The first graph of the Mandelbrot set was achieved by Robert Brooks and Peter Matelski in 1978.
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o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o . . . . . . . . . o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O @ @ @ @ @ @ @ . . . . . . . @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o . . . . . . o o o o o o o o o o o o o o o O O O O O O O O O O @ @ @ @ @ @ @ @ @ @ . . o o o o o @ o O @ o o . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o . . . . o o o o o o o o o o O O O O O O O O O O O @ @ @ @ @ @ @ @ @ @ @ . o o O O . @ . @ o O . @ @ O o o . . . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o . o o o o o O O O O O O O O O O O O @ @ @ @ @ @ @ @ @ @ . . . . . o o O @ @ . o O @ O . . O . . @ O O o o . . . . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o O O O O O O O O O O O O O @ @ @ @ @ @ @ @ . . . . . . . . o o O O . . . O @ @ @ @ @ @ @ @ . . @ O . @ O o o o o o . . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o O O O O O O O O O O O O @ @ @ @ . . . . . . o o o o o o o O O @ @ . O o @ @ @ @ @ @ @ @ @ @ @ @ . O . @ @ O O O O O o o o o o o o o . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o O O O O O O O O O @ @ @ @ . o O o @ . @ O O O O @ @ . O o @ @ @ @ . O . O . o @ @ @ @ @ . @ @ o @ . @ o o @ @ . o . . o O O . . @ @ o @ O o o . . . . . . . @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O @ @ @ . . o O o @ o O @ @ @ O . . O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O o . @ O O o o o o o . . . . . . . . . . . . . @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O @ @ @ @ . . . o o O @ @ @ O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ o @ @ O . @ . @ O o o o o o o o . . . . . . . . . . . . . . . . . . @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O @ @ @ @ . . o O @ @ @ o O @ O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ o O . @ @ O O O O O O O o o o o o o o o o o o o o o o o . . @ O o . . @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O @ @ @ @ @ . o O O O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ o . O . . @ @ @ @ @ @ @ . . . O . . o . . . o o . o O . @ o O o o o . . . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O @ @ @ @ @ @ . . o O . o . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ o o @ O O O @ O @ O @ O @ O @ @ @ @ @ . @ @ @ o . O . . @ O O O o o . . . . . . . . . @ @ @ @ @ @ @ @ @ @ O O @ @ @ @ @ @ @ . . . o O . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ o . . @ o o @ @ @ @ O o o . . . . . . . . . . @ @ @ @ @ @ O O @ @ @ @ @ @ @ . . . o o O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ o o O o . . o . @ o @ @ O O O O o o o o o o o o . @ @ O @ @ @ @ @ @ @ @ . . . . o o O O @ . O O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ . o @ o . o O o . . . o @ o . @ . . O O O @ @ @ @ @ @ @ @ . . . . o o O @ o o @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ . @ @ @ @ o @ . O . . @ O O O O O o o o o o O O O @ O O @ @ @ @ @ @ @ . . . o O . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ . o @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ o o O O . . . o @ O @ O o o o o o . . . . . . @ @ @ @ O O @ @ @ @ @ @ @ . . o O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ . o . . O . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ . O @ O o . O O O O o o . . . . . . . . . . @ @ @ @ @ @ @ @ O O O @ @ @ @ @ @ . o o O @ o O O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O @ o o . . . . o O @ @ . @ O O o . O @ . @ @ o @ @ o . @ O O o o . . . . . . . . @ @ @ @ @ @ @ @ @ @ @ @ O O O O @ @ @ @ @ . . O o . @ o @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O @ @ O o . @ @ @ O O O O O O O O @ @ o o @ O O O O O O @ @ O o @ O o o . . . . @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O @ @ @ @ . . . o o O O @ o O O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O @ o @ O . . @ O O o o o o o o o . . . . . . . . . . . . o o o o o . . @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O @ @ @ . . . o o @ o @ O @ @ @ @ @ o @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O . o . @ @ O O o o o o o o . . . . . . . . . . . . . . . . @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O @ @ @ @ . . @ . O @ @ O O @ O o o o O O @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ . @ . @ . @ O O o O O O O o o o o . . . . . . . . . . @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O @ @ @ @ . . o o @ O o o o o o o O O @ . . . . o O . o @ @ @ @ @ @ @ @ @ . O @ O o o . . . @ @ @ @ @ @ O . @ @ O O o . . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O @ @ @ @ @ @ . . . . . . . . . o o O O O . O o O @ @ @ @ @ @ @ @ @ @ . o o @ @ O O O o o o o o o o . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o O O O O O O O O O O O O @ @ @ @ @ @ @ @ @ @ . . . . . . o o @ . @ O O o @ @ @ @ o . . . . @ o @ o o o . . . . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o O O O O O O O O O O O O @ @ @ @ @ @ @ @ @ @ @ . . . o o O @ . o @ @ o O o @ @ O O o o . . . . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O @ @ @ @ @ @ @ @ @ @ . . O O @ O O @ o O O O O o o o . . . . . @ @ @ @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o . . o o o o o o o o o o o o o o o o o o O O O O O O O O O O @ @ @ @ @ @ @ @ @ . . . . . O @ . O O o . . @ @ @ @ @ @ @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O O @ @ @ @ @ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o . . . . . . . . o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o . . . . . . . . . . . |
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The Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial z*n+1 = z*n2 + c remains
bounded. That is, a complex number, c, is part of the Mandelbrot set if, when starting with z*0 = 0 and applying the iteration repeatedly, the absolute value of z*n
never exceeds a certain number (that number depends on c) however large n gets.