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1. Fractal Images


The four pictures in the first group are 24 bit color images of tiny parts of the Mandelbrot set, under extremely high magnification. The magnification is indicated in parentheses. For example, m=2.25x10^15 means that the picture is centered at a point near the boundary of the M set and is magnified about 2,250,000,000,000,000 times. That's very close to the limit of what can be computed using conventional data types (i.e., extended reals in Pascal or double precision reals in C). View 4 is a simulated 3-dimensional view in grayscale.

You should view these pictures on a machine capable of displaying 24 bit video ("Millions of colors" on a Mac). If you view them in 8 bit color (256 colors) they will appear to have false color boundaries. If you're running in black-and-white then don't bother: in black-and-white they're just not that interesting.

  • The images were generated by the Macintosh program ColorFractals. I've posted this program, along with its THINK Pascal source, in the Mac_programs directory of this site.

  • View 1. (m=8.59x10^9)

  • View 2. (m=2.25x10^15)

  • View 3. (m=5.37x10^8)

  • View 4. (m=1.00x10^15)

    If you have a fast connection and good color depth on your machine, here are some larger images that give a better sense of the variability in structure seen under high magnification at different points near the the boundary of the Mandelbrot set. These images are large enough that you may have to scroll your window to see the edges.

  • mp1.

  • mp2.

  • mp3. (m=4.50x10^15)

  • mp4. (m=1.13x10^15)

  • mp5. (m=3.52x10^13)

  • mp6. (m=1.68x10^7)

  • mp7. (m=3.36x10^7)

    What the colors represent.

    The colors in these pictures represent how "close" you are to the Mandelbrot set. If you're not already familiar with what that means, here is how it's done. For purposes of computation, the Mandelbrot set M is defined as the set of all points p in the plane for which a certain sequence of points (associated with the point p and which depends on p in a simple nonlinear way) remains bounded. In order to compute an approximation to M one has to give computable meaning to the phrase "remains bounded". Then one can plot the set of points in the picture that satisfy this condition to produce an approximate drawing of M. The simplest way of doing this is to specify a "large" constant R > 0 and a "large" integer N > 0. One then considers a point p to be in M if all of the first N terms of the sequence associated with p belong to the disc of radius R about the origin. In practice, R doesn't have to be large for good results (R = 2.0 works nicely) and the value N = 100 produces a fairly good picture of M (though these images were generated using much larger values than N=100).

    Thus every pixel p of the picture you see has a number N(p) associated with it: N(p) = N if all of the first N terms of the sequence associated with the location of p remain in the disc of radius R, or else N(p) is the first n < N for which the next term of the sequence is outside the central disc of radius R. If N(p) = N then we consider p to be a pixel belonging to M and we may color it black. Otherwise N(p) < N, we consider that p does not belong to M, and we may color it white. This produces a black-and-white picture of M.

    On the other hand, we are also free to use the value of N(p) to choose more interesting colors than white for a pixel p when N(p) < N...and that is what this program does. For example, a pixel p is colored red if N(p) is very close to N, and for smaller values of N(p) the colors vary "smoothly" with the value of N(p). Most of the pixels with colors other than black are not in M, and most of the black pixels are in M. But because R and N are finite there can be small scale misrepresentations in a picture. For example, a few isolated black pixels in a tiny field of 100 or so red pixels can't be accurate, because M is a connected set.

    2. More Fractal Images

  • The images in this second group of four relate to Julia sets. Views 5 and 6 are garden-variety Julia "dragons". Views 6 and 8 together illustrate the self-similarity of Julia sets.

    Highly magnified views of any point near the boundary of a Julia set are not very different from an unmagnified view of the entire set. This interesting fact is a reflection of the self-similarity of Julia sets under changes of scale; it is in strong contrast to the Mandelbrot set, whose general "look" changes very dramatically under high magnification at various locations (see views 1, 2, 3 above).

    You can see a clear demonstration of the self-similarity of Julia sets in views 6 and 8. If you're handy with your browser, you should open a second window and view them simultaneously. View 6 is a standard Julia set, composed of fattened black "S" shapes and colored spirals. View 7 is a somewhat closer look at one of the spirals. View 8 is an extreme blowup of the upper tip of the picture shown in view 6. The similarity of this enormously magnified bit to the total Julia set shown in view 6 is quite apparent. The magnification in view 8 is about 5,000,000,000,000,000 times that of view 6 (whatever this number is, Webster's calls it five quadrillion).

  • View 5. (m=1.0)

  • View 6. (m=1.0)

  • View 7. (m=5.0)

  • View 8. (m=5.00x10^15)

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