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Language; High C   The Mandelbrot set is the fine example of the fractal theory related to the modern computer graphics. Roughly speaking, a fractal is a property called self-similarity, "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," by Mandelbrot, B.B. Have you looked at yourself in two mirrors held against each other? Then you can see lots of the reduced-size copies of the two mirrors toward far away. This is similar to a fractal. In this example, we can see the reduced-size copies in one direction. Furthermore if we can see them in the several directions, that is called a fractal.

One of the interesting features of a fractal is that the complicated structures arise from a simple definition. It would be hard to take any interest in a fractal if all the complicated structures would arise from a complicated definition, like a Blancmange curve. In fact, we can see a lot of fractals which are defined simply. In the above example, a simple action “two mirrors held against each other” turns out to a fractal. We define the Mandelbrot set as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial z = z^2 + c remains bounded. Assigning black color for the bounded values and the other colors for the number of executed iterations at the non-bounded values, then we can see colorful images of the Mandelbrot set.

Now the Mandelbrot set is a fractal so that its computed image is never simplified or deteriorated at any given magnification. We can see the pretty characteristic images from place to place magnified. Furthermore how to assign colors turns the images completely. These two parameters, where to be magnified and how to assign colors, let us see the several unique images of the Mandelbrot set.

The following images of the Mandelbrot set are computed by my software. The coding is very simple but I supplement the artificial technique on it. In order to publish them on my website, I put them some image processing like decreasing color.

↓This is the overview image of the Mandelbrot set. The heart‐shaped domain located in the center means the set of complex values which remain bounded. ↓The following two images are ones magnified with the red square area on the above image.  The Mandelbrot set is defined by the complex quadratic polynomial ｚ＝ｚ^2＋Ｃ, then what can we see by ｚ＝ｚ^3＋Ｃ or more? This is just a natural question. We obtain the following images by the complex quadratic polynomial ｚ＝ｚ^4＋Ｃ. Generally we can see n+1 branched heart‐shaped images by ｚ＝ｚ^n＋Ｃ. ↓The following two images are ones magnified with the red square area on the above image.     