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libeskind s euclidean non euclidean transformational geometry 2024

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Textbook in PDF format 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

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Libeskind S. Euclidean, Non-Euclidean, and Transformational Geometry...2024.pdf 20.1 MB

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libeskind s euclidean non euclidean transformational geometry 2024

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