Articoli correlati a The Foundations of Geometry and the Non-Euclidean Plane
Sinossi
1. Equivalence Relations.- 1.1 Logic.- 1.2 Sets.- 1.3 Relations.- 1.4 Exercises.- Graffiti.- 2 Mappings.- 2.1 One-to-One and Onto.- 2.2 Composition of Mappings.- 2.3 Exercises.- Graffiti.- 3 The Real Numbers.- 3.1 Binary Operations.- 3.2 Properties of the Reals.- 3.3 Exercises.- Graffiti.- 4 Axiom Systems.- 4.1 Axiom Systems.- 4.2 Incidence Planes.- 4.3 Exercises.- Graffiti.- One Absolute Geometry.- 5 Models.- 5.1 Models of the Euclidean Plane.- 5.2 Models of Incidence Planes.- 5.3 Exercises.- Graffiti.- 6 Incidence Axiom and Ruler Postulate.- 6.1 Our Objectives.- 6.2 Axiom 1: The Incidence Axiom.- 6.3 Axiom 2: The Ruler Postulate.- 6.4 Exercises.- Graffiti.- 7 Betweenness.- 7.1 Ordering the Points on a Line.- 7.2 Taxicab Geometry.- 7.3 Exercises.- Graffiti.- 8 Segments, Rays, and Convex Sets.- 8.1 Segments and Rays.- 8.2 Convex Sets.- 8.3 Exercises.- Graffiti.- 9 Angles and Triangles.- 9.1 Angles and Triangles.- 9.2 More Models.- 9.3 Exercises.- Graffiti.- 10 The Golden Age of Greek Mathematics (Optional).- 10.1 Alexandria.- 10.2 Exercises.- 11 Euclid'S Elements (Optional).- 11.1 The Elements.- 11.2 Exercises.- Graffiti.- 12 Pasch's Postulate and Plane Separation Postulate.- 12.1 Axiom 3: PSP.- 12.2 Pasch, Peano, Pieri, and Hilbert.- 12.3 Exercises.- Graffiti.- 13 Crossbar and Quadrilaterals.- 13.1 More Incidence Theorems.- 13.2 Quadrilaterals.- 13.3 Exercises.- Graffiti.- 14 Measuring Angles and the Protractor Postulate.- 14.1 Axiom 4: The Protractor Postulate.- 14.2 Peculiar Protractors.- 14.3 Exercises.- 15 Alternative Axiom Systems (Optional).- 15.1 Hilbert's Axioms.- 15.2 Pieri's Postulates.- 15.3 Exercises.- 16 Mirrors.- 16.1 Rulers and Protractors.- 16.2 MIRROR and SAS.- 16.3 Exercises.- Graffiti.- 17 Congruence and the Penultimate Postulate.- 17.1 Congruence for Triangles.- 17.2 Axiom 5: SAS.- 17.3 Congruence Theorems.- 17.4 Exercises.- Graffiti.- 18 Perpendiculars and Inequalities.- 18.1 A Theorem on Parallels.- 18.2 Inequalities.- 18.3 Right Triangles.- 18.4 Exercises.- Graffiti.- 19 Reflections.- 19.1 Introducing Isometries.- 19.2 Reflection in a Line.- 19.3 Exercises.- Graffiti.- 20 Circles.- 20.1 Introducing Circles.- 20.2 The Two-Circle Theorem.- 20.3 Exercises.- Graffiti.- 21 Absolute Geometry and Saccheri Quadrilaterals.- 21.1 Euclid's Absolute Geometry.- 21.2 Giordano's Theorem.- 21.3 Exercises.- Graffiti.- 22 Saccherfs Three Hypotheses.- 22.1 Omar Khayyam's Theorem.- 22.2 Saccheri's Theorem.- 22.3 Exercises.- Graffiti.- 23 Euclid's Parallel Postulate.- 23.1 Equivalent Statements.- 23.2 Independence.- 23.3 Exercises.- Graffiti.- 24 Biangles.- 24.1 Closed Biangles.- 24.2 Critical Angles and Absolute Lengths.- 24.3 The Invention of Non-Euclidean Geometry.- 24.4 Exercises.- Graffiti.- 25 Excursions.- 25.1 Prospectus.- 25.2 Euclidean Geometry.- 25.3 Higher Dimensions.- 25.4 Exercises.- Graffiti.- Two Non-Euclidean Geometry.- 26 Parallels and the Ultimate Axiom.- 26.1 Axiom 6: HPP.- 26.2 Parallel Lines.- 26.3 Exercises.- Graffiti.- 27 Brushes and Cycles.- 27.1 Brushes.- 27.2 Cycles.- 27.3 Exercises.- Graffiti.- 28 Rotations, Translations, and Horolations.- 28.1 Products of Two Reflections.- 28.2 Reflections in Lines of a Brush.- 28.3 Exercises.- Graffiti.- 29 The Classification of Isometries.- 29.1 Involutions.- 29.2 The Classification Theorem.- 29.3 Exercises.- Graffiti.- 30 Symmetry.- 30.1 Leonardo's Theorem.- 30.2 Frieze Patterns.- 30.3 Exercises.- Graffiti.- 31 HOrocircles.- 31.1 Length of Arc.- 31.2 Hyperbolic Functions.- 31.3 Exercises.- Graffiti.- 32 The Fundamental Formula.- 32.1 Trigonometry.- 32.2 Complementary Segments.- 32.3 Exercises.- Graffiti.- 33 Categoricalness and Area.- 33.1 Analytic Geometry.- 33.2 Area.- 33.3 Exercises.- Graffiti.- 34 Quadrature of the Circle.- 34.1 Classical Theorems.- 34.2 Calculus.- 34.3 Constructions.- 34.4 Exercises.- Hints and Answers.- Notation Index.
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