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The quaternion (4-dimensional) Mandelbrot set is simply a solid of revolution of the 2-dimensional Mandelbrot set (in the j-k plane), and is therefore uninteresting to look at. [43] Taking a 3-dimensional cross section at d = 0 ( q = a + b i + c j + d k ) {\displaystyle d=0\ (q=a+bi+cj+dk)} results in a solid of revolution of the 2-dimensional ...
Analogous to the exterior case, once b is found, we know that all points within the distance of b/4 from c are inside the Mandelbrot set. There are two practical problems with the interior distance estimate: first, we need to find z 0 {\displaystyle z_{0}} precisely, and second, we need to find p {\displaystyle p} precisely.
The difference between this calculation and that for the Mandelbrot set is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. [1] The mapping is non-analytic because its real and imaginary parts do not obey the Cauchy–Riemann equations .
Kalles Fraktaler focuses on zooming into fractals. This is possible in the included fractal formulas such like the Mandelbrot set, Burning ship or so called "TheRedshiftRider" fractals. Many tweaks can visualize phenomena better or solve glitches concerning the calculation issues.
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An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. [1] Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External rays are used in complex analysis , particularly in complex dynamics and geometric function theory .
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Starting in the 1950s Benoit Mandelbrot and others have studied self-similarity of fractal curves, and have applied theory of fractals to modelling natural phenomena.Self-similarity occurs, and analysis of these patterns has found fractal curves in such diverse fields as economics, fluid mechanics, geomorphology, human physiology and linguistics.