The Mandelbrot Set  

The Mandelbrot Set

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The most well-known fractal object is probably the Mandelbrot Set, named for Benoit Mandelbrot, who first investigated its fractal properties in the late 1970s. Unlike some of my own algorithms, the Mandelbrot set takes a lot of time to compute.

The Algorithm

Each complex number can be considered as a point on the plane. Imagine that you take such a number, square it, add it to itself, then square the new number, add it to itself, and repeat the process indefinitely. The resulting sequence of points will either converge towards a limit point, or diverge - the numbers get larger and larger. If they converge, the original point is considered to be in the Mandelbrot set.

When the computer calculates, it tests each point by following the above process. It can never be certain that a point is in the set, so it will stop after a certain number of repetitions and declare the point to be part of the set. If the calculated number passes a certain threshold (an absolute magnitude of 2) then the point cannot possibly be in the set, and is "thrown out".

In the Mandelbrot images, points in the set are shown as black areas. The interesting parts of the images are the points which are not in the set. They are assigned different colors according to how quickly they diverge - that is, how many repeated calculations have occurred before the point is "thrown out".

Zooming in on the Maps

I have assigned arbitrary names to the images according to aspects of their appearance. Each named section features a monochrome map of the area, with rectangles showing further zooms which you may investigate. If you click on a blue rectangle, you will zoom further and be shown to another map. If you click on a red rectangle, you will see the expanded color image itself, since no further zooming in on that area is allowed.

If you click any of the thumbnail images, you will see an enlarged color image of the same area.



 
Main Main        Back to Top of Page    Back to Fractals Page
Click the blue rectangles to zoom further (red rectangles indicate that no further zooming is possible). Click a thumbnail for the image itself.
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Top Side
Peninsula End
Point Fold
Main
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Top Top        Back to Main Main
Click the blue rectangles to zoom further (red rectangles indicate that no further zooming is possible). Click a thumbnail for the image itself.
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Tornado
Top
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Side Side        Back to Main Main
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Meander
Side
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Peninsula Peninsula        Back to Main Main
Click the blue rectangles to zoom further (red rectangles indicate that no further zooming is possible). Click a thumbnail for the image itself.
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Starfish Spinner
Inferno Hydra
Spokes
Peninsula
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Spinner Spinner        Back to Peninsula Peninsula
Click the blue rectangles to zoom further (red rectangles indicate that no further zooming is possible). Click a thumbnail for the image itself.
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Soap
Spinner
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End End        Back to Main Main
Click the blue rectangles to zoom further (red rectangles indicate that no further zooming is possible). Click a thumbnail for the image itself.
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Rays
End
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Rays Rays        Back to End End
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Sun
Rays
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Sun Sun        Back to Rays Rays
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Stripes
Sun
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Point Point        Back to Main Main
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Pits
Point
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Pits Pits        Back to Point Point
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Fisheyes
Pits
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Fisheyes Fisheyes        Back to Pits Pits
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Crosseye Eyeball
Fisheyes
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Fold Fold        Back to Main Main
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Comet Beads
Plumes
Fold
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Comet Comet        Back to Fold Fold
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Droplets
Comet
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Beads Beads        Back to Fold Fold
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Curls
Beads
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Curls Curls        Back to Beads Beads
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Lace
Curls
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Plumes Plumes        Back to Fold Fold
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Claw
Plumes
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Claw Claw        Back to Plumes Plumes
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Pinwheel
Claw
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Pinwheel Pinwheel        Back to Claw Claw
Click the blue rectangles to zoom further (red rectangles indicate that no further zooming is possible). Click a thumbnail for the image itself.
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Spin
Pinwheel
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The Mandelbrot Set

Last Updated October 6th, 2001
Web Page by Ned May (nedmay@chromatism.net)
URL http://chromatism.net/chromat/mandelb.htm
All images and text are ©2001 by Edward S. May unless otherwise noted. All rights reserved.