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Introduction | |

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The Geometry of Our World | |

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A Review of Terminology | |

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Notes on Notation | |

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Notes on the Exercises | |

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Euclidean Geometry | |

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The Pythagorean Theorem | |

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The Axioms of Euclidean Geometry | |

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SSS, SAS, and ASA | |

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Parallel Lines | |

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Pons Asinorum | |

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The Star Trek Lemma | |

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Similar Triangles | |

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Power of the Point | |

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The Medians and Centroid | |

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The Incircle, Excircles, and the Law of Cosines | |

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The Circumcircle and the Law of Sines | |

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The Euler Line | |

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The Nine Point Circle | |

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Pedal Triangles and the Simson Line | |

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Menelaus and Ceva | |

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Geometry in Greek Astronomy | |

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The Relative Size of the Moon and Sun | |

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The Diameter of the Earth | |

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The Babylonians to Kepler, a Time Line | |

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Constructions Using a Compass and Straightedge | |

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The Rules | |

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Some Examples | |

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Basic Results | |

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The Algebra of Constructible Lengths | |

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The Regular Pentagon | |

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Other Constructible Figures | |

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Trisecting an Arbitrary Angle | |

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Geometer's Sketchpad | |

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The Rules of Constructions | |

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Lemmas and Theorems | |

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Archimedes' Trisection Algorithm | |

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Verification of Theorems | |

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Sophisticated Results | |

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Parabola Paper | |

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Higher Dimensional Objects | |

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The Platonic Solids | |

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The Duality of Platonic Solids | |

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The Euler Characteristic | |

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Semiregular Polyhedra | |

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A Partial Categorization of Semiregular Polyhedra | |

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Four-Dimensional Objects | |

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Hyperbolic Geometry | |

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Models | |

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Results from Neutral Geometry | |

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The Congruence of Similar Triangles | |

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Parallel and Ultraparallel Lines | |

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Singly Asymptotic Triangles | |

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Doubly and Triply Asymptotic Triangles | |

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The Area of Asymptotic Triangles | |

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The Poincare Models of Hyperbolic Geometry | |

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The Poincare Upper Half Plane Model | |

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Vertical (Euclidean) Lines | |

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Isometries | |

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Inversion in the Circle | |

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Inversion in Euclidean Geometry | |

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Fractional Linear Transformations | |

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The Cross Ratio | |

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Translations | |

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Rotations | |

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Reflections | |

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Lengths | |

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The Axioms of Hyperbolic Geometry | |

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The Area of Triangles | |

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The Poincare Disc Model | |

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Circles and Horocycles | |

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Hyperbolic Trigonometry | |

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The Angle of Parallelism | |

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Curvature | |

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Tilings and Lattices | |

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Regular Tilings | |

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Semiregular Tilings | |

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Lattices and Fundamental Domains | |

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Tilings in Hyperbolic Space | |

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Tilings in Art | |

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Foundations | |

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Theories | |

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The Real Line | |

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The Plane | |

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Line Segments and Lines | |

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Separation Axioms | |

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Circles | |

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Isometries and Congruence | |

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The Parallel Postulate | |

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Similar Triangles | |

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Spherical Geometry | |

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The Area of Triangles | |

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The Geometry of Right Triangles | |

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The Geometry of Spherical Triangles | |

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Menelaus' Theorem | |

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Heron's Formula | |

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Tilings of the Sphere | |

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The Axioms | |

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Elliptic Geometry | |

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Projective Geometry | |

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Moving a Line to Infinity | |

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Pascal's Theorem | |

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Projective Coordinates | |

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Duality | |

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Dual Conics and Brianchon's Theorem | |

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Areal Coordinates | |

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The Pseudosphere in Lorentz Space | |

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The Sphere as a Foil | |

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The Pseudosphere | |

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Angles and the Lorentz Cross Product | |

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A Different Perspective | |

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The Beltrami-Klein Model | |

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Menelaus' Theorem | |

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Finite Geometry | |

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Algebraic Affine Planes | |

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Algebraic Projective Planes | |

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Weak Incidence Geometry | |

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Geometric Projective Planes | |

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Addition | |

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Multiplication | |

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The Distributive Law | |

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Commutativity, Coordinates, and Pappus' Theorem | |

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Weak Projective Space and Desargues' Theorem | |

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Nonconstructibility | |

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The Field of Constructible Numbers | |

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Fields as Vector Spaces | |

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The Field of Definition for a Construction | |

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The Regular 7-gon | |

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The Regular 17-gon | |

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Modern Research in Geometry | |

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Pythagorean Triples | |

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Bezout's Theorem | |

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Elliptic Curves | |

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A Mixture of Cevians | |

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A Challenge for Fermat | |

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The Euler Characteristic in Algebraic Geometry | |

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Lattice Point Problems | |

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Fractals and the Apollonian Packing Problem. S | |