THE MAGIC LAND OF JULIA-SETS

Marcus Herold • Werner Maier

Berlin University of Technology

'What is the use of a book'

thought Alice,

'without pictures'

(Lewis Carrol, Alice in Wonderland)

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Chapter 1 DIMENSIONS OFGEOMETRIC FIGURES

On the one hand it is known that the geometric figure "sphere" has a surface which is

two-dimensional, on the other hand it has a three-dimensional expansion. The term

dimension has many (equivocal) denotations in the modern mathematic.

Particularly fractals are marked. Depending on the point of view the dimensions have

different, also fractional numbers. As an introduction we present the three definitions of

the term "dimension" one has to know to understand the "Magic Land of Julia-sets".

1.1 The Euclidean Dimension (E)

A geometric figure is called E-dimenslonal , if E coordinates

are sufficient to describe all its points.

The Euclidean definition means informal and descriptiv:

1. a pOint has no length;

2. a line has no width;

3. the both ends of a line are points;

4. a surface has only length and width;

5. lines are the border of a surface;

6. a body has length, width and heigth;

7. surfaces are the extremities of a body.

1.2 The Topological Dimension Topological dimensions can be derived recurrently from each other.

A set of points has the topological dimension t, when it is possi­

ble to subdivide it by sets of points having the topological dimen­

sion t-1.

The initial point is that geometric figures are being also sets of points, too. An incoherent set of points always has the dimension O. The dimension 1 is given if it is possible to subdivide· a coherent set of points by

eliminating an endless number of pOints. The sets of points which can be divided by CUNes have the dimension 2 [1].

Analogously the sets of points have the dimension 3 when they can be subdivided

by surfaces.

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1.3 Hausdorff-Besicovitch-DimenSion (D) Assuming from the topological dimension Dr 1 for curves, 2 for surfaces and 3 for bodies

a standard measure fitting to the dimensions is defined:

a distance of length 1 for curves;

a square of the sidelength 1 for surface;

a hexadron of the edgelength 1 for bodies.

To measure little, complex figures exactly the sides of the standard measure are divided

into parts of equal size.

This leads to the sidelength r:

1 r

b

The other way around: if one knows r one needs in order to complete

the standard distance

N = b

the standard surface

N = b 2

the standard hexadron

N

or generally

= respectively

D In N

-D r

1 In-­

r

= In N

In r

(1. 1)

(1. 2)

(1. 3)

(1. 4)

(1. 5)

(1. 6)

D is named Hausdorff-Besicovitch-Dimension or dimension of similarity. There is a cor­

relation between the three mentioned terms of dimension which is expressed in the for-

mula:

(1. 7)

1.4 Fractals

A fractal is a set which Hausdorff-Besicovitch-Dimension is really greater than

the topological dimension

D > DT (1.8)

D doesn't have to be a fractional number[2].

540

~'. 1;­,

Year

1918

1963

1975

1977

1980

1980

1982

1984

Event

Chapter 2 HISTORY

Fractals occured for the first time in a work of Fatow and Julia who

don't give any illustrations.

The first step to a development of a systematical fractal geometry including its graphic aspects were taken at the IBM T. J. Watson Research Center by 8. 8. Mandelbrot.

Mandelbrot coined the term "fractal" in order to be able to give a title to this first essay on this topic.

Mandelbrot published the book: "Fractals: Form, Chance and Dimension".

Mandelbrot discovered the set which now bears his name.

Star Trek II: The Wrath of Khan. Several computer-generated sequences of this film involve fractal landscapes and have also become classics in the core computer graphics community.

First color illustrations published in "Fractal Geometry of Nature" by 8. B. Mandelbrot.

First exhibition of pictures of Julia- and Mandelbrot-sets in Germany (Bremen). The pictures are produced by Peitgen, Richter and

Saupe.

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Chapter 3 FRACTALS

Mandelbrot's fractal geometry provides both a description and a mathematical model for

many of the seemingly complex forms found in nature. You can use fractals to describe

the fragmentary aspects of nature. Random fractals can simulate landscapes and

objects in nature, such as trees and mountains.

Mathematical modelers can use them to simulate shoreline decay and its effect on

fisheries.

Biochemists can use fractals to characterize the irregularity of protein surfaces and

its influences on molecular interactions.

You can use them to discover any order where no order was previously believed to exist.

