The Ghostly Imagery of Strange Attractors
Year: 2004 Authors: Robert J. Krawczyk
Core claim
Expanding parameter variation and visitation-time coloring reveals new visually complex strange-attractor forms, including shell-like imagery.
Topics
strange attractors, chaotic imagery, parameter variation, visual rendering
Domains
dynamical systems, iterative mapping, chaos theory, parametric equations, generative art, computer graphics, visual pattern making, exhibition imagery
Methods
parameter sweeps, random variations, iterative rendering, time-as-density coloring
Media
custom software, grayscale images, red-on-black images, shell imagery
Paper text
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BRIDGES Mathematical Connections in Art, Music, and Science
The Ghostly Imagery of Strange Attractors
Robert J. Krawczyk College of Architecture Illinois Institute of Technology 3360 South State Street Chicago, IL, 60616, USA E-mail: krawczyk@iit.edu
Abstract
Expanding on the previous investigation of strange attractors, a new series of image are produced by generating and reviewing a wider range of potential parameters. The concept of the ghostly imagery is pursued further with additional methods being developed to review a greater variety of parameters to determine which images lay within this chaotic dimension.
1. Introduction
Clifford Pickover [1] extended some of his previous writings on three-dimensional chaotic or strange attractors by including a series of two-dimensional attractors based on a simple equation consisting of sine functions. The most intriguing aspect about these images was the variations possible by the execution of a simple iterative mapping with minor changes in parameters. The visual complexity of the interior detail within the image and the visual perception of a third-dimension as curves began to suggest surfaces became the starting point for this development. Joel [2], Sprott [3], and Bourke [4] also demonstrated this type insight into strange attractors.
2. The Element of Time
In Krawczyk [7] the element of time as density was investigated. Pickover had suggested the rendering the strange attractors be based on the number of times a point was visited. An alternate method was developed that tracked the last time a point was visited, this became the basis for assigned a color to that point. This method began to uncover some of the subtle curves that were apparently found within the attractors.
The initial method used to generate the strange attractors was to exercise the equations from Pickover and Bourke and vary values, functions, and possible parameters. Figure 1 displays the general parameters for the final equations developed. The first set of images was generated with each having twelve-to-fifty-four million computed points. During this process nineteen other equations were developed and investigated. To better understand the scope of these equations, a second review of them was performed. Custom software was written to take these equations and within a set range of values generate random variations. A total of 24,000 images were generated from twenty-four equations. Each image was reviewed and then approximately 200 were selected for further rendering. From these a group of sixteen has been developed for exhibit.
Figure 2 displays two of these images in grayscale; the final ones are red on a black background. They seem to be related to a shell. Natural growth over long periods of time results in patterns that are
2004 Bridges Proceedings
normally not seen. Simple growth controlled by time begins to make these complex patterns visible. The shell is just such a container to hide such wonder, a wonder that is normally not considered because we only view a shell’s surface markings. Under this surface lies a possible complexity that can only be imagined, never followed logically, never fully understood. The complete 2003 series is in Gallery Two of: www.netcom.com/~bitart/chaos
Xptt = v1(m1(f1(v2p1)^v3)) + v4(m2(f2(v5p2)^v6)) + v7(m3(f3(v8p3)^v9))
Yptt = v10(m4(f4(v11p4)^v12)) + v13(m5(f5(v14p5)^v15)) + v16(m6(f6(v17p6)^v18))
v1-v18 = values m1-m6 = As is, ABS, NEG, SIN, or COS f1-f6 = As is,
SIN, or COS
p1-p6 = Xptt-1, or Yptt-1 + = +, -, *, or /
Within Shell I, 2003
Figure 2: Shell imagery.
Figure 1: Strange Attractor equation parameters.
Within Shell II, 2003
References
[1] C. Pickover, Chaos in Wonderland, St. Martin’s Press, New York, pp. 11-13, 55-58, 75-77, 86-87, 209, 210, 213, 1994. [2] W. Joel, A Simple Attraction, Computers & Graphics, Vol. 16, No. 1, pp. 41-43, 1992. [3] J. Sprott, Strange Attractors, Creating Patterns in Chaos, M&T Press, New York, 1993. [4] P. Bourke, Swirls, The Pattern Book: fractals, art, and nature, edited by C. Pickover, World Scientific Publishing, Singapore, pp. 197-198. 1995. [5] R. Krawczyk, “Dimension of Time in Strange Attractors”, in ISAMA and Bridges 2003 Joint 2003 Conference, edited by R. Sarhangi