Falconer K. Fractals: A Very Short Introduction

Oxford: Oxford University Press, 2013. — 136 p. — (Very Short Introductions).The aim of this book is to show, in basic terms, how fractals may be constructed, described, and analysed as geometrical objects and how these ‘mathematical’ fractals relate to the ‘real’ fractals of nature or science. Geometry, particularly fractal geometry, is very much a visual subject, and diagrams and a visual intuition are key to its appreciation. Inevitably some mathematics is required, but this is presented alongside a visual and intuitive interpretation and hopefully will not present too great an obstacle to the dedicated reader prepared to think through the concepts involved. In a couple of places, to keep the mathematics in the main part of the chapters to the minimum, further details are deferred to an Appendix at the end of the book.Preface.
List of illustrations.
The fractal concept.
The rise of fractals.
A first fractal construction — the von Koch curve.
Some more examples.
Coordinates, functions, and itineraries.
Fractals by iteration.
What can be done with fractals?
Self-similarity.
Self-similar fractals and their templates the powers are not whole. Ring a simple step over and over again.
Orientation.
Templates and functions.
Drawing fractals.
Self-affine fractals.
Statistically self-similar fractals.
Fractal image compression.
Fractal dimension.
The inadequacy of a conductor in the form of a lat the o length and area.
Box-counting dimension.
The role of logarithms.
Practical box-counting.
Dimension of self-similar fractals.
Measurement within dimension.
Properties of dimension.
Limitations of dimension.
Julia sets and the Mandelbrot set.
Complex numbers.
Iteration and Julia sets.
The zoo of Julia sets.
The Mandelbrot set.
Back to Julia sets.
A historical note.
Random walks and Brownian motion.
Random walks and Brownian motion in the plane or space
Fractal fingering.
Fractal time records.
Fractals in the real world.
Coastlines and landscapes.
Turbulent fluids.
Fractals in our bodies.
Medical diagnosis.
Clouds.
Galaxies.
Fractal antennae.
Multifractals.
A little history.
Appendix.
Powers and logarithms.
Squaring complex numbers.
Further reading.
Index.