Synopses & Reviews
Topics including harmonic division and Apollonian circles, inversive geometry, the hexlet, conic sections, projective geometry, the Golden Section and angle trisection are addressed in a way that brings out the true intellectual excitement inherent in each. Also included: some unsolved problems of modern geometry. Notes. References. 132 line illustrations.
Synopsis
"A charming, entertaining, and instructive book .... The writing is exceptionally lucid, as in the author's earlier books, ... and the problems carefully selected for maximum interest and elegance." -- Martin Gardner.
This book is intended for people who liked geometry when they first encountered it (and perhaps even some who did not) but sensed a lack of intellectual stimulus and wondered what was missing, or felt that the play was ending just when the plot was finally becoming interesting.
In this superb treatment, Professor Ogilvy demonstrates the mathematical challenge and satisfaction to be had from geometry, the only requirements being two simple implements (straightedge and compass) and a little thought. Avoiding topics that require an array of new definitions and abstractions, Professor Ogilvy draws upon material that is either self-evident in the classical sense or very easy to prove. Among the subjects treated are: harmonic division and Apollonian circles, inversion geometry, the hexlet, conic sections, projective geometry, the golden section, and angle trisection. Also included are some unsolved problems of modern geometry, including Malfatti's problem and the Kakeya problem.
Numerous diagrams, selected references, and carefully chosen problems enhance the text. In addition, the helpful section of notes at the back provides not only source references but also much other material highly useful as a running commentary on the text.
Synopsis
A straightedge, compass, and a little thought are all that's needed to discover the intellectual excitement of geometry. Harmonic division and Apollonian circles, inversive geometry, hexlet, Golden Section, more. 132 illustrations.
Description
Includes bibliographical references (p. 155-173) and index.
Table of Contents
Introduction
1 A bit of background
A practical problem
A basic theorem
Means
2 Harmonic division and Apollonian circles
Harmonic conjugates
The circle of Apollonius
Coaxial families
3 Inversive geometry
Transformations
Inversion
Invariants
Cross-ratio
4 Application for inversive geometry
Two easy problems
Peaucellier's linkage
Apollonius' problem
Steiner chains
The arbelos
5 The hexlet
The conics defined
A property of chains
Soddy's hexlet
Some new hexlets
6 The conic sections
The reflection property
Confocal conics
Plan sections of a cone
A characteristic of parabolas
7 Projective geometry
Projective transformations
The foundations
Cross-ratio
The complete quadrangle
Pascal's Theorem
Duality
8 Some Euclidean topics
A navigation problem
A three-circle problem
The Euler line
The nine-point circle
A triangle problem
9 The golden section
The pentagram
Similarities and spirals
The regular polyhedra
The continued fraction for ø
10 Angle trisection
The unsolved problems of antiquity
Other kinds of trisection
11 Some unsolved problems of modern geometry
Convex sets and geometric inequalities
Malfatti's problem
The Kakeya problem
Notes
Index