| The breathtaking beauty and intrigue that surrounds the subject of fractals has captivated both layman and professional alike. We seem to have an inbuilt fascination with these fractal images, as they exhibit an eerie 'familiarity' with the natural world around us. | |
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I chose this assignment to further my existing knowledge of fractals. Earlier this year I worked on a project about recursive graphics for my computer science half of my degree. This investigation sparked my interest into the subject of fractals. |
In this assignment I hope to provide the following:
Such striking fractal images can be created by the use of very simple mathematics, however the definition of fractals is far from being trivial. One key feature that lies behind all fractals is the concepts of recursion, which produces the appearance of self similarity in these images. This is best illustrated by an early type of fractal published in 1904 called Koch's curve.
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The pictures above are generated by applying a simple algorithm through increasing levels of iterations. The 'curve' starts off as an equilateral triangle and then is created by applying the geometric transformation of replacing a ? of the central part of each side of a triangle, with a further 2 segments having the same length as the part being taken away. After the first iteration the image obtained resembles 'David's star'. After successive iterations the result becomes more complex and looking like a snowflake. In summary the image is created by breaking up the overall image, and then breaking these subsequent parts down into smaller versions of the bigger image. This breaking into self similar parts demonstrates the recursive nature of fractals.
The ideas of creating fractal images using recursive techniques have been around long before the work of Benoit Mandelbrot. The true extent of this new branch of mathematics could not be realised until the processing power of modern computers could be used to compute the vast numbers of (albeit simple) calculations involved.
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The term 'fractal' was first coined by Benoit Mandelbrot (pictured left) in 1975. It is derived from the Latin 'fractus', meaning an irregular surface. |
| The Mandelbrot set was first discovered on March 1st 1980 by its creator Benoit Mandelbrot at IBM's Thomas J. Watson research centre (shown right). |  |
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Benoit was using computers to investigate into the previous work of another French mathematician Gaston Julia (pictured left). Gaston had originally worked on creating his own fractal type back in 1917, namely the Julia set. His findings were hampered by being restricted to manual methods of calculation. |
| Benoit then became curious as to what would happen if he could program the most basic Julia set, the resulting image became known as the Mandelbrot set. The image shown on the right is the first output created of the Mandelbrot set. |  |
Benoit Mandelbrot continues to be the foremost pioneer in fractals to date. He has published many books on his work, which have a strong following around the world.
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Using the concepts of complex numbers, imagine that each of the array members co-ordinates are representative of a point on an Argand diagram.
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For example, the array member highlighted in the diagram, with co-ordinates
[11,5] could be used to represent the complex number 3 + 4 i . Now for each
array member you simply enter the respective complex number into a simple
iterative process, using the following equation: 
where z and c are both complex numbers. |
The process is now defined like so:
1. For each member of the array do:
a. Initialise the value of 
and c as the respective complex number.
b. Calculate 
and place
the value into 
.
c. If the modulus of the new 
,
or the number of iterations done so far value n is greater than the number
of colours (eg 256) then proceed to step 'd', else repeat steps 'a' to 'c'.
d. Place the number n into the array member, if more array members are to be initialised return to step 'a'.
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Once all the array members have been set to the appropriate numeric values, the image is generated by colouring a grid of pixels on screen according to the colour codes defined earlier. |
| The true beauty of the fractal is finally revealed, producing the elaborate images that are shown on the next few pages. It is important to note that the actual mandelbrot set is the less interesting central black area. |  |
| These processes can be tweaked to either manipulate the area of the fractal that you are concentrating on, or zoom in closer to view the image with more detail. |  |
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If the size of the array is increased, then the resolution of the image generated will increase. Images are usually created from arrays of size 640X480 as this is the average monitor resoultion on a standard PC. |
| If the number of colours used is increased, then the colours will blend more smoothly in the image. Images only really need around 256 colours as these are the standard VGA pallete settings that are used by monitors. Increasing the colours will slow down the process of generating the image, as more iterations will be performed for each array member. The number of colours used can also be thought of as a default bailout value, as some iterations for certain complex numbers will continue to converge forever. |  |
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If the size of the array is kept constant, and the sample of the complex plane is decreased, then the algorithm will be preformed on smaller increments in the plane. This will give the effect of the fractal being magnified and revealing more detail. As you take smaller sample sizes of the plane, the calculations become more precise, generating the image on a smaller scale. This ability to zoom into the image can be performed indefinitely without losing any detail. |
Choosing different areas on the plane will shift the the focus of the display
onto a particular area of the set. Plotting the mandelbrot set over certain
areas on the plane will create no patterns at all. It is important to remember
the condition that if the modulus of the the function 
is greater than or equal to 2 then the image stops iterating. Therefore if you
were to select a positive area on the complex plane with a real value larger
than 2, then the array members would all have a modulus larger or equal to 2,
causing the process to bail out after only one iteration. This would then result
in a solid block of the same colour. It is therefore vital to pick the right
areas to create the more interesting images you have seen so far.
