This book proposes new notions of coherent geometric structure. Fractal patterns have emerged in many contexts, but what exactly is a `pattern' and what is not? How can one make precise the structures lying within objects and the relationships between them? The foundations laid herein provide a fresh approach to a familiar field. From this emerges a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics.

1: Basic definitions
2: Examples
3: Comparison
4: The Heisenberg group
5: Background information
6: Stronger self-similarity for BPI spaces
7: BPI equivalence
8: Convergence of metric spaces
9: Weak tangents
10: Rest stop
11: Spaces looking down on other spaces
12: Regular mappings
13: Sets made from nested cubes
14: Big pieces of bilipschitz mappings
15: Uniformly disconnected spaces
16: Doubling measures and geometry
17: Deformations of BPI spaces
18: Snapshots
19: Some sets that are far from BPI
20: A few more questions
References
Index

Guy David , Professor of Mathematics, University Paris XI and Institut Universitaire de France, France

Stephen Semmes , Professor of Mathematics, Rice University, Texas, USA

`The book contains a great variety of concepts, examples, results, and open problems...the presentation is both intuitive and precise.' Zentralblatt für Mathematik, 887

Most of the material in this book is completely new and the style, though unusual, is a refreshing change from convetional texts. The authors have taken a natural but not too stront notion relating to sets of fine structure, and follwed through its properties, relationships and applications. They freely admit that their framework is not theonly possible one, but by the end of the book they have more than justified theri claim that their approach is both rich and flexible. The book is recommended not only for those interested in the broad subject of he geometry of fractal sets and measures but also as a fine insight into how two eminent mathematicians explore and develop a new area.