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Description:

This book emphasizes the beauty of geometry using a modern approach. Models & computer exercises help readers to cultivate geometric intuition. Topics include Euclidean Geometry, Hand Constructions, Geometer's Sketch Pad, Hyperbolic Geometry, Tilings & Lattices, Spherical Geometry, Projective Geometry, Finite Geometry, and Modern Geometry Research. Ideal for geometry at an intermediate level.

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Book details

List price: $119.99 Copyright year: 2001 Publisher: Prentice Hall PTR Publication date: 12/20/2000 Binding: Paperback Pages: 384 Size: 6.25" wide x 9.25" long x 0.75" tall Weight: 1.496 Language: English

AuthorTable of Contents

Introduction

The Geometry of Our World

A Review of Terminology

Notes on Notation

Notes on the Exercises

Euclidean Geometry

The Pythagorean Theorem

The Axioms of Euclidean Geometry

SSS, SAS, and ASA

Parallel Lines

Pons Asinorum

The Star Trek Lemma

Similar Triangles

Power of the Point

The Medians and Centroid

The Incircle, Excircles, and the Law of Cosines

The Circumcircle and the Law of Sines

The Euler Line

The Nine Point Circle

Pedal Triangles and the Simson Line

Menelaus and Ceva

Geometry in Greek Astronomy

The Relative Size of the Moon and Sun

The Diameter of the Earth

The Babylonians to Kepler, a Time Line

Constructions Using a Compass and Straightedge

The Rules

Some Examples

Basic Results

The Algebra of Constructible Lengths

The Regular Pentagon

Other Constructible Figures

Trisecting an Arbitrary Angle

Geometer's Sketchpad

The Rules of Constructions

Lemmas and Theorems

Archimedes' Trisection Algorithm

Verification of Theorems

Sophisticated Results

Parabola Paper

Higher Dimensional Objects

The Platonic Solids

The Duality of Platonic Solids

The Euler Characteristic

Semiregular Polyhedra

A Partial Categorization of Semiregular Polyhedra

Four-Dimensional Objects

Hyperbolic Geometry

Models

Results from Neutral Geometry

The Congruence of Similar Triangles

Parallel and Ultraparallel Lines

Singly Asymptotic Triangles

Doubly and Triply Asymptotic Triangles

The Area of Asymptotic Triangles

The Poincare Models of Hyperbolic Geometry

The Poincare Upper Half Plane Model

Vertical (Euclidean) Lines

Isometries

Inversion in the Circle

Inversion in Euclidean Geometry

Fractional Linear Transformations

The Cross Ratio

Translations

Rotations

Reflections

Lengths

The Axioms of Hyperbolic Geometry

The Area of Triangles

The Poincare Disc Model

Circles and Horocycles

Hyperbolic Trigonometry

The Angle of Parallelism

Curvature

Tilings and Lattices

Regular Tilings

Semiregular Tilings

Lattices and Fundamental Domains

Tilings in Hyperbolic Space

Tilings in Art

Foundations

Theories

The Real Line

The Plane

Line Segments and Lines

Separation Axioms

Circles

Isometries and Congruence

The Parallel Postulate

Similar Triangles

Spherical Geometry

The Area of Triangles

The Geometry of Right Triangles

The Geometry of Spherical Triangles

Menelaus' Theorem

Heron's Formula

Tilings of the Sphere

The Axioms

Elliptic Geometry

Projective Geometry

Moving a Line to Infinity

Pascal's Theorem

Projective Coordinates

Duality

Dual Conics and Brianchon's Theorem

Areal Coordinates

The Pseudosphere in Lorentz Space

The Sphere as a Foil

The Pseudosphere

Angles and the Lorentz Cross Product

A Different Perspective

The Beltrami-Klein Model

Menelaus' Theorem

Finite Geometry

Algebraic Affine Planes

Algebraic Projective Planes

Weak Incidence Geometry

Geometric Projective Planes

Addition

Multiplication

The Distributive Law

Commutativity, Coordinates, and Pappus' Theorem

Weak Projective Space and Desargues' Theorem

Nonconstructibility

The Field of Constructible Numbers

Fields as Vector Spaces

The Field of Definition for a Construction

The Regular 7-gon

The Regular 17-gon

Modern Research in Geometry

Pythagorean Triples

Bezout's Theorem

Elliptic Curves

A Mixture of Cevians

A Challenge for Fermat

The Euler Characteristic in Algebraic Geometry

Lattice Point Problems

Fractals and the Apollonian Packing Problem. S

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