The ability to define these complex fractal images from such simple mathematical equations has lead to great interests into the applications of these concepts. One major application that is just beginning to be exploited is the ability to use fractal technology for compression of images.
The American based company Iterated Systems has recently developed revolutionary software that converts digital images into fractal formulas. This then means that the amount of data that is stored about the image is much smaller than standard methods of data storage.
This in itself is not a groundbreaking feature, as various compression techniques such as jpeg and gif formats have been around for a few years now, however the real benefits of fractal compression are realised when the image is reconstructed.
Due to the property of being able to zoom into a fractal image and never lose any of the original detail, when the fractal compressed image is enlarged, the gaps of the detail are filled in by the fractal formulas. This means that the image can be enlarged to any desired resolution, from postage stamp size to outdoor billboards.
If the same process were attempted with standard compressed files then the image would become grainy and unclear.
The jury is still out on the true extent of which this powerful area of mathematics can be used. It is thought by many leading professionals that using these techniques may help to solve such big problems as estimating the size of the universe, and unravelling the secrets behind DNA coding sequences.
Fractals are one of the foremost important discoveries in mathematics of the last century. Benoit Mandelbrot managed to pull together all the scattered concepts that had been thought up over the years from previous mathematicians such as Julia and Koch, and build a new concrete foundation for this new branch of mathematics.
The unique properties of fractals have lead to widespread interest in trying to harness the power of these complex geometric objects. Their bewildering simplicity of the processes used to create these images, begs the question into whether the true potential and understanding them can ever be fully realised. It is as if they are almost organic in structure, somehow knowing how to define themselves.
I feel that I have achieved my goal of providing a general overview of fractals and the Mandelbrot set. When researching the topic I was spoilt for choice from the plentiful sources available, mainly from the web. The hardest bit of information to find was a clear explanation of how the mathematics is used to create the fractals. I feel that I have learnt a comprehensive knowledge of the subject of fractals. I have enjoyed working on this project and I feel I have developed a further insite into the applications of mathematics.
Tommorows World TV prgoramme - Segment on Iterated Systems software that incorporated fractal technology.
Colours of Infinity - TV programme presented by Arthur C. Clarke giving an overview of fractals and Benoit Mandelbrots work.
http://library.thinkquest.org/12740/msie4/discover/page19.html - this website gave details about fractal compression and other genreal information.
http://www.cribx1.u-bordeaux.fr/fractals/history.html - this web site had very easy explanations on how fractals are generated.
http://www.daa.com.au/~james/fractals/mandel/formula.html - this web site conatined well programmed java applets that demonstrated different types of fractals, and also provided the source code used to create them.
http://www.apc.pgh.pa.us/fractal.shtml - this contained previous student submissions on an essay assignment about fractals.
http://www.willamette.edu/~sekino/fractal/fractal.htm - this contained in depth information about the history of fractals.
http://www.facstaff.bucknell.edu/udaepp/090/w3/matthewz.html - this website gave a general overview of fractals.
http://www.iglobal.net/lystad/fractals/docfiles/higher-dimensions.html - this web site explained very clearly how matheamtics is used to create fractal images.
http://www.cut-the-knot.com/do_you_know/dimension.html - this web site contains some good general information on fractals as well as good links to other sources.
Fractal Cosmos : The Art of Mathematical Design by Jeff Berkowitz - this book contained a great variety of different fractal images.
The Fractal Geometry of Nature by Benoit Mandelbrot - this book conatined a lot of detail which I found hard to understand.
