ISSN 0964-5640


And Innovative Computer Applications

Editorial and Announcements 2

Letters 4

Fractal Cosmos: The Art of Mathematical Design Book Review Dr Ian Entwistle 6

Fractal DNA Imager Software Review 7

Apollonian Circles Marius-F. Danca 8

The Pattern Book: Fractals, Art and Nature Book Review Cecil J. Freeman 13

Newton.frm Jules Verschueren 14

The Well Tempered Fractal Robert Greenhouse 20

More on Gravitational Lenses, Please Yvan Bozzonetti 21

A Fractal Window on Mars Yvan Bozzonetti 22

Images From a Three Point Attractor Marius-F. Danca 23

Magic not Squares John Sharp 28

Fractal Carpets Gabriel Landini 29

Fractal 97 Call for Papers Miroslav M. Novak 30

More Amiga Fractals Malcolm Lichtenstein 31

Spidermorph (Image only) Dr Ian Entwistle 32

Fractal Report is published by Reeves Telecommunications Laboratories,

West Towan House, Porthtowan, Truro, Cornwall TR4 8AX, United Kingdom.

Internet: [email protected], CompuServe: 100431,3127

Volume 7 no 39 First published September 1995. ISSN 0964-5640.

Editorial and Announcements


This issue comes at a time when it is often difficult to get articles. Many thanks to those who contributed. We really have very little held over for next time, so please get those thinking caps on. Remember that we accept articles on innovative computer experimentation other than fractals now, so if you have some ideas on alternative physics or sone such subject that can be expressed in a simple BASIC computer listing then please let me see it.

Many of our readers like an interesting computer project that can be experimented on in the space of a couple of hours or so.


Mike Kirk, from Geneva, sent in a photocopy of an advertisement for Fractal MusicLab for the Amiga. It costs a shade under $100 from a company calling itself Digital Expressions Research W6400 Firelane 8 Menasha WI 54952 USA.

It generates sounds from IFS patterns. These sounds can then be edited into music using a graphical editor. A review in Amiga World said that you need to be well versed in both graphics and music to get the best from the program. The reviewer wasn't very happy with the tutorials and thought the program had "design peculiarities". He concluded on the positive note that it is an interesting curiosity for the explorer of new musical frontiers.

Maybe some of our Amiga readers have this program and would like to give their two cents' worth as to whether its valuation of $100 is accurate!

News of Contemporaries

Again we have received a few exchange copies from those publishers who are holding their own against the vast demands of reading time made by the Internet. - apologies to any left out.

Roger Bagula continues in the same vein with the only small circulation newsletter reproduced in colour that I know of. The issue dated August contained an editorial about rejecting work and then publishing it as your own - the reason Roger gives for running his journal. Not that he does it, I may hasten to add, but the does not want it to happen to him.

[Fractal Translight Newsletter $20/yr ($50 overseas) from R.L. Bagula 11759, Waterhill Road Lakeside CA 92949 USA]

Unfortunately it is all too easy for this "virtual conspiracy" to appear. Many people reach the same idea concurrently, because everyone has access to the same facts. the Internet newsgroups may even increase this phenomena. Often those who present their idea in the best way get published, and this takes time and effort. Of course there may be plenty of people who have reached the same conclusion earlier, but described them in an obscure manner scruffily presented, and were rejected on those grounds.

On the Internet, of course, everyone is a self publisher. A point may be raised in a newsgroup and four or five people may give the same answer. You can call up the message headers and see who was first if such matters interest you.

Personally I am not too bothered. I often suggest apparently eccentric ideas only to see them happen. The latest is the radio-internet, which I predicted a year or so ago based on seeing an IEE article on radio LANs. The radio local area network is designed to avoid wiring problems in buildings and on campuses. However if most people in the world were on a radio internet telephone connection would be obviated and the last vestige of expense and government control would go. Apple Computers have now "stolen" (by the thinking of conspiracy theorists) this idea and propose the first 300 MHz of the 5GHz radio band be turned over to the public for a free-for-all global radio network. It would provide a 24 Mbps network, a great improvement over 14.4kBs. According to PC Magazine (Sept95) the equipment would cost a few hundred dollars and provided enough people were on the net messages could propagate all over the USA with no charging or controlling body. The "few hundred dollars" would be a one of payment with nothing more to pay - for ever! Whether it would come to the more regulated Europe is open to question, say the magazine, but once the technology exists it may be impossible to keep it out. (as with CB radio.)