Even processes of ontogeny - the course of an individual organism's development - that

have proved elusive might be explorable using fractal geometry. With fractal geometry,

the quest for scientific understanding and realistic computer graphic imagery can return

to the everyday natural world.

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[3]

Figure 4. 1:

[4]

Chapter 4 JULIA-SETS

Typical Julia-Sets

The most simple formula of a Mandelbrot's process is

x n+1 = f(x) 2 = x + c n

xeR,ceC

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(4.1)

Figure 4.2: Mandelbrot's process

[5] The most simple version of that formula is to put c = o. Depending on xO' there are three possibilities for the sequence

The sequence Xo H ~ I~ x~ H.xg ... approaches zero; zero is an' attractor.

(All pOints distant from this attractor less than 1 are drawn into it); the sequence tends towards infinity. Infinity is also an attractor for this process.

(All points distant from zero more than 1 are drawn into infinity);

the pOints sited exactly at a distance of 1 from zero the sequence lies on the

boundary between the two domains of attraction.

When c is non-zero, i. e.

c = -0.12375 +0.56508i ,

the sequence here, too, has the choice between the three possibilities, but the inner

attractor is no longer zero and the boundary is no longer smooth.

One of the peculiar things about the boundary is its self-similarity. If you look at anyone

of the corners or bay, you notice that the same shape is found at another place in

another size. Boundaries of this kind have been known in mathematics as Julia-sets

Which kind of Julia set a given choice of c implies is depending on a principle called

Mandelbrot set. Aided by Figure 4.3 one can tell whether a given complex number c

belongs to the black structure M or it does not.

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~ ~

~ i:

~ ~ ~ t, ;~ ~' t~ ,. ~ ;: ~, ~~

~ Chapter 5 ~ HARD-AND SOFTWARE REPORT ~

f 5.1 Hardware To construct the Julia-sets there are available on mainframe:

IBM 4381-2 mainframe;

graphic terminal (IBM 3179-G2Y)

with the colors (blue. cyan. green, red. pink. white and yellow). The resolution of the monitor is 383 * 719 points.

hardcopy-printer (IBM 3852-2)

the format is DIN A5;

plotter (HP7550)

the formats are DIN A4 and DIN A3. Up to 8 colors are available (black. blue. gold. greim. orange. pink. red and yellow).

on Personal Computer (PC):

PC AT 386 20 MHz

TAXAN Multisync Monitor incl. VGA-graphic card (with up to 256 colors). The res­olution of the monitor is 640*480 points.

5.2 Software The authors use the SAS[7j-Base and SAS/GRAPH products (Vers.5.18) and PC-SAS (Vers. 6.03)

545

i I

5.3 SAS-Program

DATA julia

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * DEFAULTS - JULIA * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

max x max-y max-n max:jul

719 383

50 1000000

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * COORDINATES . * * * * * * * * * * * * * * * * * * '* * * * .* * * * * * * *

X LOW X-HIGH Y-LOW Y-HIGH

,..1.5000000000 1.5000000000

-0.9120603015 0.9120603015

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * FIXPOINt * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

* * * * * * * * *

DO DO

CREAL -0.7241001565 ~IMAG 0.3467336683

'* * * * * * * * * * * * * * * * * * * .* ALGORITHM

* * *' * * * * * * * * * * * * * * * * *

:i.- 0 TO ~max_x'-ll J 0 TO max3-1.

x ~ !i+1ll max x l

* 100 j+1 I ma~y Y * 100

dx f! !x high - x lowl ~ 100l * dy y:high x-low 100 * x julia x low + dx Y:Julia y-low + dy x 2 x julia ** 2 y-2 Y:Julia ** 2 n 0 DO WHILE «n LT max nl

AND «x_2 + y_2) LT max_jul) )

END

xx y jUl:j.a x-Jul~a x-2 y:2 n + 1

modulo = MOD IF modulo IF modulo IF modulo IF modulo IF modulo IF modulo IF modulo OUTPUT

x 2 - Y 2 + creal 2-* x JUlia *-y julia + xx - -x julia ** 2 Y:Julia ** 2

(N,7) = 0 THEN

1 THEN 2 THEN 3 THEN 4 THEN 5 THEN 6 THEN

color color color color color color color

'green' 'red' 'yellow' 'green' 'red' 'yellow' 'cyan"

END END

* * * * * * *

* * * *

~l

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * KEEP SAS-DATA-SET * * * * * * * * * ~ * * * * * * * * * * * * * * * * * * * *

KEEP XSYS YSYS X Y FUNCTION COLOR

Figure 4.3: Typical Mandelbrot-Set

The corresponding Julia sets of the process

x I~ i+c are totally different.

