Four Pillars of Geometry

ISBN-10: 0387255303
ISBN-13: 9780387255309
Edition: 2005
Authors: John Stillwell
List price: $54.95 Buy it from $31.24
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Description: This new textbook demonstrates that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start,  More...

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Book details

List price: $54.95
Copyright year: 2005
Publisher: Springer
Publication date: 8/9/2005
Binding: Hardcover
Pages: 228
Size: 6.25" wide x 9.25" long x 0.75" tall
Weight: 1.320
Language: English

This new textbook demonstrates that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. In this book, two chapters are devoted to each approach, the first being concrete and introductory, while the second is more abstract. Geometry, of all subjects, should be about taking different viewpoints, and geometry is unique among mathematical disciplines in its ability to look different from different angles. Some students prefer to visualize, while others prefer to reason or to calculate. Geometry has something for everyone, and students will find themselves building on their strengths at times, and working to overcome weaknesses at other times. This book will be suitable for a second course in geometry and contains more than 100 figures and a large selection of exercises in each chapter.

Preface
Straightedge and compass
Euclid's construction axioms
Euclid's construction of the equilateral triangle
Some basic constructions
Multiplication and division
Similar triangles
Discussion
Euclid's approach to geometry
The parallel axiom
Congruence axioms
Area and equality
Area of parallelograms and triangles
The Pythagorean theorem
Proof of the Thales theorem
Angles in a circle
The Pythagorean theorem revisited
Discussion
Coordinates
The number line and the number plane
Lines and their equations
Distance
Intersections of lines and circles
Angle and slope
Isometries
The three reflections theorem
Discussion
Vectors and Euclidean spaces
Vectors
Direction and linear independence
Midpoints and centroids
The inner product
Inner product and cosine
The triangle inequality
Rotations, matrices, and complex numbers
Discussion
Perspective
Perspective drawing
Drawing with straightedge alone
Projective plane axioms and their models
Homogeneous coordinates
Projection
Linear fractional functions
The cross-ratio
What is special about the cross-ratio?
Discussion
Projective planes
Pappus and Desargues revisited
Coincidences
Variations on the Desargues theorem
Projective arithmetic
The field axioms
The associative laws
The distributive law
Discussion
Transformations
The group of isometries of the plane
Vector transformations
Transformations of the projective line
Spherical geometry
The rotation group of the sphere
Representing space rotations by quaternions
A finite group of space rotations
The groups S[superscript 3] and RP[superscript 3]
Discussion
Non-Euclidean geometry
Extending the projective line to a plane
Complex conjugation
Reflections and Mobius transformations
Preserving non-Euclidean lines
Preserving angle
Non-Euclidean distance
Non-Euclidean translations and rotations
Three reflections or two involutions
Discussion
References
Index

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