The other of my eccentric ideas is virtual travel (VT). (VR+Internet=VT, work it out!) I am sure it will come, and soon.

The September issue of The Fractal Translight Newsletter contains a questionnaire that is responded to is rewarded by an 8 x 10 colour print of a fractal image selected from a choice of 8. It is not that easy to pick out the "best" one. However the other questions have to be answered and some of these require a knowledge of the previous issues of the newsletter... The answers will also be used to create limited edition signed sets of the most popular images, which will be offered for sale together with certificates of authenticity of the limited issue. 8" x 10" sizes would cost $15 with various ranges up to 36" x 60" for $1k. An interesting marketing concept, and I wish him the best of luck!

Another topic of interest in this issue was a discussion about the possibility of an artificial language based on fractals or chaos theory.

Fractalia the Romanian language fractal newsletter reached its 13th issue in June 1995. A couple of articles from its editor Marius-F. Danca appear in this issue of Fractal Report. It contained articles on Lorenz, Fractal Dimensions, Julia Sets, and others. As it is in a foreign language it is difficult to be sure what it is all about, but I noticed an intriguing picture of what looked like a computer generated 3D representation of a root structure in one of the articles. There aren't as many listings as in some western newsletters, but the layout of the whole thing is very professional and I imagine that the content is more academic than many journals here that are aimed more at the computer hobbyist.

[Fractalia C.P. 524 Cluj Romania]

REC Volume 9 nos 7 & 8 is a double issue that starts with Dr Ecker's news that he has bought a 75MHz Pentium, as like the rest of us he is no longer able to cope with the lack of speed of earlier models. There is quite a lot of further discussion of bugs in various versions of BASIC, and whether there are any more bugs in the Pentium. Apparently the famous bug was due to a lack of five entries in a lookup table of 4,000 entries! He says that he is unlikely to upgrade to the P6 chip for a while ... (Which is just as well as it can only run better with code written specifically for it.) Plus of curse, the usual diet of mathemagical black holes, puzzles etc. ... and a little editorial comment on the use of calculators by children to evaluate computations such as ten times three divided by two. This is a subject Dr Ecker is qualified to pontificate on, as he is both an experienced "player" with such equipment and a teacher.

REC volume 10 no 1 has just arrived and has a programming challenge to write a program to generate two identical images shown on the front cover. I seem to recall publishing a program that generates those images in the early days of Fractal Report. Inside there are the usual mix of mathematical puzzles and graphics, the most notable of which is a BASIC program that produces an excellent picture of bubbles in very few lines. If you need a prime number generator, you may like various speed-up options that were discussed in another article.

[REC 909 Violet Terrace Clarks Summit PA 18411 USA, $36 pa worldwide, $28 Canada, $27 USA.]


from Mr John Reece

Dear John,

First of all, thanks for the mention of frmtutor.txt in your last issue of Fractal Report. I've got this file now and it is excellent. You may find the following newsworthy.

The Amiga port of Fractint has been updated. The newer version, Fractint26 is available on Aminet/gfx/frac. Although it runs more slowly than the earlier version, it is much more stable and easy to use. It does require the use of MUI. If you have a co-processor, etc, the author, Terje Pedersen, will supply a customised version for what I consider to be a very reasonable shareware fee. This of course, runs much faster. Warning: I don't think the e-mail address quoted in the program is any longer valid. The other thing perhaps worth mentioning is that there is a fractals conference on - one of the topics is Fractint, while another has got some interesting images. Fractal Report was mentioned there the other day. Surprise was expressed that Fractal Report was still going, your address was given, etc. I think that there's about 100 members of this conference, so you might be getting one or two new subscribers. [we got one enquiry!]

Looking forward to the next issue, John Reece

An exchange on chaos and theory of government:

This Internet exchange was badly documented, so I have lost the name of the correspondent:

In <[email protected]> John de Rivaz <[email protected]> writes:

Anyone who has read David Ruelle's Chance and Chaos, particularly the chapter on economics, will realise government has a great deal to do with fractals. (ISBN 0-691-02100-7, Princetown Science Library.)

Chaos theory tells us that it is very difficult to predict and control complex systems, and an economy is probably the most complex system that humans try and control.

Harry Browne's presidential campaign is headed by the slogan "Government does not work", and although he does not quote Ruelle, his arguments do suggest fractals and chaos by what they say. I have a couple of files on this subject available by email. Dark Ages and The Breakdown of Government.

the reply:

Chaos theory says that many systems (including some simple ones) can not be predicted for long. The problem of control is a different (and frequently simpler) problem from prediction.

