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PGD's Quick Guide to the Mandelbrot Set

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This guide assumes your primary interest in the M-set is as an explorer, and that you have a fractal-drawing program such as FractInt and know how to zoom with it.   (If you are looking for a fractal-generating program click here.   If you do not know how to bring up an image of the M-set and zoom into it read the program's documentation.)


What Is The Mandelbrot Set?

Mandelbrot Set Frequently Asked Questions

The Mandelbrot Set is a well-known fractal, with an infinitely detailed structure that is highlighted by computer generated colors.   It can be magnified by enormous amounts and still keep its detail.   (If you have tried zooming in with your program and the detail smooths out after a while click here.)   Formula Details

Here is a guide to the locations of the most commonly visited parts of the Set and what may be found there.


World Map and Popular Tourist Areas

The large image below is the Mandelbrot Set.   To the left and below are the popular names of the areas indicated on the map, and small thumbnails of typical creatures from this area.

1. Seahorse Valley
-0.75, 0.1
2. Elephant Valley
0.275, 0
3. Triple Spiral Valley
-0.088,0.654
4. Quad-Spiral Valley
0.274,0.482
7. Double Scepter Valley
-0.1002,0.8383
5. Scepter Valley
-1.36,0.005
8. Mini Mandelbrot
-1.75,0
6. Scepter Variant
-1.108,0.230
9. Another Mandelbrot
-0.1592,-1.0317

One may notice a pattern with the seahorse, the two-headed seahorse of triple spiral valley, and the three-headed seahorse of quad-spiral valley, and the corresponding spirals on the opposite cleft with two, three, and four arms.   Indeed, if one takes a cleft beneath a bud further along the series that starts with the front and top buds and proceeds in a stately row down the cleft at the rear of the set, one will find five, six, and seven and so on armed spirals, and seahorses with an ever increasing number of heads.

Another pattern is in the scepters:  pick any bud with a dendrite.   Note the number of arms of the dendrite and any spirals it has.   Now pick the bud's biggest bud.   As the biggest bud of the Mandelbrot core has a cleft of seahorses, so the cleft of this bud has seahorses that have scepters attached.   The scepters radiate in a group from the seahorse's 'crown' and are the same in number as the arms of the dendrite.   If the dendrite was itself an elephant or seahorse with a spiral, the scepters have spirals and have the form of an elephant or seahorse.   And other buds beside the biggest bud of the larger bud give rise to three-armed spirals, four headed seahorses, and other creatures familiar from elsewhere that are here adorned with similar scepters.


Seahorses Change As You Descend A Cleft

Not all seahorses in Seahorse Valley are alike.   In fact depending on how far down the cleft you go to zoom up a seahorse its characteristics may change remarkably.

Anatomy of a seahorse

A seahorse consists of a stalk protruding from a 'bud' of the Mandelbrot core, which reaches into a central 'eye' from which radiate an assortment of curved 'arms'.   Two of these 'arms' terminate in spirals, a 'head' spiral and the large 'tail' spiral.

Growth and development of a seahorse

If one examines the seahorses attached to buds along the row descending the Valley cleft, one shall find that each one in turn has two arms more than the previous, one inserted between the stalk and the head spiral and one between the head spiral and tail spiral.   The spirals also become wrapped more tightly and the 'eye' pattern narrows in the center and becomes oval.   Lastly, the head spiral stretches closer to touching the seahorse's 'belly' and the tail spiral of one seahorse comes closer to touching the 'back' of the previous one up the valley's side.

The first two seahorses are a good example.   The large branch at the top of the set attached to the large bud there is the first seahorse.   It doesn't look much like one.   From the branch radiate the stalk, one arm ending in a slight curve which is a harbinger of tail spirals, and one arm that has a would be head spiral.   (Zoom one of these at the tip again and again and you can see it turn 360 degrees around over an order of magnitude of magnification.) The next seahorse is the five-branched 'starfish' between the y-branch bud and the front spike bud.   It has two additional arms.

The images below show five buds further along the trail into the Valley, and their attendant seahorses.   Note how the seahorse takes shape.

A pattern very similar to this happens with the two and three-headed seahorses, the elephant, and the multi-spiral creatures.   Spiral arms wind tighter and spirals become more spiral the further into the cleft one zooms.


Mini Mandelbrots

They're everywhere:  miniature images of the Mandelbrot set.   Not only do they crop up inside of the large Mandelbrot set, they turn up in completely unrelated formulas too.

Mini Mandelbrots have every feature of the main set replicated, including all the buds, the seahorses, and the spirals.   Each mini Mandelbrot also sports "dendrites" whose shape is dependent on the particular mini Mandelbrot.   These "dendrites" may be spirals, lightning bolts, and that and attach in twos and fours and eights to the mini Mandelbrot and its seahorses and other antennae.
Mini Mandelbrot in a magnetic phase transition formula.
A mini Mandelbrot can often be found where there is a symmetrical pair of spirals or other objects.

Mini Mandelbrots raise interesting questions:  why do many different unrelated formulas have them?   What does this universal presence mean about these formulas?

A pair of symmetrical
seahorses

A mini Mandelbrot lies
within


Mini Mandelbrots And 'Replication'

Mini mandelbrots have an amazing ability in fact.   They can take their surroundings and produce a pair of replicas joined together.   The above minibrot came from near the central "eye" of a seahorse, and lo and behold the large structure surrounding it is a pair of seahorses joined near their "eyes".
The familiar double spiral is another example of this phenomenon.   Inside a seahorse spiral there is a mini Mandelbrot; near the mini Mandelbrot there are two spirals joined exactly at the point where the minibrot is.
Here are some other examples of this phenomenon.   The top image of each pair shows a Mandelbrot structure; the bottom image shows the surroundings of a mini Mandelbrot found near the marked point in the top image.   The bottom image is invariably two of the top image joined at the marked point, sometimes in a convoluted fashion.


Julia Sets

There is a whole family of fractals called Julia sets that are intimately related to the Mandelbrot set.   In fact for each point in the M-set there is a Julia set fractal.   Formula Details
The Julia set associated with a given point in the M-set usually resembles in some way the area of the Mandelbrot set around the point.

Map Of Julia Sets

Here are a few Julia sets and their points in the M-set:

As can be seen there is a lot of variety to Julia sets.   Certain patterns are evident.   The Julia sets belonging to points outside the set are in an infinity of pieces and have no black interior lake.   Julias from points on the tips of dendrites are themselves dendritic in form.   Those belonging to points inside the set have areas of black interior lake.
There are patterns in Julia sets that are related to mini Mandelbrots.   Observe:

Miniature Julia sets can be found in the Julia sets whose points lie near a mini Mandelbrot, and in the Mandelbrot set itself in the dendrites of a mini Mandelbrot!


So Begins Your Journey

This is just a taste of what lies in these fractal sets.   Zoom and explore around the M-set, and you'll quickly discover many more interesting areas and many more patterns such as the pattern of spirals growing more arms as you go down the cleft, the pattern of miniature Julia sets, and the pattern of "replication" near mini Mandelbrots.

If you find anything really outrageous, or a fascinating new area of the Mandelbrot set far from any of the areas indicated on the map, mail me at ao950@freenet.carleton.ca and tell me about it.   I may well add the new area to the list here.


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