Measure, Topology, and Fractal Geometry (Undergraduate Texts in Mathematics) [Hardcover]

Gerald Edgar (Author)

Book Description

Publication Date: November 26, 2007 | ISBN-10: 0387747486 | ISBN-13: 978-0387747484 | Edition: 2nd

Based on a course given to talented high-school students at Ohio University in 1988, this book is essentially an advanced undergraduate textbook about the mathematics of fractal geometry. It nicely bridges the gap between traditional books on topology/analysis and more specialized treatises on fractal geometry. The book treats such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. It takes into account developments in the subject matter since 1990. Sections are clear and focused. The book contains plenty of examples, exercises, and good illustrations of fractals, including 16 color plates.

Editorial Reviews

Review

From the reviews of the second edition: "As a non-specialist, I found this book very helpful. It gave me a better understanding of the nature of fractals, and of the technical issues involved in the theory. I think it will be valuable as a textbook for undergraduate students in mathematics, and also for researchers wanting to learn fractal geometry from scratch. The material is well-organized and the proofs are clear; the abundance of examples is an asset for a book on measure theory and topology." (Fabio Mainardi, MathDL, February, 2008) "This is the second edition of a well-known textbook in the field … . The book may serve as a textbook for a one-semester (advanced) undergraduate course in mathematics. … the book is also suitable for readers interested in theoretical fractal geometry coming from other disciplines (e.g. physics, computer sciences) with a basic knowledge of mathematics. The presentation of the material is appealing … and the style is clear and motivating. … the book will remain as a standard reference in the field." (José-Manuel Rey, Zentralblatt MATH, Vol. 1152, 2009)

From the Back Cover

For the Second Edition of this highly regarded textbook, Gerald Edgar has made numerous additions and changes, in an attempt to provide a clearer and more focused exposition. The most important addition is an increased emphasis on the packing measure, so that now it is often treated on a par with the Hausdorff measure. The topological dimensions were rearranged for Chapter 3, so that the covering dimension is the major one, and the inductive dimensions are the variants. A "reduced cover class" notion was introduced to help in proofs for Method I or Method II measures. Research results since 1990 that affect these elementary topics have been taken into account. Some examples have been added, including Barnsley leaf and Julia set, and most of the figures have been re-drawn.   From reviews of the First Edition:  "...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..."  - 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. [For such 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

Product Details

• Hardcover: 288 pages
• Publisher: Springer; 2nd edition (November 26, 2007)
• Language: English
• ISBN-10: 0387747486
• ISBN-13: 978-0387747484
• Product Dimensions: 9.2 x 6.6 x 0.8 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #444,704 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

8 of 8 people found the following review helpful:

4.0 out of 5 stars Good starting point to study fractal geometry., April 12, 2000

By 

Bernardo Vargas - See all my reviews

This review is from: Measure, Topology, and Fractal Geometry (Undergraduate Texts in Mathematics) (Hardcover)

This book could be used as a bridge between traditional books on topology-analysis and the speciallized treatises on fractal geometry. More a catalog of definitions, methods, and references than a course text, it covers the fundamental topological and measure-theoretic concepts needed to understand the principles of some of the different dimension theories that exist. But warning: the book is far away of being a complete exposition on any of the subjects it includes.

Suitable for 3rd-year undergrads. Interesting examples and exercises. Extensive bibliography.

Please check my other reviews in my member page (just click on my name above).

4.0 out of 5 stars some new material->same approach, January 22, 2010

By 

R. Bagula "Roger L. Bagula" (Lakeside, Ca United States) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Measure, Topology, and Fractal Geometry (Undergraduate Texts in Mathematics) (Hardcover)

I bought the first edition of this in the early 90's
and was disappointed that it didn't have the Mandelbrot or other complex dynamics
in it. Dr. Edgar has updated the older book with Julias, multifractals
and Superfractals, but has stayed true to his topological measure theory
Hausdorff space approach. He never updates his Biscovitch-Ursell functions
to 2d and 3d parametrics or the unit Mandelbrot cartoon method.
Some of his definitions are still so minimal
that duplicating the fractals needs much more information?!
The text is still the good place to begin, but
it is a shame that Dr. Edgar has not kept up
with many of the developments in the field. Zipf and Per Bak
are left out, but my double V L-system made the index as a picture.

2 of 4 people found the following review helpful:

4.0 out of 5 stars A nice book, September 14, 2000

By 

Todd Ebert (Long Beach California) - See all my reviews

This review is from: Measure, Topology, and Fractal Geometry (Undergraduate Texts in Mathematics) (Hardcover)

I liked this book because it provided me with a new perspective on metric spaces, in using them as a basis learning about fractals. I think it serves as a nice book for an undergraduate to read and get enthused about studying fractals at a higher level.