Mandelbrot set

English: The Mandelbrot set, a fractal, named after its creator the French mathematician Benoît Mandelbrot. The set is a map of the Julia set.

Polski: Zbiór Mandelbrota, fraktal, nazwany imieniem francuskiego matematyka. Zbiór ten jest mapą zbiorów Julii.

Slovenščina: Mandelbrotova množica je fraktal, imenovan po avtorju francoskem matematiku Mandelbrotu. Gre za karto Juliajeve množice.

Українська: Множина Мандельброта

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  Benoît Mandelbrot and the set bearing his name

General

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  Hi-resolution Mandelbrot set with axes
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  Mandelbrot set and periodicities of orbits
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  Mandelbrot set and colorcoded periodicities of orbits
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  Mandelbrot set with well defined colour stripes
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  Mandelbrot set with irregular colour stripes
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  Mandelbrot set in grayscale
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  Mandelbrot set with smooth color gradient
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  Buddhabrot
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  Mandelbrot zoom
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  Inside colour-mapping, (B&W version).
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  Inside colour-mapping, (colour version).
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  Screenshot von RFL Mandelbrot Set Exploration Tool v0.0.4
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  Visualization of Mandelbrot set in complex plane
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  Command-line depiction of the Mandelbrot set.
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  Yet another image of the Mandelbrot Set.
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  Representation of Inner Structure
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  colors

Structure

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  Boundaries of hyperbolic components of mandelbrot set
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  Lemniscates - boundaries of level sets of escape time
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  Centers of hyperbolic components
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  All boundaries of level sets of escape time up from n=1

Rays

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  External and internal rays, center and root
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  External rays of Misiurewicz point
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  External ray of Misiurewicz point c=-2
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  Uniformization of complement of Mandelbropt set
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  Wakes near the period 1 continent in the Mandelbrot set
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  Wakes along the main antenna in the Mandelbrot set

Fractalizer

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  Example 01
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 02
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 03
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 04
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 05
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 06
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 07
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 08
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 09
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example 10
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example (video) 01
  www.fractalizer.de/en archive copy at the Wayback Machine
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  Example (video) 02
  www.fractalizer.de/en archive copy at the Wayback Machine

Zoom

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  Initial image of a zoom sequence with 14 steps
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  Initial image of a corresponding zoom sequence with frames
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  Zoom step 1 of 14
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  Zoom step 1 of 13
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  Zoom step 2 of 14
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  Zoom step 2 of 13
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  Zoom step 3 of 14
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  Zoom step 3 of 13
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  Zoom step 4 of 14
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  Zoom step 4 of 13
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  Zoom step 5 of 14
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  Zoom step 5 of 13
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  Zoom step 6 of 14
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  Zoom step 6 of 13
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  Zoom step 7 of 14
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  Zoom step 7 of 13
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  Zoom step 8 of 14
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  Zoom step 8 of 13
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  Zoom step 9 of 14
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  Zoom step 9 of 13
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  Zoom step 10 of 14
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  Zoom step 10 of 13
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  Zoom step 11 of 14
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  Zoom step 11 of 13
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  Zoom step 12 of 14
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  Zoom step 12 of 13
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  Zoom step 13 of 14
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  Zoom step 13 of 13
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  Zoom step 14 of 14
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  Mandelbrot (Ausschnitt)
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  Mandelbrot (Ausschnitt)
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• Zooming Movie 03
• Zooming movie 04
• Zooming movie 06
• Zooming movie 15
• High-resolution zoom
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  Featured golden gradient zoom on the Mandelbrot set by more than 31 orders of magnitude.

Iteration

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  At a count of 32, the whole image is black, since it is completely inside the false-negative contour.
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  If we allow 64 iterations, some points are no longer falsely inside the set.
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  At 128 iterations, the image is blobby, but recognizable as a fractal.
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  256
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  At 512, we get a nice image. The black dots up and to the left of each "wart" contain tiny cardioids.
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  1024
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  2048
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  Diminishing returns are quite obvious when we use a million iterations. Even with periodicity checking, this one took 10–15 seconds to generate on an Athlon XP 2000+.
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  Number of iterations changing from 1 to 50.

Some details of the Mandelbrot set

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  side=0.582; lower-left-point=-0.4+0.5i (made using a JAVA applet archive copy at the Wayback Machine)
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  side=0.0017815; lower-left-point=-0.75+0.06i (made using a JAVA applet archive copy at the Wayback Machine)
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  side=0.004402; lower-left-point=0.28+0.0084i (made using a JAVA applet archive copy at the Wayback Machine)
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  side=0.000191; lower-left-point=-0.78-0.136i (made using a JAVA applet archive copy at the Wayback Machine)
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  side=0.00004; lower-left-point=-1.595+0.000095i (made using a JAVA applet archive copy at the Wayback Machine)
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  side=0.0001558; lower-left-point=-0.75+0.064i (made using a JAVA applet archive copy at the Wayback Machine)
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  side=0.0000829; lower-left-point=0.253-0.0031i (made using a JAVA applet archive copy at the Wayback Machine)
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  side=0.0166; lower-left-point=-1.042-0.0346i (made using a JAVA applet archive copy at the Wayback Machine)
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  center=-0.745-0.1i

Art] and the Mandelbrot set

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  Calm Mandelbrot, magnified 100,458,337,236 times, 256 iterations
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  Cool Mandelbrot, magnified 248,034,982,258 times, 256 iterations
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  Hot Mandelbrot, magnified 261,880 times, 1024 iterations
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  Galaxy of Galaxies
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  Rendered with a scenery generator
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  Spiral, Color cycling animation
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Universality of Mandelbrot set

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  Mandelbrot sets occurring in the analysis of Newton's method
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  Mandelbrot set occurring in the analysis of Newton's method (detail)
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  Mandelbrot set occurring in the analysis of Newton's method (detail)
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  Mandelbrot set occurring in a model of phase transitions (detail)
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  Mandebrot set embedded in cos(z)+c.
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  Mandebrot set embedded in gaussian exp(-z^2)+c.

Miscellaneous

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  Comparing functions, Mandel/Bifurcation.
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  Periodic cycles in the Mandelbrot set
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  Mandelbrot, plane: 1/lambda
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  Relation between Mandelbrot set and en:logistic map
• Multibrot sets
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  Map of Julia sets showing the resemblance with the Mandelbrot set
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  Mandelbulber v2.23
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  High-resolution Buddhabrot rendering of the Mandelbrot set

First steps in Mandelbrot set

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  1.step of Ms
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  2.step of Ms
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  3.step of Ms
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  4.step of Ms
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  5.step of Ms
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  9.step of Ms
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  30.step of Ms
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  part of 30.step of Ms
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  step 1
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  step 2
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  step 3
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  step 4
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  step 5
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  step 6
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  step 7
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  step 8
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  step 9
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  step 10
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  step 11
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  step 12
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  step 13
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  step 14
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  step 15
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  step 16
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  step 17
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  step 18
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  step 19
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  step 20