| Since the ancient days of Plato and Euclid of classical Greece, mathematicians have worked with "pure" geometric figures and solids, such as circles, rectangles, spheres, cubes, and pyramids . But these are human-made and do not generally occur in nature, which is rough-edged and chaotic . In the 20th century, mathematicians began to study shapes generated by functions that are iterated. In the past 30 years or so, their ideas have come together into a new branch of mathematics called fractals. Mathematicians now use fractals to model dynamical systems and to describe shapes and patterns found in nature. We saw iterated functions in the last part of Chapter 5 when we used fixed-point iteration to find the roots of a function. In this chapter, we'll use that as a starting point, then move on to the generation of fractal images of Julia sets, and finally to the Mandelbrot set. This chapter can only graze the surface of this new mathematics. Explaining why iterated functions generate recursive images is beyond the scope of this book, but we can still appreciate the intricacy and beauty of the fractal images.  |
