Mandelbrot set

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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

  • Benoît Mandelbrot and the set bearing his name

Contents

[edit] General

  • Hi-resolution Mandelbrot set with axes

  • Mandelbrot set and periodicities of orbits

  • Mandelbrot set and colorcoded periodicities of orbits

  • Mandelbrot set with well defined colour stripes

  • Mandelbrot set with irregular colour stripes

  • Mandelbrot set in grayscale

  • Mandelbrot set with smooth color gradient

  • Mandelbrot zoom

  • Inside colour-mapping, (B&W version).

  • Inside colour-mapping, (colour version).

  • Mandelbrot and Julia.png

  • Screenshot von RFL Mandelbrot Set Exploration Tool v0.0.4

  • Visualization of Mandelbrot set in complex plane

  • Command-line depiction of the Mandelbrot set.

  • Yet another image of the Mandelbrot Set.

  • Mandelbrot 70000x.png

  • Representation of Inner Structure

[edit] Structure

  • Boundaries of hyperbolic components of mandelbrot set

  • Lemniscates - boundaries of level sets of escape time

  • Centers of hyperbolic components

  • Uniformization of complement of Mandelbropt set

  • All boundaries of level sets of escape time up from n=1

[edit] Rays

[edit] Fractalizer

[edit] Zoom

  • Mandelbrot-similar1.png
  • Mandelbrot-similar2.png
  • Mandelbrot-similar3.png
  • Mandelbrot-similar4.png

  • Initial image of a zoom sequence with 14 steps

  • Initial image of a corresponding zoom sequence with frames

  • Zoom step 1 of 14

  • Zoom step 1 of 13

  • Zoom step 2 of 14

  • Zoom step 2 of 13

  • Zoom step 3 of 14

  • Zoom step 3 of 13

  • Zoom step 4 of 14

  • Zoom step 4 of 13

  • Zoom step 5 of 14

  • Zoom step 5 of 13

  • Zoom step 6 of 14

  • Zoom step 6 of 13

  • Zoom step 7 of 14

  • Zoom step 7 of 13

  • Zoom step 8 of 14

  • Zoom step 8 of 13

  • Zoom step 9 of 14

  • Zoom step 9 of 13

  • Zoom step 10 of 14

  • Zoom step 10 of 13

  • Zoom step 11 of 14

  • Zoom step 11 of 13

  • Zoom step 12 of 14

  • Zoom step 12 of 13

  • Zoom step 13 of 14

  • Zoom step 13 of 13

  • Zoom step 14 of 14

  • Mandelbrot (Auschnitt)

  • Mandelbrot (Auschnitt)

  • Mandelbrot Animation1.gif
  • Fractal-zoom-1-03-Mandelbrot Buzzsaw.ogg

    Zooming Movie 03

  • Fractal-zoom-1-04-Snowflake.ogg

    Zooming movie 04

  • Fractal-zoom-1-06-inside a moth.ogg

    Zooming movie 06

  • Fractal-zoom-1-15-rupture.ogg

    Zooming movie 15

  • Featured golden gradient zoom on the Mandelbrot set by more than 31 orders of magnitude.

[edit] Iteration

  • At a count of 32, the whole image is black, since it is completely inside the false-negative contour.

  • If we allow 64 iterations, some points are no longer falsely inside the set.

  • At 128 iterations, the image is blobby, but recognizable as a fractal.

  • 256

  • At 512, we get a nice image. The black dots up and to the left of each "wart" contain tiny cardioids.

  • 1024

  • 2048

  • 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+.

  • Number of iterations changing from 1 to 50.

[edit] Some details of the Mandelbrot set

  • side=0.582; lower-left-point=-0.4+0.5i (made using a JAVA applet)

  • side=0.0017815; lower-left-point=-0.75+0.06i (made using a JAVA applet)

  • side=0.004402; lower-left-point=0.28+0.0084i (made using a JAVA applet)

  • side=0.000191; lower-left-point=-0.78-0.136i (made using a JAVA applet)

  • side=0.00004; lower-left-point=-1.595+0.000095i (made using a JAVA applet)

  • side=0.0001558; lower-left-point=-0.75+0.064i (made using a JAVA applet)

  • side=0.0000829; lower-left-point=0.253-0.0031i (made using a JAVA applet)

  • side=0.0166; lower-left-point=-1.042-0.0346i (made using a JAVA applet)

  • center=-0.745-0.1i

[edit] Art and the Mandelbrot set

  • Calm Mandelbrot, magnified 100,458,337,236 times, 256 iterations

  • Cool Mandelbrot, magnified 248,034,982,258 times, 256 iterations

  • Hot Mandelbrot, magnified 261,880 times, 1024 iterations

  • Fractal-zoom-1-03-Mandelbrot Buzzsaw.png
  • Fract027.jpg

  • Galaxy of Galaxies

  • Mandelbrot Budding turbines.jpg
  • Mandelbrot Cypress Underbrush.jpg
  • Mandelbrot Islands of Consciousness.jpg
  • Mandelbrot Spiral Cleft.jpg
  • Mandelbrot Spiral Cleft 2.jpg
  • Mandelbrot set from moire patterns.jpg
  • Mandelbrot Deep Re-Twisted.jpg

  • Rendered with a scenery generator

  • Mandelbrot Contours.jpg

[edit] Universality of Mandelbrot set

  • Mandelbrot sets occuring in the analysis of Newton's method

  • Mandelbrot set occuring in the analysis of Newton's method (detail)

  • Mandelbrot set occuring in the analysis of Newton's method (detail)

  • Mandelbrot set occuring in a model of phase transitions (detail)

  • Mandebrot set embedded in cos(z)+c.

  • Mandebrot set embedded in gaussian exp(-z^2)+c.

[edit] Miscellaneous

  • Comparing functions, Mandel/Bifurcation.

  • Periodic cycles in the Mandelbrot set

  • Mandelbrot, plane: 1/lambda

  • Relation between Mandelbrot set and en:logistic map

  • Multibrot.ogg

    Multibrot sets

  • Map of Julia sets showing the resemblance with the Mandelbrot set

[edit] First steps in Mandelbrot set

  • 1.step of Ms

  • 2.step of Ms

  • 3.step of Ms

  • 4.step of Ms

  • 5.step of Ms

  • 9.step of Ms

  • 30.step of Ms

  • part of 30.step of Ms

  • Mandelbrot seq.png

  • step 1

  • step 2

  • step 3

  • step 4

  • step 5

  • step 6

  • step 7

  • step 8

  • step 9

  • step 10

  • step 11

  • step 12

  • step 13

  • step 14

  • step 15

  • step 16

  • step 17

  • step 18

  • step 19

  • step 20

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