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Sunday, June 29, 2014

Try this Mandelbrot program I have written

During the last three weeks I have been amusing myself learning the java programming language.  It hasn't been so easy as at the age of 84 learning and remembering new stuff is harder than it used to be! I have had some help from my son Claude.

To give me inspiration I decided to use my interest in the Mandelbrot set to get to grips with the java language.  I created a program which displays the whole set and you can outline any area you think might be interesting to examine it more closely.

There is a limit to the minimum size of area, due to the limitations of computer precision and it's unavoidable.  However there is plenty of interesting stuff there.

I believe that to see the Mandelbrot set as anything more than a pretty picture or a boring mathematical formula, you need to be mathematically inclined.  The fascination, for me, lies in the mysterious ability of this formula to produce extraordinary patterns from apparently humdrum repetition of a formula.  Looking at the patterns, I am struck by the similarity to many natural forms: Could it be that natural forms, such as a leaf or a flower, have as their formal basis a mathematical formula? Is mathematics more pervasive in the universe than we have heretofore imagined?

There is also the weird phenomenon of little pieces of the set cropping up at some distance from the main chunk, similar in shape but facing in different directions and connected to it by untraceably fine filaments (there are no isolated pieces, they say, and I also suspect that every distant piece is connected to the main by only one path.)
In case anyone wants to explore for themselves, this is a link to the file of my program which I have placed online for anyone to download and try.  To run it, or any other java-based program, you need "java runtime environment" which you probably already have on your computer, but you can download it at the Oracle website if you don't have it.
Once you have opened my program, use your mouse to draw a rectangle somewhere interesting, then press ENTER.  Drawing the rectangle is largely automatic as it has to be horizontal and have certain proportions, so you only need draw the top side of it.  A little practice will soon give you the idea. Email me if you need help.  Read the info on the left of the screen. "Page Up" increase the precision at high magnifications, but also takes longer to draw the picture - perhaps ten seconds or more depending on your  machine's speed..

Possibly you can also open the file in a text editor such as Notepad or Wordpad, if you want to fiddle with it and make alterations.

Friday, June 20, 2014

More Mandelbrot

The Mandelbrot set holds endless fascination for me.  Who would have thought that repeated iteration of a simple mathematical formula could show such intricate, repeating patterns?  For the first picture I used 5,000 iterations as the maximum necessary for determining whether a given point in the complex plane is a member of the set (shown black) or not.  This means that every tiny pixel in the black areas is black because the formula has been iterated 5,000 times to make it so.  It takes 3 or 4 seconds for my laptop to generate this picture, and roughly, say, a quarter of the picture is black and there are nearly a million pixels in the picture. So that alone necessitates 1,250,000,000 iterations, and each iteration means a lot of work for the computer.  What an amazing thing a modern laptop is!  Without such devices at our disposal, we never could have guessed at the intricacies depicted here.

The parts which are not black are points which have been rejected before 5,000 iterations have been performed for them, and their colour is dependent on the number of iterations necessary to decide they should be rejected.

By the way, I read somewhere that it has been proved there are no isolated pieces of the set, every little black bit is connected to the main chunk, even though we may not be able to see the connection because screen resolution is not fine enough or perhaps because we have not used enough iterations, resulting in a pixel being shown black when it shouldn't be. 

Tuesday, June 03, 2014

Do I exist?

The question of the near-universal restricting sense of "I" afflicting humans is a highly interesting and puzzling one. I say "near" universal because there are some people - let's call them "No-I"s - who don't have this restricting sense of "I", they have lost it either intentionally or fortuitously; and "afflicting" because it seems to be at the bottom of nearly all the suffering in human society. The No-Is can still say "I" but it doesn't mean the same thing for them as it does for us unregenerates.

Taking myself as an example, how is it that I have this particular sense of "I" and not some other person"s sense of "I"?  Of course, if I had some other person's sense of "I" I would not be aware of that, I would believe I was that other person, so it seems like a very silly question.  But give it some thought. It can have an effect rather like some of those drawings by M C Escher.

The sense of "I" seems to go with a particular body. As far as I am concerned, this body I am associated with is the centre of my universe.  But is this restricted sense of "I" really necessary?  Some - those who have lost it - say that it's a handicap, and life is infinitely better without it.  Sure, the body-mind organism is still there for use as necessary, but the sense of their personhood has gone.  It's an illusion we would be better discarding.  Each of us thinks that he/she is a person, with goals, problems, etc. but, say the No-Is, this is simply an illusion which dramatically restricts your life and is the cause of all suffering. It sticks, too.  You can't get rid of it easily, because you really do think you are a person - don't you?  It would  be like committing suicide!

The Buddha, it is said, sat under a bodhi tree and vowed not to get up until that illusion had released its hold on him. It did so release, and he did get up, to teach others. He is famously supposed to have said, "Events happen, deeds are done, but there is no doer thereof."

Many will become very indignant when I suggest that the belief that they are a person may be mistaken. It's just a suggestion, but it deserves some thought.

Sunday, June 01, 2014


This picture is a very tiny detail from an image of the Mandelbrot set, a set of complex numbers defined by a simple recursive formula. I made it in the course of modifying Daniel Schiffman's program written in the "Processing" programming language ( I love playing around with programming, especially graphics.)

The Mandelbrot set has fascinated mathematicians and scientists - and other kinds of people, too -  for some decades now, especially since the advent of powerful computers made producing its image very simple.  Magnifying any interesting-looking area will simply produce more fascinating detail, and there's no end to it - in theory.  In practice, it's limited by the ability of the computer to deal with many decimal places.  The image above is a portion of the image below, much magnified (and colour added.)

When I first tried to make the Mandelbrot set image, 23 years ago, my PC had to be left on all night in order to give it time to draw it. I programmed it in the language called "C."  Now, my very ordinary laptop takes just a few seconds to produce a colour image.  Here is an image of the complete set: Those filaments wandering out from the main mass can be magnified to show the kind of detail in the first picture.


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