fractals are really beautiful mathematics

8/14/2019 Fractals Are Really Beautiful Mathematics

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Fractals are reallybeautiful mathematics

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Who started ?

When Benoit B.Mandelbrot in 1975 publishedhis frst book about fractals, the interest

increased rapidly. Few years later (1978)came his book The Fractal Geometri ofNature .

This book is by no wayeasy to read, and youshould be well skilled inmathematics and itsformulations to get aprofound advantage fromreading this book. With

its 468 pages anextensive job waits for you !

If you prefeer a more spontanous 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 you 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 :

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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 have got in on your screen, just click in upper left corner and then on

File and Open file. Then you get 30 different fractals you can play with. Choseone of them, and Open it. Wait for some seconds, and then click in upper rightcorner. This should bring you a menu, and click on Data. Now you can enter intothe formula, change iterations etc. etc., and then click on Start to see the result.

Do you just 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 :

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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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The computation of 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 the 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 theirunique properties, they are

often used in fractals that aregraphed in complex planes.

The so called Mandelbrot set isone example of a fractal that isgraphed in the complex plan.

Looking closely with amagnifying glass along theperiferical border (sharppicture), one will see just thesame structure as in the mainpicture, a unik kind of a

repetition.

Julia sets also exist in the complex plane,where the horizontal axis represent the realnumbers, and the vertical axis representsimginary numbers. An assortment of Juliasets here sourrounds theMandelbrotndelndelbrot set.

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In the equation (x=x2+ C ), the C for Juliasets are more sophisticated, having acomplex number involved. This imply

infinite possibilities for the developing offractals.

The two fractal examples shown (right)was achieved by different values for the C in the equation, and shows whatinfluence this had for the image of the fractalpictures.

More thrilling pictures can be achieved by layingin colours , and the colour distribution willdepend on how many iterations used.

Flashbacks

Mandelbrots set, discovered and joined in the therm Fractals in the early1970s is one of the most beautiful and profound discoveries in the history ofmathematics. Not since Pythagoras and the Greeks (ca. 600 B.C) whodocumented all their mathematical breakthrough has there been such arevolutionary discovery.

Fractals were not discovered until the invention of computors. It was virtuallyimpossible to discover fractals before the advent of the computor, because oftheir complexity and gargantuan output.

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Some geometric figures in a basic form was given in 1918 by Gaston Julia. Inthe 1960s early experiments with computor graphics, primarely at MIT, gavesome more advanced figures. The interest in fractals now increased rapidly.

In 1992 , I quite unexpected discovered an article in a nordic mathematical

periodical (NORMAT) with the heading : Matematiska bilder av fraktaler ochkaos (Mathematical pictures of fractals and chaos) written by Hans Wallin,Anders Fllstrm and Mats Wallin. Hans Wallin was then a math processor bythe University of Ume in Sweeden. Laterly he retired (professor emeritus), nowstill going strong. He happens to be my only sweedish second cousin, but I nevermet him. We have only sent and got Christmas greeting for some years, andHans has given me important family relatives information His son Mats is aprofessor of theoretical physics at KTH (Royal Institute of Technology), inStockholm.

I managed several years later to get this article (13 p, 23 images) sent from

KTH, and could study what was presentet as new image materials for fractals,intended as a contribution to the matematics and phyics training in the sixth -former scool.Programs for three experimental tasks was given, written in the programlanguage Turbo Pascal 4.0. My curiosity for such an opportunity lead me to putnew life into my IBM Aptiva from 1994, and try to download the languageTurbo Pascal 6.0 , which I had accessible. I didnt succeed , dont know why.

Instead, the sections concerning Real iteration and Complex iteration was veryinstructive, and had references to several typical images/figures. The computorfor the more simple figures was an IBM PS2, but for more advanced images abigger computor (Apollo) was used.

Concerning Real Iterations this article shows an examples with the following textgiven :

This picture shows whathappens for r=0.7 The row x0 ,x| x2 ...... converge to x*

The point x* is called anattractor to the polynom1 rx2

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Concerning Complex iteration , the iteration of the polynom z2 + c , where c is aconstant, and z is a complex number z= x + iy, where x and y are real numbersand i2 = -1.

This picture illustrate the Julia amount for the equation

c = -0.74543 + 0.11301i

The Julia amount is related to pictures in the physics , illustrating magnetism.

This picture shows an enlargementof the framed part in picture 13

A picture describing chaos fromthe physics.

A differential equation here describe themovement for a pendelum beeing exposed toboth damped and energetic forces

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Skien, 20.februar 2010

Kjell W. Tveten

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fractals are really beautiful mathematics

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