Mandelbrot set facts for kids
The Mandelbrot set is an example of a fractal in mathematics. It is named after Benoît Mandelbrot, a Polish-French-American mathematician. The Mandelbrot set is important for chaos theory. The edging of the set shows a self-similarity, which is not perfect because it has deformations.
The Mandelbrot set can be explained with the equation zn+1 = zn2 + c. In that equation, c and z are complex numbers and n is zero or a positive integer (natural number). Starting with z0=0, c is in the Mandelbrot set if the absolute value of zn never becomes larger than a certain number (that number depends on c), no matter how large n gets.
Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovering the Mandelbrot set in 1979. That was because he had access to IBM's computers. He was able to show how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess" or "chaotic", like clouds or shorelines, actually had a "degree of order". The equation zn+1 = zn2 + c was known long before Benoit Mandelbrot used a computer to visualize it.
Images are created by applying the equation to each pixel in an iterative process, using the pixel's position in the image for the number 'c'. 'c' is obtained by mapping the position of the pixel in the image relative to the position of the point on the complex plane.
The shape of the Mandelbrot Set is represented in black in the image on this page.
For example, if c = 1 then the sequence is 0, 1, 2, 5, 26,…, which goes to infinity. Therefore, 1 is not an element of the Mandelbrot set, and thus is not coloured black.
On the other hand, if c is equal to the square root of -1, also known as i, then the sequence is 0, i, (−1 + i), −i, (−1 + i), −i…, which does not go to infinity and so it belongs to the Mandelbrot set.
When graphed to show the entire Set, the resultant image is striking, pretty, and quite recognizable.
There are many variations of the Mandelbrot set, such as Multibrot, Buddhabrot, and Nebulabrot.
Multibrot is a generalization that allows any exponent: zn+1 = znd + c. These sets are called Multibrot sets. The Multibrot set for d = 2 is the Mandelbrot set.
Images for kids
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The Mandelbrot set (black) within a continuously colored environment
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Mandelbrot animation based on a static number of iterations per pixel
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Mandelbrot set detail
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Zooming into the boundary of the Mandelbrot set
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The first published picture of the Mandelbrot set, by Robert W. Brooks and Peter Matelski in 1978
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Correspondence between the Mandelbrot set and the bifurcation diagram of the logistic map
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With z_{n} iterates plotted on the vertical axis, the Mandelbrot set can be seen to bifurcate where the set is finite
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External rays of wakes near the period 1 continent in the Mandelbrot set
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Periods of hyperbolic components
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Attracting cycles and Julia sets for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs
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Self-similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-x direction. The display center pans left from the fifth to the seventh round feature (-1.4002, 0) to (-1.4011, 0) while the view magnifies by a factor of 21.78 to approximate the square of the Feigenbaum ratio \delta.
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A 4D Julia set may be projected or cross-sectioned into 3D, and because of this a 4D Mandelbrot is also possible.
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Image of the Tricorn / Mandelbar fractal
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Start. Mandelbrot set with continuously colored environment.
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Gap between the "head" and the "body", also called the "seahorse valley"
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Double-spirals on the left, "seahorses" on the right
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"Seahorse" upside down
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Animated gradient structure inside the Mandelbrot set
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Animated gradient structure inside the Mandelbrot set, detail
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Rendering of progressive iterations from 285 to approximately 200,000 with corresponding bounded gradients animated
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Thumbnail for gradient in progressive iterations
See also
In Spanish: Conjunto de Mandelbrot para niños