Regarding any region resulting from a Julia set process 2

x I~ x + c one can distinguish four shapes of characteristic structures:

if c is in the interior of the main body of the Mandelbrot set, a fractally deformed

circle surrounds one attractive fixed point;

if c is in the interior of one of the buds, then the Julia set consists of infinitely many

fractally deformed circles which surround the points of a periodic attractor and their

pre-images;

if c is the germination point of a bud, then you have a parabolic case: the bound­ary has tendrils that reach up to marginally stable attractor;

if c is any other boundary point of the cardioid or a bud[6], then the result is a

Siegel disk.

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Figure 6.1:

[8]

Chapter 6 PLOTS

Example 1 - Julia-Set

·548

Figure 6.2: Example 2 - Julia-Set

[9]

549

Figure 6.3: Example 3 - Julia-Set

[10]

550

~.'

Figure 6.4: Example 4 - Julia-Set

[11 ]

551

.\

!

BIBLIOGRAPHY

DEWD88a DEWDNEY, A. K. (1988):

Computer-Kurzwei/; Mit einem Computer-Mikroskop untersuchen wir ein

Objekt von faszinierender Struktur in der Ebene komplexer Zahlen. Spek­

trum dar Wissenschaft, Oktober 1985.

DEWD88b DEWDNEY, A. K. (1988):

Computer-Kurzwei/; Augenweide und unauslotbares Geheimnis: . die

Mandelbrot-Menge und eine Schar ihrer Cousinen namens Julia, die eine

BrOcke zum Chaos schlagen; Spektrum der Wissenschaft, Februar 1988.

JULlAt8 JULIA, G .. (1918):

Memoire sur /'iteration des functionsrationefles; Journal de mathematiques

pures at appUquees 4, p. 47-245.

MAND87 MANDELBROT, B. B. (1987):

Die fraktale Geometrie der Natur; BirkMuser Verlag,. Basel.

PEITG86 PEITGEN, H. 0., RICHTER, P. H. (t986):

The Beauty of Fractals; Springer Verlag, Berlin.

PEITG88 PEITGEN, H. 0., SAUPE, D. (1988):

The Science of Fractal Images; Springer Verlag, Berlin.

SHROE87 SCHROEDER, P. B. (1987):

Plotting the Mandelbrot set; BYTE, December.

URI88 URIAN, R. (1988):

Wir bauen uns ein Monster; c't, Heft 5.

WORT8? McWORTER, W. A. Jr., TAZELAAR, J. M. (1987):

Creating Fractals; BYTE, August.

552

'G, r

"

I >~. --... - .. - '. -~- -~ -- --.

[1] so the surface of the solid has the topological dimension 2.

[2] i. e. the tetrahedron of Sierpinsky has the dimension D = 2 and Dr"" 1

[3] named by the french mathematician Gaston Julia (1893-1978)

[4] [PEITG86], p. 14.

[5] [PEITG86], p. 5.

[6] there are some technical conditions regarding the irrationality of the point

[7] SAS is registered trademark of SAS Institute Inc., Cary NC, USA.

[8] The coordinates for example 1· are:

-1.50000 < x < + 1.50000

-0.91206 < y < +0.91206

creal = -0.7241001565 Cimag = +0.3467336683

[9] The coordinates for example 2 are:

-0.03521 < x < +0.20422

-0.58040 < Y < -0.21859

creal =-0.610172144 Cimag = -0.3969849246

[10] The coordinates for example 3 are:

-0.28522 < x < -0.20411

+0.52527 < y < +0.59600

Creal = +0.5 Cimag = -0.3

[11] The coordinates for example 4 are:

-0.96009 < x <: +0.96483

-1.25879 < Y < + 1.25879

creal =+0.4414710485 Cimag = +0.2261306533

For current information and how to obtain the program for generating JULIA-graphs,

write to:

or:

Dip!. Inform. Marcus Herold & Dip!. Inform. Werner Maier

c/o Technische Universitat Berlin

Secr. FR 6-4

Franklinstrasse 28/29

1000 Berlin 10

EARN-ID: HEROLD@DBOTUIl1 phone: (+49l 30-31425817 MAIER@DBOTUIl1 (+49 30-31423698

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