Fractals #1: the Cantor and Mandelbrot Sets

Oh, no - not *another* page of Mandelbrot set pictures?

Well, yes, it is - but with a difference. Please bear with me; I think you'll be pleasantly surprised.

Actually, I don't have a problem if I happen to run across a web-page full of fractals of various kinds, including Mandelbrot set pictures. In fact, I go looking for them sometimes. It may be some 20 years since the "M-set" burst into public consciousness; but I'm still happy to see it. I can't imagine I'll ever get sick of it.

Although Mandelbrot set graphics are the main thrust of this page, there are also a few other matters mentioned here by way of introduction. If some of them encourage you to scratch your brain, I'm doing something right!

[Just while I think of it - if you like the Mandelbrot set, and if you also like science fiction, may I recommend "The Ghost from the Grand Banks" by Arthur C. Clarke (Orbit, 1990), in which the M-set plays a key rôle.]

In the early 1980's, there were rumblings in the mathematical/scientific community that some sort of major paradigm-shift was under way. Our understanding of how the universe is put together was changing.

A century earlier, in 1883, the German mathematician Georg Cantor had published a paper in which he described what would come to be known as the "Cantor set".

Suppose you have a straight line segment. (Usually, we consider the line segment joining 0 and 1 on the real number line.) Now, remove the middle third - the bit between 1/3 and 2/3 - but leave the actual points 1/3 and 2/3 behind. Then remove the middle third of each of the two remaining pieces, again leaving the end-points. This leaves four short pieces. Remove the middle third of each of these...

If we continue in this way "forever", what is left?

Well, of course, we can't continue forever, because we don't have enough
time. But we can *imagine* what would happen.
"Eventually", there will be no line segments left at all; there will just be
a *dust* of isolated points.

There are some surprises among those isolated points. You'd expect all the
end-points of the various line-segments generated by removing bits of the
original line to be included - as they indeed are. But there are other
points in there too. Guess what: one of them is the point corresponding
to 1/4 on the number line - and that is very definitely *not* one of
the end-points! How about that? Can you see which of the tiny line-segments
at the bottom of the above diagram must contain that point?

__Even better:__ Can you *prove* that 1/4 is in the Cantor set? Spend a bit
of time thinking about it; if you really can't figure it out, you can click
here
to see a proof. (Here's a clue: there's a *geometric progression*
involved.)

NOTE: There is always a chance that Word documents may pick up viruses or other nasties in their travels over the internet. Please see the suggestions on my home page under SECURITY before downloading this if you have any doubts.

The Cantor set is the simplest example of what Benôit Mandelbrot would call
a *fractal*, in his book "Fractals: Form, Chance and Dimension",
published in 1977 by W.H.Freeman and Company.

Since Cantor's time, many fractals have been "discovered". If you're not too familiar with the subject, I recommend that you use a WWW search engine to see what you can find about the following: Koch snowflake curve; Peano space-filling curve; Sierpinski gasket; Menger sponge; Alexander horned sphere. The following links are helpful in this regard, and also contain further information about the Cantor set:

http://en.wikipedia.org/wiki/Cantor_set

http://mathworld.wolfram.com/CantorSet.html

http://mathworld.wolfram.com/TotallyDisconnectedSpace.html

All the fractals mentioned in the previous paragraph are close relatives of the Cantor set, except perhaps the second - and even then, it's possible to make a connection. In Scientific American's "Mathematical Games" column, in the April 1978 issue, Martin Gardner discusses a link between Peano's ideas and the Koch snowflake curve. Well worth a read if you can find a copy.

During the early 1980's, Dr. Mandelbrot placed the emerging science of chaos theory and fractals on a firm foundation. Most importantly, he pointed out that fractals do indeed have a lot to do with nature, from coastlines and cloud formations to the way plants grow. Appropriately, the most famous and popular fractal of all, the Mandelbrot set (which he discovered), is named after him.

The basic concept of a fractal is *self-similarity* - the idea that if
you "zoom in" (or "zoom out", for that matter), the thing still looks pretty
much the same at any level of magnification. Thus, when you remove the
middle thirds at a particular stage while "generating" the Cantor set, the
result at the next stage looks - well, *similar*.

The same is true of coastlines and clouds, and also many plants (fern leaves and broccoli are classic examples). It really does seem to be a law of nature. In another of his books, "The Fractal Geometry of Nature" (1982), also published by W.H.Freeman and Company, Mandelbrot develops this theme.

