A Course in Modern Geometries

by Cederberg, Judith N.

• ISBN13: 9780387989723
• ISBN10: 0387989722
• eBook ISBN(s): 9781475734904
• Edition: 2nd
• Format: Hardcover
• Copyright: 2000-12-01
• Publisher: Springer Nature
• More Book Details

Summary

This is a new edition of a highly accessible presentation on modern geometry. Each chapter includes a list of suggested resources for applications or related topics in areas such as art and history. This second edition contains new material throughout, including a new chapter on chaos theory and fractal geometry. The book now includes pointers to the author's website, which contains guides for dynamic software explorations of many concepts, using both Cabri Geometry and Geometers Sketchpad.

Author Biography

Judith N. Cederberg is an associate professor of mathematics at St. Olaf College in Minnesota.

Table of Contents

Preface to the Second Edition	v
Preface to the First Edition	xiii
Axiomatic Systems and Finite Geometries	1	(32)
Gaining Perspective*	1	(1)
Axiomatic Systems	2	(7)
Finite Projective Planes	9	(9)
An Application to Error-Correcting Codes	18	(7)
Desargues' Configurations	25	(5)
Suggestions for Further Reading	30	(3)
Non-Euclidean Geometry	33	(66)
Gaining Perspective	33	(1)
Euclid's Geometry	34	(13)
Non-Euclidean Geometry*	47	(4)
Hyperbolic Geometry---Sensed Parallels	51	(10)
Hyperbolic Geometry---Asymptotic Triangles	61	(7)
Hyperbolic Geometry---Saccheri Quadrilaterals	68	(6)
Hyperbolic Geometry---Area of Triangles	74	(6)
Hyperbolic Geometry---Ultraparallels	80	(4)
Elliptic Geometry	84	(9)
Significance of the Discovery of Non-Euclidean Geometries	93	(1)
Suggestions for Further Reading	93	(6)
Geometric Transformations of the Euclidean Plane	99	(114)
Gaining Perspective	99	(4)
Exploring Line and Point Reflections*	103	(5)
Exploring Rotations and Finite Symmetry Groups*	108	(8)
Exploring Translations and Frieze Pattern Symmetries*	116	(5)
An Analytic Model of the Euclidean Plane	121	(8)
Transformations of the Euclidean Plane	129	(7)
Isometries	136	(8)
Direct Isometries	144	(10)
Indirect Isometries	154	(11)
Frieze and Wallpaper Patterns	165	(8)
Exploring Plane Tilings*	173	(10)
Similarity Transformations	183	(7)
Affine Transformations	190	(8)
Exploring 3-D Isometries	198	(9)
Suggestions for Further Reading	207	(6)
Projective Geometry	213	(102)
Gaining Perspective	213	(1)
The Axiomatic System and Duality	214	(7)
Perspective Triangles*	221	(2)
Harmonic Sets*	223	(6)
Perspectivities and Projectivities*	229	(11)
Conics in the Projective Plane*	240	(10)
An Analytic Model for the Projective Plane	250	(8)
The Analytic Form of Projectivities	258	(6)
Cross Ratios	264	(6)
Collineations	270	(13)
Correlations and Polarities	283	(15)
Subgeometries of Projective Geometry	298	(13)
Suggestions for Further Reading	311	(4)
Chaos to Symmetry: An Introduction to Fractal Geometry	315	(74)
A Chaotic Background*	316	(18)
Need for a New Geometric Language	334	(13)
Fractal Dimension	347	(13)
Iterated Function Systems*	360	(17)
Finally---What Is a Fractal?	377	(3)
Applications of Fractal Geometry	380	(2)
Suggestions for Further Reading	382	(7)
Appendices	389	(24)
A Euclid's Definitions, Postulates, and the First 30 Propositions of Elements, Book I	389	(6)
B Hilbert's Axioms for Plane Geometry	395	(4)
C Birkhoff's Postulates for Euclidean Plane Geometry	399	(2)
D The SMSG Postulates for Euclidean Geometry	401	(4)
E Some SMSG Definitions for Euclidean Geometry	405	(4)
F The ASA Theorem	409	(4)
References	413	(14)
Index	427