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This undergraduate textbook provides a comprehensive treatment of Euclidean and transformational geometries, supplemented by substantial discussions of topics from various non-Euclidean and less commonly taught geometries, making it ideal for both mathematics majors and pre-service teachers. Emphasis is placed on developing students' deductive reasoning skills as they are guided through proofs, constructions, and solutions to problems. The text frequently emphasizes strategies and heuristics of problem solving including constructing proofs (Where to begin? How to proceed? Which approach is more promising? Are there multiple solutions/proofs? etc.). This approach aims not only to enable students to successfully solve unfamiliar problems on their own, but also to impart a lasting appreciation for mathematics.
The text first explores, at a higher level and in much greater depth, topics that are normally taught in high school geometry courses: definitions and axioms, congruence, circles and related concepts, area and the Pythagorean theorem, similarity, isometries and size transformations, and composition of transformations. Constructions and the use of transformations to carry out constructions are emphasized. The text then introduces more advanced topics dealing with non-Euclidean and less commonly taught topics such as inversive, hyperbolic, elliptic, taxicab, fractal, and solid geometries. By examining what happens when one or more of the building blocks of Euclidean geometry are altered, students will gain a deeper understanding of and appreciation for Euclidean concepts.
To accommodate students with different levels of experience in the subject, the basic definitions and axioms that form the foundation of Euclidean geometry are covered in Chapter 1. Problem sets are provided after every section in each chapter and include nonroutine problems that students will enjoy exploring. While not necessarily required, the appropriate use of freely available dynamic geometry software and other specialized software referenced in the text is strongly encouraged; this is especially important for visual learners and for forming conjectures and testing hypotheses.
Preface
Surprising Results and Basic Notions
Surprising Results and Unexpected Answers
Basic Notions
Congruence, Constructions, and the Parallel Postulate
Angles and Their Measurement
Triangles and Congruence of Triangles
The Parallel Postulate and Its Consequences
Parallel Projection and the Midsegment Theorem
More on Constructions
Circles
Central and Inscribed Angles
Inscribed Circles
More on Constructions
Area and the Pythagorean Theorem
Areas of Polygons
The Pythagorean Theorem
Similarity
Ratio, Proportion, and Similar Polygons
Further Applications of the Side-Splitting Theorem and Similarity
Areas of Similar Figures
The Golden Ratio and the Construction of a Regular Pentagon
Circumference and Area of a Circle
Isometries and Size Transformations
Reflections, Translations, and Rotations
Congruence and Euclidean Constructions
More on Extremal Problems
Similarity Transformation with Applications to Constructions
Composition of Transformations
Introduction
In Search of New Isometries
Composition of Rotations, the Treasure Island Problem, and Other Treasures
More Recent Discoveries
The Nine-Point Circle and Other Results
Complex Numbers and Geometry
Inversion
Introduction
Properties of Inversions
Applications of Inversions
The Nine-Point Circle and Feuerbach's Theorem
Stereographic Projection and Inversion
Hyperbolic Geometry
Introduction
Hyperbolic Geometry
Models of Hyperbolic Geometry
Compass and Straightedge Constructions in the Poincaré Disc Model D
Hyperbolic Tessellations
Elliptic Geometries
Introduction and Basic Results
Models of Elliptic Geometry
Projective Geometry
Introduction and Early Results
Projective Planes
The Real Projective Plane
Homogeneous Coordinates
Duality: Poles, Polars, and Reciprocation
Polar Circles and Self-Polar Triangles
Taxicab Geometry
Introduction
Taxicab Versus Euclidean
Distance from a Point to a Line
Taxicab Midsets
Circle(s) Through Three Points
Conics in Taxicab Geometry
Taxicab Incircles, Circumcircles, Excircles, and Apollonius' Circle
Inversion in Taxicab Geometry
Fractal Geometry
Introduction
Fractal Dimension
Affine Transformations
Iterated Function Systems
The Julia and Mandelbrot Sets
Linear Transformations and Matrices: A Brief Summary
Solid Geometry
Objectives
Fundamental Concepts
Polyhedra
Descartes' Lost Theorem
Euler's Formula and Its Consequences
Bibliography
Index