Dr. BENOIT B MANDELBROT, whilst an IBM Fellow and Visiting Professor at Harvard University, coined the word ‘FRACTAL’ from the Latin ‘FRACTUS’ = ‘broken’ as the name for the Geometry dealing with lines having FRACTIONAL DIMENSIONS: ‘Fractal Geometry’.

Fractal Geometry may be used to create pictures of natural objects.

A fractional dimension is used to describe how ‘wiggly’ a line is. The edge of a leaf or a tree has a fractal dimension of about 1.3, because it is quite jagged or ‘wiggly’. A smooth line has a dimension of 1.

Fractal Geometry contrasts with Euclidean Geometry which is used to create pictures having smooth edges, such as circles, cones and rectangles.

Statement of Artistic Beliefs

To me, the essence of art is that the viewer should feel emotionally

moved - and ideally happier - after looking at any work of art.

Impressionist paintings usually fulfil that criterion - which explains

their ceaseless popularity. Just as Impressionism added to

our culture in the 19th century, so Fractal Art is an addition to our 20th Century culture. Both were

enabled by new technology.

Fractal Art could never have happened before in the history of

mankind.

THE FAMOUS MANDELBROT SET is the black area of the picture shown. It was first seen on a computer in 1980 by Dr. BENOIT MANDELBROT, whilst an IBM Fellow and Visiting Professor at Harvard University.

Dr Mandelbrot's famous set is created byiterating the complex quadratic equation

Znew = Zold2 + C, where C & Z are

complex numbers having two components x and iy, until the magnitude

of Z is 2. The number of iterations is used to

colour of the pixel C points at.

E V E N S O M E Q U A D R A T I C E Q U A T I O N S H A V E C O M P L E X R O O T S ! T h e s t a n d a r d f o r m f o r a q u a d r a t i c e q u a t i o n i s 2axy + cbx T h e s o l u t i o n i s g i v e n b y t h e f o r m u l a w e r e m e m b e r f r o m s c h o o l :

a

acbbx

2

42 w h i c h i s t h e v a l u e

o f x w h i c h m a k e s y e q u a l t o 0 .

I f a = b = c = 1 , t h e n

2

411 2 x s o

2

31 x ,

w h i c h i s u s u a l l y w r i t t e n 2

31 ix

W e l l l o o k h e r e ! T h e s o l u t i o n r e q u i r e s t h e s q u a r e r o o t o f a n e g a t i v e n u m b e r ! ! ! ! ! ! F o r y o u r h o m e w o r k p u t t h i s b a c k i n t h e s t a n d a r d f o r m a n d E u r e k a ! y o u w i l l f i n d t h a t y = 0 .

Euler’s Formula ejz = cosz + jsinz(acknowledgements to Wikepedia)

The next slide demonstrates that the successive values of Z in the the equation

Z n+1 = Zn2 + C can change utterly

for small changes in the value of C.

This property defines this equation as ‘Chaotic’.

You may recall the ‘Butterfly’s wing effect’.

If we put C = 0.3 + i0.12, the values of Z versus n do not increase rapidly.

If we put C = 0.4 + i0.12 then the values of Z do increase rapidly.

Z n+1 = Zn2 + C

Note the change of scale of Zn between the left and right hand diagrams

If the Mandelbrot Set were electrically charged, then the lines outside are also

the equipotential lines!

The Creation of a Fractal Art Picture

Run the program to iterate the equation to colour each pixel.

Often about 80 million iterations (calculations) are required

for a complete picture, which may take a few minutes, or even a few hours

on a Personal Computer.

"Zoom in" to magnify interesting areas of the initial picture (the "opening chorus") by a Million Million Times, or more.

This magnifies the screen to give a picture a few million kilometres wide, which can correspond to many billion million square kilometres of "image space".

Search for beautiful pictures in this

enormous image space,and choose colours until an emotional

connection with the image is established. (That is, until you adore it!)

Sob with a curious mixture of pain and joy when it's sold.

I use a DOS program called FRACTINT to create FRACTAL ART.

A WINDOWS program calledUltra Fractal Five can googled &

downloaded.

If anyone would like a copy of my leaflet and the DOS FRACTINT program

would they please email me:davidjwalter@btinternet.com

BOOKS Search for Fractals, or Chaos in Amazon books: ‘Chaos’ by James Gleick ‘Does God play Dice?’ by Ian Stewart. If you search Wikepedia for ‘FRACTALS’ and MANDELBROT SET you will be fascinated.

The Colours of Infinity: The Beauty and Power of Fractals - Paperback (1 Mar 2004) by Ian Stewart, Prof. Michael Barnsley, Nigel Lesmoir-Gordon, Benoit B. Mandelbrot, et al. 5 used from £7.00 DVDs only even though it says a book!!!!!!!

This is the most fascinating beautiful DVD

showing the Mandelbrot set and it’s creator

PIXELS - AND HOW THEY ARE COLOURED.

A picture on a computer screen is made up of coloured dots - just like TV or

newspaper pictures. The dots are called PICTURE ELEMENTS, or PIXELS for

short. The computer stores a range of colours and refers to them by a number, such as

4 for red. The number of times the equation is iterated for a particular pixel is the number used to colour that pixel.

Many pictures have 1024 pixels across the top and 768 pixels down, totalling 786,432 pixels. Often 100 calculations are done to colour one pixel, so a complete picture may need about 80 million calculations, which can take seconds or hours, depending on the equation and the speed of the computer.

For example, if C ‘points at’ the pixel at x = -1, y = 0, (which is where the little white arrow on the Mandelbrot Set is pointing) then successive values of Znew are -1, 0, -1, 0, -1, 0, -1, 0,... and so on for ever, or until stopped after a certain number of iterations. This pixel (-1, 0) belongs to the Mandelbrot Set, which is coloured black, and defined as the set of pixels whose successive values of Znew do NOT increase all the time.

For other values of C such as C = 2, the successive values of Znew are 2, 6, 38, 1446,... , and Znew gets very big very quickly. These pixels form the coloured bands around the outside of the Mandelbrot set.

I = VjwC I = V/jwL = -jV/wL

For each pixel, starting at the top left of the screen (where C is x = -2.5, y = 1.5), the computer sets Zold = 0 and iterates the equation Znew = Zold

2 + C to calculate a series of values for Znew.

Put another way, Z n+1 = Zn2 + C,

for n = 0 to 10 say.

Print from CorelDraw to a fabulously good printer -

Giclée of Course!Rush to the Framer and spend a delightful 2 hours selecting

a suitable frame and mounting.

Sob with a curious mixture of pain and joy

when it's sold.

SOME DETAILS ABOUT COMPUTING THE

MANDELBROT SET

The equation for Dr Mandelbrot's famous set is the quadratic

Znew = Zold2 + C, where C is a complex

number having two components x and iy, which are the co-ordinates of a pixel on the computer screen - just like a graph.