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

    Benoît Mandelbrot and the set bearing his name

General[edit]

  • Hi-resolution Mandelbrot set with axes

    Hi-resolution Mandelbrot set with axes

  • Mandelbrot set and periodicities of orbits

    Mandelbrot set and periodicities of orbits

  • Mandelbrot set and colorcoded periodicities of orbits

    Mandelbrot set and colorcoded periodicities of orbits

  • Mandelbrot set with well defined colour stripes

    Mandelbrot set with well defined colour stripes

  • Mandelbrot set with irregular colour stripes

    Mandelbrot set with irregular colour stripes

  • Mandelbrot set in grayscale

    Mandelbrot set in grayscale

  • Mandelbrot set with smooth color gradient

    Mandelbrot set with smooth color gradient

  • Mandelbrot zoom

    Mandelbrot zoom

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

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

  • Inside colour-mapping, (colour version).

    Inside colour-mapping, (colour version).

  • Mandelbrot and Julia.png
  • Screenshot von RFL Mandelbrot Set Exploration Tool v0.0.4

    Screenshot von RFL Mandelbrot Set Exploration Tool v0.0.4

  • Visualization of Mandelbrot set in complex plane

    Visualization of Mandelbrot set in complex plane

  • Command-line depiction of the Mandelbrot set.

    Command-line depiction of the Mandelbrot set.

  • Yet another image of the Mandelbrot Set.

    Yet another image of the Mandelbrot Set.

  • Mandelbrot 70000x.png
  • Representation of Inner Structure

    Representation of Inner Structure

  • colors

    colors

Structure[edit]

  • Boundaries of hyperbolic components of mandelbrot set

    Boundaries of hyperbolic components of mandelbrot set

  • Lemniscates - boundaries of level sets of escape time

    Lemniscates - boundaries of level sets of escape time

  • Centers of hyperbolic components

    Centers of hyperbolic components

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

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

Rays[edit]

  • External and internal rays, center and root

    External and internal rays, center and root

  • External rays of Misiurewicz point

    External rays of Misiurewicz point

  • External ray of Misiurewicz point c=-2

    External ray of Misiurewicz point c=-2

  • Uniformization of complement of Mandelbropt set

    Uniformization of complement of Mandelbropt set

  • Part of parameter plane with external 5 rays landing on the Mandelbrot set.png
  • Part of parameter plane with Mandelbrot set and bifurcation point 13 over 34 of the main cardioid.png
  • Wakes near the period 1 continent in the Mandelbrot set

    Wakes near the period 1 continent in the Mandelbrot set

  • Wakes along the main antenna in the Mandelbrot set

    Wakes along the main antenna in the Mandelbrot set

Fractalizer[edit]

Zoom[edit]

  • 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 zoom sequence with 14 steps

  • Initial image of a corresponding zoom sequence with frames

    Initial image of a corresponding zoom sequence with frames

  • Zoom step 1 of 14

    Zoom step 1 of 14

  • Zoom step 1 of 13

    Zoom step 1 of 13

  • Zoom step 2 of 14

    Zoom step 2 of 14

  • Zoom step 2 of 13

    Zoom step 2 of 13

  • Zoom step 3 of 14

    Zoom step 3 of 14

  • Zoom step 3 of 13

    Zoom step 3 of 13

  • Zoom step 4 of 14

    Zoom step 4 of 14

  • Zoom step 4 of 13

    Zoom step 4 of 13

  • Zoom step 5 of 14

    Zoom step 5 of 14

  • Zoom step 5 of 13

    Zoom step 5 of 13

  • Zoom step 6 of 14

    Zoom step 6 of 14

  • Zoom step 6 of 13

    Zoom step 6 of 13

  • Zoom step 7 of 14

    Zoom step 7 of 14

  • Zoom step 7 of 13

    Zoom step 7 of 13

  • Zoom step 8 of 14

    Zoom step 8 of 14

  • Zoom step 8 of 13

    Zoom step 8 of 13

  • Zoom step 9 of 14

    Zoom step 9 of 14

  • Zoom step 9 of 13

    Zoom step 9 of 13

  • Zoom step 10 of 14

    Zoom step 10 of 14

  • Zoom step 10 of 13

    Zoom step 10 of 13

  • Zoom step 11 of 14

    Zoom step 11 of 14

  • Zoom step 11 of 13

    Zoom step 11 of 13

  • Zoom step 12 of 14

    Zoom step 12 of 14

  • Zoom step 12 of 13

    Zoom step 12 of 13

  • Zoom step 13 of 14

    Zoom step 13 of 14

  • Zoom step 13 of 13

    Zoom step 13 of 13

  • Zoom step 14 of 14

    Zoom step 14 of 14

  • Mandelbrot (Ausschnitt)

    Mandelbrot (Ausschnitt)

  • Mandelbrot (Ausschnitt)

    Mandelbrot (Ausschnitt)

  • Mandelbrot Animation1.gif

  • Zooming Movie 03

  • Zooming movie 04

  • Zooming movie 06

  • Zooming movie 15

  • High-resolution zoom

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

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

Iteration[edit]

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

    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.

