations of Geometry and the Non-Euclidean PlaneSpringer 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
Resultat 1-5 av 66
Side 6
... tion on Z by a b if a - b is even . So ( a , b ) E G iff a and b are either both even or both odd . Example 4 ~ Let Z be the set of integers . Define an equivalence rela- tion on Z by a - biff a = b = 0 or ab > 0 . ~ Example 5 ...
... tion on Z by a b if a - b is even . So ( a , b ) E G iff a and b are either both even or both odd . Example 4 ~ Let Z be the set of integers . Define an equivalence rela- tion on Z by a - biff a = b = 0 or ab > 0 . ~ Example 5 ...
Side 15
... tion allows us to say that fmaps / to m ; we write " f ( l ) = m " rather than the more formal " f . ( l ) = m . ” 2.2 COMPOSITION OF MAPPINGS Given mappings f : D → C and g : B → A such that the range of ƒ is a subset of the domain ...
... tion allows us to say that fmaps / to m ; we write " f ( l ) = m " rather than the more formal " f . ( l ) = m . ” 2.2 COMPOSITION OF MAPPINGS Given mappings f : D → C and g : B → A such that the range of ƒ is a subset of the domain ...
Side 16
... tion of the composition of gf followed by h . The second equality follows from the definition of the composition of ... tion on S ? Yes ! Since permutations on S are injections , we have al- ready seen that their product is an injection ...
... tion of the composition of gf followed by h . The second equality follows from the definition of the composition of ... tion on S ? Yes ! Since permutations on S are injections , we have al- ready seen that their product is an injection ...
Side 17
... tion f ' on S such that ff ' == f'f . 2.3 EXERCISES • 2.1 Which of the following functions are injections and which are surjections ? f1 : R → R , f1 : x → 2x ; + f2 : z → z , f2 : x → 2x ; fg : Z → R , fs : x + 2x ; f1 : R → R ...
... tion f ' on S such that ff ' == f'f . 2.3 EXERCISES • 2.1 Which of the following functions are injections and which are surjections ? f1 : R → R , f1 : x → 2x ; + f2 : z → z , f2 : x → 2x ; fg : Z → R , fs : x + 2x ; f1 : R → R ...
Side 25
... tion the rationality of a person just because that person used an irra- tional number such as V2 , we should be aware that 1 + 2i is no more real or imaginary , in the everyday use of these words , than is -3 . Since negative numbers ...
... tion the rationality of a person just because that person used an irra- tional number such as V2 , we should be aware that 1 + 2i is no more real or imaginary , in the everyday use of these words , than is -3 . Since negative numbers ...
Innhold
10 | |
17 | |
24 | |
38 | |
PART ONE ABSOLUTE GEOMETRY | 48 |
INCIDENCE AXIOM AND RULER POSTULATE | 65 |
BETWEENNESS | 75 |
ANGLES AND TRIANGLES | 100 |
SACCHERIS THREE HYPOTHESES | 255 |
19 | 261 |
EUCLIDS PARALLEL POSTULATE | 269 |
BIANGLES | 292 |
26 | 302 |
EXCURSIONS | 317 |
31 | 318 |
PART TWO NONEUCLIDEAN GEOMETRY | 332 |
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 |
CIRCLES | 226 |
ABSOLUTE GEOMETRY AND SACCHERI | 239 |
15 | 253 |
BRUSHES AND CYCLES | 347 |
22228 | 356 |
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 |
Hints and Answers | 494 |
Notation Index | 503 |
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The Foundations of Geometry and the Non-Euclidean Plane G.E. Martin Ingen forhåndsvisning tilgjengelig - 2011 |
Vanlige uttrykk og setninger
AABC absolute geometry Acute Angle ADEF affine plane Assume axiom system biangle Bolyai Cartesian plane CHAPTER circle with center collinear congruent contains convex set coordinate system Corollary cosh defined definition distance scale distinct points elements equal equation equidistant equivalence relation Euclid's Parallel Postulate Euclid's Proposition Euclidean geometry Euclidean plane exactly FIGURE fixes follows Gauss given glide reflection H₁ H₂ halfplane Hence horocircle horoparallel horopencil hyperbolic geometry hyperparallel Hypothesis implies Incidence Axiom Incidence Plane integer interior angles isometry isomorphic LAVB line intersects Lobachevsky mapping mathematics midpoint non-Euclidean geometry obtuse opposite sides PASCH perpendicular bisector Proof Let Proof Suppose Protractor Protractor Postulate prove radius real numbers right angle Ruler Postulate SABCD Saccheri quadrilateral segment set of points sinh statement straight line symmetry tanh Taxicab Geometry theory of parallels three-space tion unique line unique point vertex y₁ y₂