Carlson's Fractal Gallery

Selected Wicked Cool Mindbonk Site of the Month  for June 1996
Selected for a Links2Go Key Resource award  for July 1998

3D Phoenix Spirals

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The images in this group were all created using Clifford Pickover's quartic variation of Ushikis's "Phoenix" Julia set equations: Z = Z*Z - .5Z + C, X = Z*Z - .5Y + C, Y = Z, Z = X (see "The World of Chaos" by Clifford A. Pickover, "Computers in Physics" Sep/Oct 1990). For these images, C = (0.563, 0.0) was used instead of C = (0.56667, 0.0) as given in Pickover's paper. The image was colored using the atan method (see "Atan Method Fractals," below). Here's a few images (480 x 640 x 256 colors) in the style of the one above selected from my very large collection:

Atan Method Fractals

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The images in this group were created using a variety of equations, most involving transcendental functions. The colormap for these images consisted of two color ranges, each range varying in intensity linearly from light to dark. Which range was used for a particular pixel depended on whether the number of iterations at bailout was odd or even. The angle formed by a line joining the last two points in the orbit and the real axis was computed. The absolute value of this angle was converted to a colormap index in the proper color range, and the pixel was plotted using that color. Here's a few images (480 x 640 x 256 colors) in the style of the one above selected from my very large collection:

3D Stalks

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The images in this group were created using a variation of a method developed by Clifford Pickover in which pixels are plotted in certain colors if the absolute value of either the real or imaginary component of Z falls below a specified value. In my variation of this method, this absolute value was converted to an index into one of two color ranges in the colormap, depending on whether the number of iterations was odd or even. A wide variety of equations were used to produce these images. Here's a few images (480 x 640 x 256 colors) in the style of the one above selected from my very large collection:

Bubbles

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The images in this group were all created using the standard Mandelbrot equation Z = Z*Z + C. The images were created in two passes. On the first pass, the minimum value of the modulus of Z obtained in the iteration loop for each pixel was converted to a colormap index, and the pixel was plotted using that color. When the first pass was complete the program would pause until I clicked the mouse with the cursor over a pixel, say pixel A, in the image. The minimum modulus of Z for pixel A was computed and then a second pass was made in which all pixels having a minimum modulus of Z less than that of pixel A's were plotted the background color. Here's a few images (480 x 640 x 256 colors) in the style of the one above selected from my very large collection:

Pokorny Cones

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The images in this group were all created using the Pokorny equation Z = 1 / (Z*Z + C). The pixels were colored using the atan method (see "Atan Method Fractals," above). This method gives many of the images the appearance of consisting of an infinite number of (sometimes quite distorted) cones. Here's a few images (480 x 640 x 256 colors) in the style of the one above selected from my very large collection:

Miscellaneous Images

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The fractals in this section were produced using a variety of equations and rendering methods. Many of the fractals were rendered by methods utilizing one or more "orbit traps." An orbit trap is a bounded area in the complex plane of some simple shape such as a circle or square. When a point in the iteration orbit falls inside an orbit trap, the iteration loop is exited and the distance from the point to the center of the trap is used to index into two or more color ranges of the colormap.

The images above are just a small sample of my work. You will find many more of these images in the ever growing collection of fractals maintained by Clint Sprott. Also check out Sprott's Fractal Gallery which contains hundreds of examples of automatically generated fractals including computer programs and a new Fractal of the Day posted daily. See also my Dragon Julia Sets and the Paul Carlson Fractal Museum.

Many people continue to ask me how I created a particular fractal. The answer is, that all the fractals I post were created with programs that I wrote myself over the past fifteen years. These programs were not designed to be used by anyone but myself and were never made available on the web.

However, I have recently completed a program designed especially to create my style of fractals, fractals with sharply defined elements having a rounded 3D appearance. The program is designed to be very easy to use, especially for beginners and requires no knowledge of the mathematics involved.

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