    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.

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

  • 256

    256

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

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

  • 1024

    1024

  • 2048

    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+.

    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.

    Number of iterations changing from 1 to 50.

Some details of the Mandelbrot set[edit]

Art and the Mandelbrot set[edit]

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

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

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

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

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

    Hot Mandelbrot, magnified 261,880 times, 1024 iterations

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

    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

    Rendered with a scenery generator

  • Mandelbrot Contours.jpg
  • Mandelbrot Set Image 01 by Aokoroko.jpg
  • Mandelbrot Set Image 02 by Aokoroko.jpg
  • Mandelbrot Set Image 04 by Aokoroko.jpg
  • Mandelbrot Set Image 11 by Aokoroko.jpg
  • Mandelbrot Set Image 08 by Aokoroko.jpg
  • Mandelbrot Set Image 10 by Aokoroko.jpg
  • Mandelbrot Set Image 03 by Aokoroko.jpg
  • Mandelbrot Set Image 06 by Aokoroko.jpg
  • Mandelbrot Set Image 05 by Aokoroko.jpg
  • Mandelbrot Set Image 07 by Aokoroko.jpg
  • Mandelbrot Set Image 09 by Aokoroko.jpg
  • Mandelbrot Set Image 12 by Aokoroko.jpg
  • Mandelbrot Set Image 13 by Aokoroko.jpg
  • Mandelbrot Set Image 14 by Aokoroko.jpg
  • Mandelbrot Set Image 15 by Aokoroko.jpg
  • Mandelbrot Set Image 16 by Aokoroko.jpg
  • Mandelbrot Set Image 19 by Aokoroko.png
  • Mandelbrot Set Image 20 by Aokoroko.png
  • Mandelbrot Set Image 21 by Aokoroko.png
  • Mandelbrot Set Image 23 by Aokoroko.png
  • Mandelbrot Set Image 18 by Aokoroko.png
  • Mandelbrot Set Image 24 by Aokoroko.png
  • Mandelbrot Set Image 25 by Aokoroko.png
  • Mandelbrot Set Image 17 by Aokoroko.jpg
  • Mandelbrot Set Image 26 by Aokoroko.png
  • Mandelbrot Set Image 27 by Aokoroko.png
  • Mandelbrot Set Image 28 by Aokoroko.png
  • Mandelbrot Set Image 29 by Aokoroko.png
  • Mandelbrot Set Image 30 by Aokoroko.png
  • Mandelbrot Set in blue 01 by Aokoroko.png
  • Mandelbrot Set Color Cycling Animation 400px.gif
  • Mandelbrot Set Color Cycling Animation 400px 2.gif
  • Mandelbrot Set Color Cycling Animation 400px 3.gif

Universality of Mandelbrot set[edit]

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

    Mandelbrot sets occurring in the analysis of Newton's method

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

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

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

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

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

    Mandelbrot set occurring in a model of phase transitions (detail)

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

    Mandebrot set embedded in cos(z)+c.

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

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

Miscellaneous[edit]

  • Comparing functions, Mandel/Bifurcation.

    Comparing functions, Mandel/Bifurcation.

  • Periodic cycles in the Mandelbrot set

    Periodic cycles in the Mandelbrot set

  • Mandelbrot, plane: 1/lambda

    Mandelbrot, plane: 1/lambda

  • Relation between Mandelbrot set and en:logistic map

    Relation between Mandelbrot set and en:logistic map

  • Multibrot sets

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

    Map of Julia sets showing the resemblance with the Mandelbrot set

  • Mandelbulber v2.23

    Mandelbulber v2.23

  • picture of a Buddhabrot

    High-resolution Buddhabrot rendering of the Mandelbrot set

First steps in Mandelbrot set[edit]

  • 1.step of Ms

    1.step of Ms

  • 2.step of Ms

    2.step of Ms

  • 3.step of Ms

    3.step of Ms

  • 4.step of Ms

    4.step of Ms

  • 5.step of Ms

    5.step of Ms

  • 9.step of Ms

    9.step of Ms

  • 30.step of Ms

    30.step of Ms

  • part of 30.step of Ms

    part of 30.step of Ms

  • Mandelbrot seq.png
  • step 1

    step 1

  • step 2

    step 2

  • step 3

    step 3

  • step 4

    step 4

  • step 5

    step 5

  • step 6

    step 6

  • step 7

    step 7

  • step 8

    step 8

  • step 9

    step 9

  • step 10

    step 10

  • step 11

    step 11

  • step 12

    step 12

  • step 13

    step 13

  • step 14

    step 14

  • step 15

    step 15

  • step 16

    step 16

  • step 17

    step 17

  • step 18

    step 18

  • step 19

    step 19

  • step 20

    step 20

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