The Mandelbrot set has this self-similar property too. It's fun to zoom in on parts of it and find - astonishingly - small "almost-copies" of itself scattered all over the place, each containing even smaller "almost-copies"...

Great fleas have little fleas upon their backs to bite 'em,

And little fleas have lesser fleas, and so on *ad infinitum*.

And the great fleas themselves, in turn, have greater fleas to grow on,

While these again have greater still, and greater still, and so on.

*- Augustus de Morgan (1806-1871), in "A Budget of Paradoxes", 1866*

(You'll find lots of "fleas" in the pictures toward the bottom of this page.)

Just before moving on:

In February 1999, Scientific American published an article by Dr. Mandelbrot entitled "A Multifractal Walk down Wall Street", in which he asserted that the behaviour of the stock market is essentially chaotic in nature. (Click here to see a short review of this article.)

To read more about this obviously important issue, click here. You may also be interested to vist my All that glisters... page to see some of my thoughts on this and related matters. (That page also contains a link to the web-page just mentioned.)

In August 1985, in Scientific American's "Computer Recreations" column, host A.K.Dewdney presented a magnificent article which has probably done more to make the Mandelbrot set known to a wide audience than any other. To their credit, Scientific American gave the cover of that issue over to a lovely graphic of part of the M-set. If you'd like a good introduction, you can't do better than to obtain a copy. (You can't have mine - it ranks among my other treasures like "The Boy Electrician" by Alfred P. Morgan, "Model Making for Young Physicists" by A.D.Bulman, and "Polyhedron Models" by Magnus J. Wenninger - and you can't have any of those either. Sorry!)

On the other hand, if you'd like a very brief overview of how the Mandelbrot set "works", you can click here to see my own article.

NOTE: There is always a chance that Word documents may pick up viruses or other nasties in their travels over the internet. Please see the suggestions on my home page under SECURITY before downloading this if you have any doubts.

As the opening of this page suggests, there are many, many web-pages which feature fractals - the M-set especially. One of the best is that by Prof. Clint Sprott, which features an enormous range of very interesting fractals of many kinds - highly recommended.

Such graphics are often produced with top-of-the-range programming software which requires the user to have considerable expertise in its use. Or they might be generated with dedicated packages like the excellent freeware program FRACTINT, which - though powerful and effective - can't really provide the user with a nitty-gritty programming experience.

Another way of exploring the Mandelbrot set is by using a dedicated M-set zoom program. There are quite a few of these around on the net; I recommend the one by Rudy Rucker .

Actually, this is part of a software package inspired by James Gleick's magnificent book "CHAOS", published in 1988 by Sphere Books. This package, also entitled "CHAOS", can be downloaded here . It's a DOS package which will run OK under Windows. In addition to the section on the Mandelbrot set (and variations), there is a host of other very intriguing modelling programs: a pendulum over magnets; strange attractors (including the famous Lorenz attractor); fractal landscapes; various cellular automata (including a "hodgepodge machine", which models the fascinating Belousov-Zhabotinsky reaction) - and lots more. It's described as shareware, but there's no cost for its non-commercial use. You'd be hard pressed to find anything better anywhere.

In common with FRACTINT, CHAOS is still more of a software package than a "programming language", but can provide you with a neat entry-point into M-set lore. You can use the Mandelbrot component to hunt around the M-set to find a region you like, read off its screen coordinates, and then - if you're so inclined - enter those coordinates into your own M-set program (feel free to modify mine, presented below) to produce highly customized views of your favourite regions.

If you've come here by way of my Mathematically-Based Computer Graphics sub-menu, you'll know that I live in what some may unkindly call a time-warp. ("It's just a jump to the left...") For programming, I prefer good old quick-and-dirty methods like a DOS BASIC interpreter, in which I can throw something together which is easy to understand, and which operates in what I regard as a reasonably non-threatening environment - and I can tweak things to work exactly as I want. And that's very satisfying!

The rest of this page is dedicated to some of my own M-set examples, with some discussion of the origin of each one and how it was "massaged" into something of which I could be proud. As you'll see, there's as much art as science in generating a really satisfying fractal.

My first attempts to produce Mandelbrot set pictures were done in BASIC on
my old Commodore 64, way back in late 1985. They took *ages* - each
picture was an overnight run (or even longer, if I was trying to "dig deep"
into the set). However, if I may say so, I obtained some pretty nice
results! (Click
here
and scroll down toward the bottom of the page to see them.)

I "got the bug" (or "flea", perhaps?) to start doing this again, on my PC, late in 2004.

