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This book emphasizes the beauty of geometry using a modern approach. Models & computer exercises help readers to cultivate geometric intuition. Topics include Euclidean Geometry, Hand Constructions, Geometer's Sketch Pad, Hyperbolic Geometry, Tilings & Lattices, Spherical Geometry, Projective Geometry, Finite Geometry, and Modern Geometry Research.
Ideal for geometry at an intermediate level.
Introduction.
The Geometry of Our World. A Review of Terminology. Notes on Notation. Notes on the Exercises.
1. Euclidean Geometry.
The Pythagorean Theorem. The Axioms of Euclidean Geometry. SSS, SAS, and ASA. Parallel Lines. Pons Asinorum. The Star Trek Lemma. Similar Triangles. Power of the Point. The Medians and Centroid. The Incircle, Excircles, and the Law of Cosines. The Circumcircle and the Law of Sines. The Euler Line. The Nine Point Circle. Pedal Triangles and the Simson Line. Menelaus and Ceva.
2. Geometry in Greek Astronomy.
The Relative Size of the Moon and Sun. The Diameter of the Earth. The Babylonians to Kepler, a Time Line.
3. Constructions Using a Compass and Straightedge.
The Rules. Some Examples. Basic Results. The Algebra of Constructible Lengths. The Regular Pentagon. Other Constructible Figures. Trisecting an Arbitrary Angle.
4. Geometer's Sketchpad.
The Rules of Constructions. Lemmas and Theorems. Archimedes' Trisection Algorithm. Verification of Theorems. Sophisticated Results. Parabola Paper.
5. Higher Dimensional Objects.
The Platonic Solids. The Duality of Platonic Solids. The Euler Characteristic. Semiregular Polyhedra. A Partial Categorization of Semiregular Polyhedra. Four-Dimensional Objects.
6. Hyperbolic Geometry.
Models. Results from Neutral Geometry. The Congruence of Similar Triangles. Parallel and Ultraparallel Lines. Singly Asymptotic Triangles. Doubly and Triply Asymptotic Triangles. The Area of Asymptotic Triangles.
7. The Poincaré Models of Hyperbolic Geometry.
The Poincaré Upper Half Plane Model. Vertical (Euclidean) Lines. Isometries. Inversion in the Circle. Inversion in Euclidean Geometry. Fractional Linear Transformations. The Cross Ratio. Translations. Rotations. Reflections. Lengths. The Axioms of Hyperbolic Geometry. The Area of Triangles. The Poincaré Disc Model. Circles and Horocycles. Hyperbolic Trigonometry. The Angle of Parallelism. Curvature.
8. Tilings and Lattices.
Regular Tilings. Semiregular Tilings. Lattices and Fundamental Domains. Tilings in Hyperbolic Space. Tilings in Art.
9. Foundations.
Theories. The Real Line. The Plane. Line Segments and Lines. Separation Axioms. Circles. Isometries and Congruence. The Parallel Postulate. Similar Triangles.
10. Spherical Geometry.
The Area of Triangles. The Geometry of Right Triangles. The Geometry of Spherical Triangles. Menelaus' Theorem. Heron's Formula. Tilings of the Sphere. The Axioms. Elliptic Geometry.
11. Projective Geometry.
Moving a Line to Infinity. Pascal's Theorem. Projective Coordinates. Duality. Dual Conics and Brianchon's Theorem. Areal Coordinates.
12. The Pseudosphere in Lorentz Space.
The Sphere as a Foil. The Pseudosphere. Angles and the Lorentz Cross Product. A Different Perspective. The Beltrami-Klein Model. Menelaus' Theorem.
13. Finite Geometry.
Algebraic Affine Planes. Algebraic Projective Planes. Weak Incidence Geometry. Geometric Projective Planes. Addition. Multiplication. The Distributive Law. Commutativity, Coordinates, and Pappus' Theorem. Weak Projective Space and Desargues' Theorem.
14. Nonconstructibility.
The Field of Constructible Numbers. Fields as Vector Spaces. The Field of Definition for a Construction. The Regular 7-gon. The Regular 17-gon.
15. Modern Research in Geometry.
Pythagorean Triples. Bezout's Theorem. Elliptic Curves. A Mixture of Cevians. A Challenge for Fermat. The Euler Characteristic in Algebraic Geometry. Lattice Point Problems. Fractals and the Apollonian Packing Problem. Sphere Packing.
16. A Selective Time Line of Mathematics.
The Ancient Greeks. The Fifth Century A.D. to the Fifteenth Century A.D. The Renaissance to the Present.
Appendix A: Quick Reviews.
2x2 Matrices. Vector Geometry. Groups. Modular Arithmetic.
Appendix B: Hints, Answers and Solutions.
Hints to Selected Problems. Answers to Selected Problems. Solutions to Selected Problems.
Bibliography.
Index.
