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Explaining Mandelbrot's fractal geometry, and describing some of its applications in the natural world, this book steers a middle course between the formality of many papers in mathematics and the informality of picture-oriented book on fractals. Fractal geometry exploits a characteristic property of the real world - self-similarity - to find simple rules for the assembly of complex natural objects. Beginning with the foundations of measurement in Euclidean goemetry, Hastings and Sugihara progress from analogues in the geometry of random fractals to illustrative applications spanning the natural sciences: the developmental biology of neurons and pancreatic islets; fluctuations of bird populations; patterns in vegetative ecosystems; and earthquake models. The final section provides a toolbox of user-ready programs. This volume should be an essential resource for all natural scientists interested in working with fractals.