Hypercomplex Iterations. Distance Estimation And Higher Dimensional Fractals, Cd-Rom Included

In 1843, after a 15-year search, Sir William Rowan Hamilton discovered the quaternions, the highly significant algebra of four-dimensional space. A century later, in his round-breaking work The Fractal Geometry of Nature (1977), Benoit Mandelbrot brought mathematical iterations and their images into the world of applications and simulations. Intricate infinite mathematical tapestries entered the domains of statistical physics and computer-generated natural scenes. In this book, we study the mathematics and graphical geometry of quaternionic iterations, combining the insights of Hamilton and Mandelbrot with our own work on the mathematics of quaternionic and hypercomplex distance estimation. Our distance estimation algorithms are generalizations of algorithms known to work over the complex numbers. Proofs that justify the original distance estimation algorithm rely on delicate and subtle complex analysis. In order to justify the generalized algorithms, we needed to find new proofs that would apply to hypercomplex numbers. Our proofs turn out to be surprisingly simple and geometrical. The book details many different points of view about distance estimation, gives algorithms for computing these images and is accompanied by an interactive CD-ROM with much additional information.