This book emphasizes the beauty of geometry using a modern approach. Models & computer exercises help readers to cultivate geometric intuition. Topics include Euclidean Geometry, Hand Constructions, Geometer's Sketch Pad, Hyperbolic Geometry, Tilings & Lattices, Spherical Geometry, Projective Geometry, Finite Geometry, and Modern Geometry Research.
Ideal for geometry at an intermediate level.
Table of Contents
Introduction.
The Geometry of Our World. A Review of Terminology. Notes on Notation. Notes on the Exercises.
1. Euclidean Geometry.
The Pythagorean Theorem. The Axioms of Euclidean Geometry. SSS, SAS, and ASA. Parallel Lines. Pons Asinorum. The Star Trek Lemma. Similar Triangles. Power of the Point. The Medians and Centroid. The Incircle, Excircles, and the Law of Cosines. The Circumcircle and the Law of Sines. The Euler Line. The Nine Point Circle. Pedal Triangles and the Simson Line. Menelaus and Ceva.
2. Geometry in Greek Astronomy.
The Relative Size of the Moon and Sun. The Diameter of the Earth. The Babylonians to Kepler, a Time Line.
3. Constructions Using a Compass and Straightedge.
The Rules. Some Examples. Basic Results. The Algebra of Constructible Lengths. The Regular Pentagon. Other Constructible Figures. Trisecting an Arbitrary Angle.
4. Geometer's Sketchpad.
The Rules of Constructions. Lemmas and Theorems. Archimedes' Trisection Algorithm. Verification of Theorems. Sophisticated Results. Parabola Paper.
5. Higher Dimensional Objects.
The Platonic Solids. The Duality of Platonic Solids. The Euler Characteristic. Semiregular Polyhedra. A Partial Categorization of Semiregular Polyhedra. Four-Dimensional Objects.
6. Hyperbolic Geometry.
Models. Results from Neutral Geometry. The Congruence of Similar Triangles. Parallel and Ultraparallel Lines. Singly Asymptotic Triangles. Doubly and Triply Asymptotic Triangles. The Area of Asymptotic Triangles.
7. The Poincaré Models of Hyperbolic Geometry.
The Poincaré Upper Half Plane Model. Vertical (Euclidean) Lines. Isometries. Inversion in the Circle. Inversion in Euclidean Geometry. Fractional Linear Transformations. The Cross Ratio. Translations. Rotations. Reflections. Lengths. The Axioms of Hyperbolic Geometry. The Area of Triangles. The Poincaré Disc Model. Circles and Horocycles. Hyperbolic Trigonometry. The Angle of Parallelism. Curvature.
8. Tilings and Lattices.
Regular Tilings. Semiregular Tilings. Lattices and Fundamental Domains. Tilings in Hyperbolic Space. Tilings in Art.
9. Foundations.
Theories. The Real Line. The Plane. Line Segments and Lines. Separation Axioms. Circles. Isometries and Congruence. The Parallel Postulate. Similar Triangles.
10. Spherical Geometry.
The Area of Triangles. The Geometry of Right Triangles. The Geometry of Spherical Triangles. Menelaus' Theorem. Heron's Formula. Tilings of the Sphere. The Axioms. Elliptic Geometry.
11. Projective Geometry.
Moving a Line to Infinity. Pascal's Theorem. Projective Coordinates. Duality. Dual Conics and Brianchon's Theorem. Areal Coordinates.
12. The Pseudosphere in Lorentz Space.
The Sphere as a Foil. The Pseudosphere. Angles and the Lorentz Cross Product. A Different Perspective. The Beltrami-Klein Model. Menelaus' Theorem.
13. Finite Geometry.
Algebraic Affine Planes. Algebraic Projective Planes. Weak Incidence Geometry. Geometric Projective Planes. Addition. Multiplication. The Distributive Law. Commutativity, Coordinates, and Pappus' Theorem. Weak Projective Space and Desargues' Theorem.
14. Nonconstructibility.
The Field of Constructible Numbers. Fields as Vector Spaces. The Field of Definition for a Construction. The Regular 7-gon. The Regular 17-gon.
15. Modern Research in Geometry.
Pythagorean Triples. Bezout's Theorem. Elliptic Curves. A Mixture of Cevians. A Challenge for Fermat. The Euler Characteristic in Algebraic Geometry. Lattice Point Problems. Fractals and the Apollonian Packing Problem. Sphere Packing.
16. A Selective Time Line of Mathematics.
The Ancient Greeks. The Fifth Century A.D. to the Fifteenth Century A.D. The Renaissance to the Present.
Appendix A: Quick Reviews.
2x2 Matrices. Vector Geometry. Groups. Modular Arithmetic.
Appendix B: Hints, Answers and Solutions.
Hints to Selected Problems. Answers to Selected Problems. Solutions to Selected Problems.
Bibliography.
Index.