The following special sections provide in depth detail for the shown images: Verhulst Dynamics, Julia Sets and Their Computergraphical Generation, Sullivan's Classification of Critical Points, The Mandelbrot Set, External Angles and Hubbard Trees, Newton's Method for Complex Polynomials: Cayley's Problem, Newtons's Method for Real Equations, A Discrete Volterra-Lotka System, Yang-Lee Zeros, Renormalization (Magnetism and Complex Boundaries).

The book also includes invited Contributions by Benoît Mandelbrot, Adrien Douady, Gert Eilenberger and Herbert W. Franke, which provide additional formality and some historically interesting detail.

Adrien Douady explains the solved and unsolved problems relating to the almost amusingly complex Mandelbrot set.

Part of the text was originally conceived as a supplemented catalogue to the exhibition Frontiers of Chaos of the German Goethe-Institut, first seen in Europe and the United States.

The cover article of the Scientific American August 1985 edition showed some of the images and provided reference to the book to be published.