Mandelbrot set

Benoît Mandelbrot and the set bearing his name

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

Buddhabrot

Mandelbrot zoom

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

Inside colour-mapping, (colour version).

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.

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

External and internal rays, center and root

External rays of Misiurewicz point

External ray of Misiurewicz point c=-2

Example 01
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Example 02
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Example 03
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Example 04
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Example 05
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Example 06
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Example 07
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Example 08
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Example 09
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Example 10
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Example (video) 01
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Example (video) 02
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Initial image of a zoom sequence with 14 steps

Initial image of a corresponding zoom sequence with frames

Zoom step 1 of 14

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Zoom step 3 of 14

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Zoom step 5 of 14

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Zoom step 9 of 14

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Mandelbrot (Auschnitt)

Mandelbrot (Auschnitt)

Zooming Movie 03

Zooming movie 04

Zooming movie 06

Zooming movie 15

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.

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

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

Galaxy of Galaxies

Rendered with a scenery generator

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)

Comparing functions, Mandel/Bifurcation.

Periodic cycles in the Mandelbrot set

Mandelbrot, plane: 1/lambda

Relation between Mandelbrot set and en:logistic map

Multibrot sets

Map of Julia sets showing the resemblance with the 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

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step 20