Exploring Geometry
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Exploring Geometry

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Description

This text 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.
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Product details

  • Hardback | 538 pages
  • 156 x 235 x 31.75mm | 1,111g
  • Productivity Press
  • Portland, United States
  • English
  • New edition
  • 2nd New edition
  • 4 Tables, black and white; 557 Illustrations, black and white
  • 1498760805
  • 9781498760805
  • 2,061,183

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
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