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The Binet formula provides a mechanism for the Fibonacci numbers to be viewed as a function of a complex variable. The Binet formula may be generalized by using other bases and multiplicative parameters that also give functions of a complex variable. Thus, filled-in Julia sets that exhibit escape time may be constructed. Moreover, these functions have computable critical points and hence we can create escape time images of the critical point based upon the underlying multiplicative parameter. Like the classic Mandelbrot set, these parameter space images provide an atlas of Julia sets.

We present a comparative study of simple parallelisation schemes for the most widely used methods for the graphical representation of Mandelbrot and Julia sets. The compared methods render the actual attractor or its complement.

This brief note and artistic statement describes the development of new software to render 3D Mandelbrot and Julia sets. This differs from others in the field in that the algebra used by most 3D-fractal software is either quaternion or hypercomplex, while complexified quaternion (cquat) algebra is neither. This results in a different perspective for 3D imagery of this type.

Complex numbers are redefined from the plane to two non-Euclidean geometries, namely the surfaces of the sphere and the torus. The properties of the iterated equation z n +1 = z n p + c are investigated in these geometries, where z , p and c are complex numbers. Plots of the number of iterations required by this equation for convergence to a fixed point appear fractal and resemble multiple copies (at different scales) of the Julia set obtained when the same equation is iterated in the complex plane.

A method for constructing the general M (Mandelbrot) set of a non-analytic mapping is presented. The equivariant mapping with symmetry of the modular group is considered as an illustration. By investigating the distribution of attractors in the upper half-plane and the assignment of colors to each attractor, an algorithm is presented for the construction of filled-in Julia sets with 2- or 3-color symmetry. Such Julia sets not only reveal the characteristics of a system, but also have high artistic appeal.

Both the computer drawing of the complement of the Mandelbrot-like set of a one-parameter-dependent complex exponential family of maps and the computer drawing of the Julia sets of the maps of this family, grow with the maximal number of iterations we choose. Some graphic examples of this growth, which evoke the image of a garden, are shown here.

Three methods are presented for obtaining 3-D effects in fractal images generated by the iteration of complex valued functions.

A method to define and generate Mandelbrot and Julia sets in more than four dimensions is presented. A doubling process is used to create from the set of real numbers a hypercomplex number system of arbitrary dimension. Since the new number system is closed under addition and multiplication, it can be used to generate Mandelbrot and Julia sets of corresponding dimension. Generation of these sets in more than four dimensions is discussed. A graphical analysis manifests the sets are fractal in these higher dimensions. Symmetrical properties of Mandelbrot and Julia sets are observed and reported.

This paper presents various patterns and computer art from three extensions to common fractal methods: Here, I use discontinuous functions (such as Boolean functions and the conditional function), plots of root-finding methods other than Halley's and Newton's methods (such as Aitken's and Muller's methods), and fractals produced using arithmetic in q -systems other than the complex number system. Still fractal images and parametrization “movies” are discussed; these allow several new properties of these algorithms (and their combinations) to be uncovered. Also, the interaction of randomness with fractals is explored.

We give examples of Iterated Function Systems where the usual affine functions are replaced by complex polynomials. Since these are not global contractions, some care must be exerted with the domain on which the functions are applied. On the other hand, the lack of unique inverse functions gives rise to multiple addresses for the same regions of the attractor, thus providing another degree of information available for determining other attributes—for example, colors or textures—of the image. Traversing a loop through parameter space reveals an “explosion” of features and also what may be a curious cascade of self-intersections.

The escape time behavior of a function associated with elliptic curves is studied via Julia sets and composite Mandelbrot sets. The Mandelbrot sets are composite in the sense that two critical points are followed.

Newton's method is sensitive to an initial guess. It exhibits chaotic behavior that generates interesting fractal images. Most of them have either a finite number of attractors (attractive fixed points) or unbounded Julia sets. In this paper, we show that Newton's method for a family of equations exp −α ζ+z ζ−z −1=0 (for α > 0 and |ζ| = 1) has infinitely many attractors and a bounded Julia set. The dynamics of Newton's method for finding their roots are also visualized.

This paper explores methods of generating fractals from the mapping z → z−n + c and discusses the structure of the resulting images. Lyapunov exponents and cycle periodicity show details of these fractals that are obscured by prior escape-time techniques. The resulting fractals are shown to have a hypocycloid shape. The associated Julia sets can be classified into a variety of types.

This article presents the use of computer graphics to gain a better understanding of the dynamics occurring in the Siegel disk Julia Sets. Traditional images of the Julia Sets usually display only the final behavior of the forward orbits of each point in the discretized complex plane. For each pixel in the image, a forward orbit must be computed—usually the information contained in the orbit is thrown away. This approach examines the orbit as well as its final behavior.

The present note explores the structure of the interior of Julia sets and of the Mandelbrot set of quadratic maps. These sets usually appear as ‘dark’ regions in the computer generated graphics of iterations of the quadratic map. Two different ways of coloring and probing into these sets are provided.

The dynamic behavior and attractors of quasi-stable orbits of 0 for the iteration z n + 1 ← z 2 n + c are explored. Relationships to their associated Julia sets are also discussed in this tutorial.

By manipulating the imaginary part of the complex-analytic quadratic equation, we obtain, by iteration, a set that we call the “burning ship” because of its appearance. It is an analogue to the Mandelbrot set (M-set). Its nonanalytic “quasi”-Julia sets yield surprizing graphical shapes.

We examine the Julia sets for certain cubic and quartic polynomials and observe the structure of the Julia set as determined by the orbits of the critical points. In examples where the orbits of some critical points diverge and others converge, or where orbits converge to different cycles, we note the local structure of the Julia set is dominated by the behavior of the nearest critical point. This information is encoded in a generalized Mandelbrot set reflecting the behavior of all the critical points.

I discuss the “Mandelbrot” set and Julia sets arising from the iteration of a map of the 2-D torus defined in terms of Jacobi's elliptic function sn. These results generalize well known behaviour of maps of the circle.

Julia sets of switched processes are introduced. Such processes are of importance in the study of dynamical systems in which several free-standing processes may be involved.

The artistic value of fractal imaging is demonstrated in a black-and-white cubic Julia set using staggered alternation.

Recent mathematical work on the dynamics of complex analytic functions has given rise to a new subject matter for computer graphics. The combination of mathematical theory and computer graphics has resulted in new insight into the nature of some of the simplest of mathematical objects. second-degree polynomials. Most of that work has focused on the possibilities within the two-dimensional complex plane. This article shows how these investigations may be extended to higher dimensions, resulting in fractals that naturally reside in the 4-dimensional quaternions. Particular attention is paid to the formula ax2 + b. A method is given for obtaining various interconnection patterns for the Julia sets in 4-space, and the results are displayed in 3-D computer graphics.

In this paper, computer graphics is used to explore the dynamical behaviour of the Julia sets and related iterated sets derived by using alternative divergence tests.