First, chaos theory sometimes applies to even simple systems (the Lorenz equations only use three variables). Central planning breaks down for much larger systems. It works on the level of a family, small businesses, and even some large businesses. (It fails on a national level.)

Second, if a system doubles differences over a time T, it will -- in the absence of control -- become unpredictable in a few multiples of T. In the presence of control, the controlling authority can act in that time T to keep matters stable.

Third, there are many artificial systems which stabilize otherwise unpredictable phenomena. An uncontrolled fire is extremely unpredictable, but controlling a fire is merely difficult. It is almost impossible to predict how a natural protein will fold, but it is theoretically possible to artificially design a protein which will fold in a given pattern.

The above arguments can also be used against anyone who uses chaos theory to defend environmentalism.

I am not claiming that governments work; just that another explanation is needed [as to why they don't].

Book Review: Fractal Cosmos

The Art of Mathematical Design

By Jeff Berkowitz Published by Amber Lotus California (1994) ISBN 1-56937-064-8

by Dr I D Entwistle

A number of subscribers to Fractal Report may be familiar with the work of Jeff Berkowitz if they have seen copies of " Fractal Cosmos " calendars for 1992 and 1994. Others may have seen the colourful catalogues of Lifesmith Classic Fractals. I have a copy of the Spring 1990 collection. Fractal Cosmos the book brings together most of these published images amid hundreds of others out of the authors own collection of more than 250,000 fractal images. The sheer range of functions that is covered by the illustrations suggests that Fractal Report authors may have a way to go before running out of inspiration for potential articles. Readers interested in strange attractors, affine transitions or Lyapunov maps and fractals other than Mandelbrots and Julia sets will be disappointed by their absence. This book is essentially about broadening the audience for the authors many and varied high quality images and the data from which they are generated. It is therefore appropriate to comment primarily on the images and the books statistics and then briefly on the relatively small amount of textual content.

This book has merit just as a collection of images for your coffee table. It is a large format soft backed glossy paged volume (9"*12"). The image content occupies around 136 out of 207 pages with image sizes from six per page to a substantial number of full page prints. Apart from a few pages of monotone illustrations all the images appear to be at least 16 bit . The smaller images range from easily recognised Julia and Mandelbrots sets from z2+c iteration to a great many produced by iterative mapping of more unusual functions. For example taking a page at random, mappings on page 6-15 include f(z)=(z5+c)/z3+z2+z+1 and f(z)=z2sin(x)+cyz+z2cos(x)+czsin(y)+c. The image pages are divided under various chapter headings such as highly magnified views , high integer values of n in zn. An example of the latter includes six views of a study of f(z)=z24+c. I have read or perused over 80 books related to the study of fractals and none of their illustrations compare for variety , quality or quantity with Fractal Cosmos. Even Peitgens "The Beauty of Fractals" is outclassed. For a cover price of $29.95 this book surely represents a good buy just to admire the images !

The second main section of the book contains the supplemental data utilised to derive the images and covers 47 pages. This data typically includes the set type, the equation, creal and cimag, screen parameters, magnification and iterations. A bibliography and a list of important complex equations complement an interesting index of the equations used for the images . Among the 77 equations are such as f(z)=z12-z11-z10+c.

Turning to the minor sections of the book namely text; there is a comprehensive historical introduction to fractals and a number of one page discussions which preface each of the picture chapters. For chapter 5 which contains 37 pages of 3D compositions most of which are full page derived from large files (12-60MB) the comment on their origin is over brief. Nice to look at but how were they generated? Although there is a comprehensive discussion on generating Julia sets there are no algorithms or code listings in the book.

My copy of Fractal Cosmos came from Media Magic and is signed by the author!!

Review: Fractal DNA Imager


Fractal DNA Imager is a windows program that produces string like images of DNA in the manner familiar to those who have read Dr Pickover on the subject. It is well set out with two groups of controls, one for inputting the DNA sequence and another for imaging it, setting different scales. The program also has a feature whereby markers in the sequence can be set.

It uses icons and not text buttons which means you have to read the help file if you just want to play, but it is quite short and you can start using the program within a couple of minutes or so.

