From reviews of the first edition:
"In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology...the book also contains many good illustrations of fractals (including 16 color plates)."
Mathematics Teaching
"The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples."
Christoph Bandt, Mathematical Reviews
"...not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out."
H.Haase, Zentralblatt
About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2--6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused.
Author(s): Gerald Edgar
Series: Undergraduate Texts in Mathematics
Edition: 2nd
Publisher: Springer
Year: 2007
Language: English
Commentary: 1613787
Pages: 293
Tags: Математика;Высшая геометрия;
Preface......Page 7
Contents......Page 14
The Triadic Cantor Dust......Page 17
The Sierpinski Gasket......Page 23
A Space of Strings......Page 27
Turtle Graphics......Page 30
Sets Defined Recursively......Page 34
Number Systems......Page 47
*Remarks......Page 51
Metric Space......Page 57
Metric Structures......Page 64
Separable and Compact Spaces......Page 73
Uniform Convergence......Page 81
The Hausdorff Metric......Page 87
Metrics for Strings......Page 91
*Remarks......Page 97
Zero-Dimensional Spaces......Page 101
Covering Dimension......Page 107
*Two-Dimensional Euclidean Space......Page 115
Inductive Dimension......Page 120
*Remarks......Page 129
Ratio Lists......Page 133
String Models......Page 138
Graph Self-Similarity......Page 141
*Remarks......Page 149
Lebesgue Measure......Page 153
Method I......Page 162
Two-Dimensional Lebesgue Measure......Page 168
Metric Outer Measure......Page 171
Measures for Strings......Page 175
*Remarks......Page 178
Hausdorff Measure......Page 181
Packing Measure......Page 185
Examples......Page 193
Self-Similarity......Page 201
The Open Set Condition......Page 206
Graph Self-Similarity......Page 215
Graph Open Set Condition......Page 221
*Other Fractal Dimensions......Page 226
*Remarks......Page 232
*Deconstruction......Page 241
*Self-Affine Sets......Page 245
*Self-Conformal......Page 250
*A Multifractal......Page 254
*A Superfractal......Page 258
*Remarks......Page 263
Terms......Page 267
Notation......Page 270
Reading......Page 271
References......Page 273
Index......Page 277
