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