page_title: Mandelbrot set
section_title: 3D images of Mandelbrot and Julia sets
hierarchical_section_title: Mandelbrot set / Geometry / 3D images of Mandelbrot and Julia sets
caption_attribution_description: English: This plot is centered at -.75,0. It has a real width of 3.5
context_page_description: The Mandelbrot set is the set of complex numbers for which the function does not diverge when iterated from, i.e., for which the sequence, etc., remains bounded in absolute value.
  Its definition is credited to Adrien Douady who named it in tribute to the mathematician Benoit Mandelbrot. The set is connected to a Julia set, and related Julia sets produce similarly complex fractal shapes.
  Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point, whether the sequence goes to infinity. Treating the real and imaginary parts of as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequence crosses an arbitrarily chosen threshold, with a special color used for the values of for which the sequence has not crossed the threshold after the predetermined number of iterations. If is held constant and the initial value of —denoted by —is variable instead, one obtains the corresponding Julia set for each point in the parameter space of the simple function.
context_section_description: In addition to creating two dimensional images of the Mandelbrot set, various techniques can be used to render Mandelbrot and Julia sets as 3D Heightmap images, where each pixel in a 2D image is given a height value, and the resulting image is rendered as a 3D graphic.
  The simplest approach to 3D rendering uses the iteration value for each pixel as a height value. This produces images with distinct "steps" in the height value.
  If instead the fractional iteration value (also known as the potential function) is used to calculate the height value for each point, steps in the resulting image are avoided. However, images rendered in 3D using fractional iteration data still look rather bumpy and visually noisy.
  An alternative approach is to use Distance Estimate (DE) data for each point to calculate a height value. Non-linear mapping of distance estimate value using an exponential function can produce visually pleasing images. Images plotted using DE data are often visually striking, and more importantly, the 3D shape makes it easy to visualize the thin "tendrils" that connect points of the set. Color plates 29 and 30 on page 121 of "The Science of Fractal Images" show a 2D and 3D image plotted using External Distance Estimates.
  Below is a 3D version of the "Image gallery of a zoom sequence" gallery above, rendered as height maps using Distance Estimate data, and using similar cropping and coloring.
  The image below is similar to "zoom 5", above, but is an attempt to create a 3D version of the image "Map 44" from page 85 of the book "The Beauty of Fractals" using a visually similar color scheme that shows the details of the plot in 3D.
caption: Mandelbrot set rendered in 3D using Distance Estimates
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