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. |
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Resultat 1-5 av 69
Side iii
... Equations . Exner : An Accompaniment to Higher Mathematics . Fine / Rosenberger : The Fundamental Theory of Algebra . Fischer : Intermediate Real Analysis . Flanigan / Kazdan : Calculus Two : Linear and Nonlinear Functions . Second ...
... Equations . Exner : An Accompaniment to Higher Mathematics . Fine / Rosenberger : The Fundamental Theory of Algebra . Fischer : Intermediate Real Analysis . Flanigan / Kazdan : Calculus Two : Linear and Nonlinear Functions . Second ...
Side 5
... equation ax + by + c = 0. Geometrically , an element is the set of all points on some line in the Cartesian plane . Thus , thinking of a line as a set of points , our for- midable looking set is the set of all lines in the Cartesian ...
... equation ax + by + c = 0. Geometrically , an element is the set of all points on some line in the Cartesian plane . Thus , thinking of a line as a set of points , our for- midable looking set is the set of all lines in the Cartesian ...
Side 23
... equation , one goes on to deduce that a and b are both even . The contradiction proves that the original assumption must be false . The details of this historically famous proof are left for Ex- ercise 3.1 . Considering the set of all ...
... equation , one goes on to deduce that a and b are both even . The contradiction proves that the original assumption must be false . The details of this historically famous proof are left for Ex- ercise 3.1 . Considering the set of all ...
Side 25
... equation z2 = -1 has no solution in R but has a solution i in C where i = 0 + 1i . Complex numbers are usually introduced in high school algebra so that all quadratic equations with real coefficients have solutions . A complex number x ...
... equation z2 = -1 has no solution in R but has a solution i in C where i = 0 + 1i . Complex numbers are usually introduced in high school algebra so that all quadratic equations with real coefficients have solutions . A complex number x ...
Side 37
... equation in x and y . The plane is named after René Descartes ( 1596-1650 ) . The Real Cartesian Inci- dence Plane is an example of the first of three types of incidence planes that we shall consider . Axiom System I An affine plane is ...
... equation in x and y . The plane is named after René Descartes ( 1596-1650 ) . The Real Cartesian Inci- dence Plane is an example of the first of three types of incidence planes that we shall consider . Axiom System I An affine plane is ...
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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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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