Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure
Fractal patterns have emerged in many contexts, but what exactly is a pattern? How can one make precise the structures lying within objects and the relationships between them? This book proposes new notions of coherent geometric structure to provide a fresh approach to this familiar field. It develops a new concept of self-similarity called "BPI" or "big pieces of itself," which makes the field much easier for people to enter. This new framework is quite broad, however, and has the potential to lead to significant discoveries. The text covers a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics. Although fractal geometries arise in many different ways mathematically, comparing them has been difficult. This new approach combines accessibility with powerful tools for comparing fractal geometries, making it an ideal source for researchers in different areas to find both common ground and basic information.
1101399091
Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure
Fractal patterns have emerged in many contexts, but what exactly is a pattern? How can one make precise the structures lying within objects and the relationships between them? This book proposes new notions of coherent geometric structure to provide a fresh approach to this familiar field. It develops a new concept of self-similarity called "BPI" or "big pieces of itself," which makes the field much easier for people to enter. This new framework is quite broad, however, and has the potential to lead to significant discoveries. The text covers a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics. Although fractal geometries arise in many different ways mathematically, comparing them has been difficult. This new approach combines accessibility with powerful tools for comparing fractal geometries, making it an ideal source for researchers in different areas to find both common ground and basic information.
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Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure

Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure

Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure

Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure

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Overview

Fractal patterns have emerged in many contexts, but what exactly is a pattern? How can one make precise the structures lying within objects and the relationships between them? This book proposes new notions of coherent geometric structure to provide a fresh approach to this familiar field. It develops a new concept of self-similarity called "BPI" or "big pieces of itself," which makes the field much easier for people to enter. This new framework is quite broad, however, and has the potential to lead to significant discoveries. The text covers a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics. Although fractal geometries arise in many different ways mathematically, comparing them has been difficult. This new approach combines accessibility with powerful tools for comparing fractal geometries, making it an ideal source for researchers in different areas to find both common ground and basic information.

Product Details

ISBN-13: 9780198501664
Publisher: Oxford University Press
Publication date: 02/19/1998
Series: Oxford Lecture Series in Mathematics and Its Applications , #7
Pages: 224
Product dimensions: 9.21(w) x 6.14(h) x 0.56(d)

About the Author

Institut Universitaire de France

Rice University

Table of Contents

1. Basic definitions2. Examples3. Comparison with rectifiability4. The Heisenberg group5. Background information6. Stronger self-similarity for BPI spaces7. BPI equivalence8. Convergence of metric spaces9. Weak tangents10. Rest stop11. Spaces looking down on other spaces12. Regular mappings13. Sets made from nested cubes14. Big pieces of bilipschitz mappings15. Uniformly disconnected spaces16. Doubling measures and geometry17. Deformations of BPI spaces18. Snapshots19. Some sets that are far from BPI20. A few more questions
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