Exploring Geometry: 2nd Edition (Hardback) book cover

Exploring Geometry

2nd Edition

By Michael Hvidsten

Chapman and Hall/CRC

538 pages | 557 B/W Illus.

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Hardback: 9781498760805
pub: 2016-12-05
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Description

Exploring Geometry, Second Edition promotes student engagement with the beautiful ideas of geometry. Every major concept is introduced in its historical context and connects the idea with real-life. A system of experimentation followed by rigorous explanation and proof is central. Exploratory projects play an integral role in this text. Students develop a better sense of how to prove a result and visualize connections between statements, making these connections real. They develop the intuition needed to conjecture a theorem and devise a proof of what they have observed.

Features:

  • Second edition of a successful textbook for the first undergraduate course
  • Every major concept is introduced in its historical context and connects the idea with real life
  • Focuses on experimentation
  • Projects help enhance student learning
  • All major software programs can be used; free software from author
  • Table of Contents

    Geometry and the Axiomatic Method

    Early Origins of Geometry

    Thales and Pythagoras

    Project 1 - The Ratio Made of Gold

    The Rise of the Axiomatic Method

    Properties of the Axiomatic Systems

    Euclid's Axiomatic Geometry

    Project 2 - A Concrete Axiomatic System

    Euclidean Geometry

    Angles, Lines, and Parallels ANGLES, LINES, AND PARALLELS 51

    Congruent Triangles and Pasch's Axiom

    Project 3 - Special Points of a Triangle

    Measurement and Area

    Similar Triangles

    Circle Geometry

    Project 4 - Circle Inversion and Orthogonality

    Analytic Geometry

    The Cartesian Coordinate System

    Vector Geometry

    Project 5 - Bezier Curves

    Angles in Coordinate Geometry

    The Complex Plane

    Birkhoff's Axiomatic System

    Constructions

    Euclidean Constructions

    Project 6 - Euclidean Eggs

    Constructibility

    Transformational Geometry

    Euclidean Isometries

    Reflections

    Translations

    Rotations

    Project 7 - Quilts and Transformations

    Glide Reflections

    Structure and Representation of Isometries

    Project 8 - Constructing Compositions

    Symmetry

    Finite Plane Symmetry Groups

    Frieze Groups

    Wallpaper Groups

    Tilting the Plane

    Project 9 - Constructing Tesselations

    Hyperbollic Geometry

    Background and History

    Models of Hyperbolic Geometry

    Basic Results in Hyperbolic Geometry

    Project 10 - The Saccheri Quadrilateral

    Lambert Quadrilaterals and Triangles

    Area in Hyperbolic Geometry

    Project 11 - Tilting the Hyperbolic Plane

    Elliptic Geometry

    Background and History

    Perpendiculars and Poles in Elliptic Geometry

    Project 12 - Models of Elliptic Geometry

    Basic Results in Elliptic Geometry

    Triangles and Area in Elliptic Geometry

    Project 13 - Elliptic Tiling

    Projective Geometry

    Universal Themes

    Project 14 - Perspective and Projection

    Foundations of Projective Geometry

    Transformations and Pappus's Theorem

    Models of Projective Geometry

    Project 15 - Ratios and Harmonics

    Harmonic Sets

    Conics and Coordinates

    Fractal Geometry

    The Search for a "Natural" Geometry

    Self-Similarity

    Similarity Dimension

    Project 16 - An Endlessly Beautiful Snowflake

    Contraction Mappings

    Fractal Dimension

    Project 17 - IFS Ferns

    Algorithmic Geometry

    Grammars and Productions

    Project 18 - Words Into Plants

    Appendix A: A Primer on Proofs

    Appendix A □ A Primer on Proofs 497

    Appendix B □ Book I of Euclid’s Elements

    Appendix C □ Birkhoff’s Axioms

    Appendix D □ Hilbert’s Axioms

    Appendix E □ Wallpaper Groups

    About the Series

    Textbooks in Mathematics

    Learn more…

    Subject Categories

    BISAC Subject Codes/Headings:
    MAT000000
    MATHEMATICS / General
    MAT012000
    MATHEMATICS / Geometry / General
    MAT025000
    MATHEMATICS / Recreations & Games