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Fractals

arebeautiful mathematic

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The formula behind fractals.

The simple formula, first used by Benoit B.Mandelbrot was this :

Z = Z2 + C

Seemingly very simple, but it contains possibilities for an extremely complictedoutput when given interation possibility, and it has also an imaginary part. Thisimaginar part involve use of complex numbers in C, in the terms of i, whichequals the square root of -1.Complex numbers follow their own rules that sometimes differ from those of realnumbers. Because of their unique properties, they are often used in fractals thatare graphed in thecomplex planes.

The so called Mandelbrotset is one example of afractal that is graphed inthe complex plan.

Looking close with amagnifying glass alongthe periferical border(sharp picture), one willsee just the same

structure as in the mainpicture, a unik kind of arepetition.

Julia sets exist in the complex plane,

where the horizontal axis represent thereal numbers, and the vertical axisrepresents imginary numbers. Anassortment of Julia sets here sourroundsthe Mandelbrot set.

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In the equation (x=x2+ C ), the C for Julia setsare more sophisticated, having a complex numberinvolved. This imply infinite possibilities for thedeveloping of fractals.

The two fractal examples shown here wasachieved by different values for the C in theequation, and shows what influence this had forthe image of the fractal pictures.

More thrilling pictures can be achieved bylaying in colours , and the colour distributionwill depend on how many iterations used.

Where to learn more.

When Benoit B.Mandelbrot in 1975 published hisfrst book about fractals, the interest increasedrapidly. Few years later (1978) came his book

The Fractal Geometri of Nature .

This book is far from easilyread, and you should be wellskilled in mathematics and itsformulations to get a profoundadvantage from reading thebook. With its 468 pages anextensive job waits for you !

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For those who prefer a more spontanus meeting with beautyful fractals and lessheavy mathematics, the book The Beauty of Fractals is recommended. It waspublishet in 1986, with 199 pages and 185 figures, many in colour.

Where can we just play with fractals ?

You have a fine opportunity for doing this by downloading a freeware programcalled Fractal Forge.

You find it in Google, just try this :

Fractovia - Fractal Forge Fractal Forge v.2.8.2 is freeware. You can use it to draw your

own fractal images, and explore Mandelbrot Set's branches. Now it's easier and

faster than ... http://www.fractovia.org/uberto/

When you has got in on your screen, just click in upper left corner and then onFile and Open file. Then you get 30 different fractals you can play with.Chose one of them, and Open it. Wait for some seconds, and then click inupper right corner. This should bring you a menu, and click on Data. Now

you can enter into the formula, and change iterations etc. etc., and thenclick on Start to see the result. Good Luck !

Do you only want to look at beautiful fractals ?

An excellent collection can be found in Sekinos Fractal Gallery, try it onthe address . http://www.willamette.edu/~sekino/fractal/annex.htm

Take a look at four of them :

http://go.startsiden.no/go/e/content_results;siteId=230;afu=verden.abcsok.noa47index.html%3Fq%3Dfractal%2520forge%252C%2520freeware%26cs%3Dlatin1/http:/www.fractovia.org/uberto/http://www.fractovia.org/uberto/http://www.willamette.edu/~sekino/fractal/annex.htmhttp://go.startsiden.no/go/e/content_results;siteId=230;afu=verden.abcsok.noa47index.html%3Fq%3Dfractal%2520forge%252C%2520freeware%26cs%3Dlatin1/http:/www.fractovia.org/uberto/http://www.fractovia.org/uberto/http://www.willamette.edu/~sekino/fractal/annex.htm

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Skien, 7. febr. 2010

Kjell W. Tveten