Here you can see the results. To date (March 2005), I have produced 18 essentially different Mandelbrot set graphics, some with variations. You can click on the links below to go to a dedicated page for each.

UPDATE, August 2005: three more have been added, taking the total to 21.

UPDATE, April 2006: four more, slotted in so that they are numbers 20, 21, 22, and 23; those previously numbered 20 and 21 are now 24 and 25.

In each of the following pages, you will find BASIC source code for each graphic, with some discussion of how I adjusted the programs to get just the effect I wanted at any stage. By clicking on a particular scaled-down graphic (320 × 240 pixels), you can see the full-size version (640 × 480 pixels).

There's enough here, hopefully, to enable you to get involved in producing your own Mandelbrot set graphics, with whatever programming system you use - if you're that way inclined.

Let the journey begin!

Mandelbrot set: Region #1

(the whole set)

Mandelbrot set: Region #4

("Winter in Mand-land" - my favourite bit of the M-set)

Mandelbrot set: Region #9

(invaded by "South Park"'s Mr. Hanky??!)

Mandelbrot set: Region #10

(here's lookin' at you, kid)

Mandelbrot set: Region #18

(trying to survive in an algal bloom?)

Mandelbrot set: Region #20

(a view from the east)

Mandelbrot set: Region #21

(more branches)

Mandelbrot set: Region #22

(still more branches)

Mandelbrot set: Region #23

(Andrew laMance's "Love Canal")

Mandelbrot set: Region #24

"The Utter West"

Mandelbrot set: Region #25

"The Last Trump"...?

Earlier in this page, I mentioned Arthur C. Clarke's novel "The Ghost from the Grand Banks". The book's main theme is an attempt to salvage the Titanic; but - intriguingly - the Mandelbrot set plays an important part. The novel was first published by Orbit in 1990, soon after the Mandelbrot set became well-known - almost certainly largely as a result of Dr. Dewdney's previously-mentioned article in Scientific American.

*ANYWAY*, about halfway into the novel (Chapter 20: "Into the M-Set"),
a quote from Albert Einstein appears. I'd like to reproduce it here:

The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand wrapt in awe, is as good as dead.

Who could disagree with that?

CONCLUSION

In this page we've looked at the the simplest fractal - the Cantor set - and the Mandelbrot set, which has been called "the most complicated object in mathematics" by John H. Hubbard, an authority on the set. Both sets are fascinating and charming in their own ways.

There's a lot of self-similarity in the Mandelbrot set - but you never see exactly the same thing twice. It's absolutely full of surprises!

If you've taken some time to have a look through the above pages, by now you've probably gained a reasonable grasp of what the M-set is all about, and perhaps picked up a few techniques for exploring and representing it yourself.

However, we've only scratched the surface. There are other ways of representing the M-set which look quite different from the somewhat "traditional" way discussed here; and they are no less valid. If you hunt around on the web, no doubt you'll find some of them. So now, it's up to you! Why don't you continue your journey?

__UPDATE, Wednesday, 6th October 2010__

Just yesterday, I discovered a series of six YouTube videos which together make up a production called "Fractals: the Colors of Infinity". This remarkable program discusses, in considerable detail, fractals in general and the Mandelbrot Set in particular. The host is Arthur C. Clarke, and the program begins with a reference to Albert Einstein's "... most beautiful thing..." comment (see above).

In this program, there are many sequences showing the M-set in all its glory, in full and fizzing colour, along with clear explanations and some speculative thought about what it all means. The soundtrack features the music of Pink Floyd's David Gilmour (click here to visit Mr. Gilmour's website). I recommend it highly. Here is the URL for the first of these six videos:

http://www.youtube.com/watch?v=qB8m85p7GsU

Videos 2 to 6 are then easily found within YouTube's own indexing system - along with many other videos in similar vein. So, what are you waiting for? Go ahead - enjoy them!

__UPDATE, Saturday, 30th October 2010__

I've just become aware that Benôit Mandelbrot died on Thursday 14th October - just 16 days ago - at the age of 85.

Of course I'm saddened; but I'd like to say that I can't help but have my spirits lifted by what this "father of fractal geometry" has achieved. Since his discovery of the M-set in 1980, his contribution has helped enormously to give mathematics a more human face, not least by drawing attention to the fact that nature itself appears to be fractal.

(Click here to read the Wikipedia article, and here for the New York Times obituary.)

R.I.P., Dr. Mandelbrot. The world is a better, more thoughtful place for your having been here.

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