Fractals On the edge of chaos

A densely wrought exploration of Mandelbrot equations, the Droste effect, the Barnsley theorem, and other thorny problems of mathematics. "How long is the coastline of Cornwall, UK?" That's a question that would have sent Newton and even Einstein into the depths of despair. It's owing to an insight by Polish-born mathematician Benoit B. Mandelbrot that we can map out the three-dimensional world in which we live to some degree of certainty thanks to fractals, which, in nature, reveal themselves to be miniature images of the larger whole: "the magnified detail," writes Cambridge econometrician Linton, "is exactly the same as the whole thing." The attendant conceptual difficulty is that any map that is sufficiently detailed to reveal the whole accurately will be the size of the whole thing itself: A map of the universe would be the size of the universe, a thought that would have pleased Jorge Luis Borges. Lacking room and the wherewithal to prove the point with that map, mathematicians have come to develop numerical shortcuts--but those shortcuts are extraordinarily demanding of data, such that the GPS in your car relies on billions of numbers even as "nature…uses fractals for reasons of economy." This is not a book for the mathematically weak of heart. Although it's admirably short, certainly as compared to what might have happened to the discussion in the hands of a Douglas Hofstadter, each page bristles with equations and heady prose: "The pattern is clear; if you need l unit objects to make it m times larger then the number of dimensions the object has is d where l = md." If that sort of writing is your cup of pi, then Linton's compact explication of fractals will be child's play; others will be flummoxed. A small treasure for those who enjoy brain teasers and mathematical formulas. Copyright (c) Kirkus Reviews, used with permission.

For 2000 years, mathematicians, scientists and philosophers, blinded by the precision of Euclidean geometry, assumed that everything in the world around us could be built up from spheres, cones, circles, smooth planes and straight lines. They were not entirely wrong: much can be learned by modelling atoms as spheres, faces as multifaceted polyhedra and hurricane winds as straight or circular.The reason for this is mathematical economy. A sphere is completely described by a single numberits radius; a triangle by threethe lengths of its three sides. Even a hurricane is largely described by just two numbersits speed of rotation at a characteristic diameter.But to describe a cloud or a coastline in detail requires millions of numbers. What would be the point? By the time you had written down all those numbers, the cloud would have long since vanished. Coastlines, however, are more permanent and more important too. Have you ever wondered how many numbers are needed to specify the map in your SatNav? The answer is literally billions.But in 1982, a brilliant Polish born mathematician named Benoit Mandelbrot showed the world that it was possiblesometimes at leastto describe complex structures like clouds and coastlines as easily as spheres and lines, and fractal geometry was born. This book attempts to describe the revolution in mathematics and art which followed. Excerpted from Fractals: On the Edge of Chaos by Oliver Linton All rights reserved by the original copyright owners. Excerpts are provided for display purposes only and may not be reproduced, reprinted or distributed without the written permission of the publisher.