This is from the help file:

DNA sequence information is usually represented and communicated via the traditional letter-based code of A, T, G and C. Although adequate for many aspects of DNA sequence analysis, this coding is less than ideal for identifying and characterizing specific or unique patterns in the ordering of the nucleotides. In response to this problem several graphical-based methods have been developed (see Selected Papers for several examples).

Fractal DNA Imager 1.0 is a Windows utility that translates letter-based DNA sequences (i.e. ATGC) into graphical images that facilitate the detection and characterization of nucleotide patterns. The algorithm used is based on a method presented previously by M. A. Gates (1986). With this approach, each of the four nucleotide types is assigned a vector direction on a 2-dimensional grid. As the DNA sequence is read one nucleotide at a time, the vectors are connected and the final result is a sequence-specific image.

Obtaining Fractal DNA Imager

Fractal DNA Imager 1.0 is available as shareware by ftp from, or your can register straight away for $20 and ask for a disk to be mailed. Say you saw this in Fractal Report.

Make checks payable to Jason H. Moore.

Jason H. Moore,

Software Visions,

615 Hidden Valley Dr., #211,

Ann Arbor, MI 48104


Apollonian Circles

by Marius-F. Danca

P.O. Box 524 Cluj Romania

This nonlinear and nearly self-similar fractal is called "Apollonius's Circles" (Greek mathematician of Perga 262-180 B.C.). B. Mandelbrot called it "Leibniz packing" of circle because Leibniz described it so: "Imagine a circle; inscribe within in three other circles other circle congruent to each other and of maximum radius; proceed similarly within each of these circles and within each internal between them, and image that the process continues to infinity ..."1. Because the autosimilarity and the extreme irregularity, this ensemble is a fractal.

Other details about this fractal can be founded in references [1] and [2].

It's mathematical construction is possible through many modes: by analytical geometry, by geometrical construction using a compass and a ruler, or by geometrical transformation of inversion. Using this last method, Mandelbrot called this class of fractals "fractals of autoinversion"2. Apollonius's Circles are nonlinear fractals because of the transformation of inversion which generate the nonlinearity. In the same way the method of analytical geometry, that we shall explore, uses the circles equation that is nonlinear. We had chosen this last method because it is very easy to followed even though the solution is complicated.

So let it be Ci(xi,yi,ri) i=1,2,3 three circles (tangent or not). We must find the C0(x0,y0,r0) circle which is tangent to the three given circles (fig.1). To solve this problem we start from the equations of the tangent C0 circle with the three given circles, easily found by using the expression of Euclidian distances between two points:

(xi-x0)2+(yi-y0)2=(ri+r0)2, i=1,2,3

which is one nonlinear system with x0,y0,r0 unknown quantities. The solution is:




where the notes can be taken from the code.

There are two ways to simulate the fractal. The first, the easiest, uses recursion.

The recursion algorithm

read C1(x1,y1,r1), C2(x2,y2,r2), C3(x3,y3,r3)


where the procedure calculus is:

procedure calculus(x1,y1,r1,x2,y2,r2,x3,y3,r3)





draw C0

if r0>6 (pixels) then




This algorithm is written in Turbo Pascal.

The second way, not recursive but more difficult is more efficient. While the recursivity goes in depth until the stop condition (r0>6) for example (C1,C2,C0)--> C0',

(C1,C2,C0')--> C0", (C1,C2,C0")--> C0"' etc (fig.2)

the second method goes circular surrounding the last circle with three new circles (fig.3).

Even the first way is very easy to program in Turbo Pascal for example, there are situations when this method failed (fig.4,5) That happen when the circles C1,C2,C3 aren't tangents (when the stop condition isn't verified). Generally the second method seems to be able to manage this situation.

Sometimes it happen that one infinite cycle appears when the recursive algorithm oscillates between C0 and C0' (fig.5) It seems that situation can be solved by dealing with the sign +/- above the radical of r0 expression.

Of course both methods failed when the circles C1,C2,C3 are colinear.

The last image was made using the unit roots located on the unit circle boundary.


1. B. Mandelbrot Les Objets Fractals Flammarion 1975 Paris,

2. B. Mandelbrot The Fractal Geometry of Nature, Freeman 1983, New York

book review

The Pattern Book: Fractals, Art and


Editor, Clifford A. Pickover

(World Scientific Publishing, 1995. Hardback, 427 pp. $51.00)

by Cecil J. Freeman

This long-awaited book, contributions for which were first invited in 1990, has at last appeared. It is an ambitious project, containing some 160 contributions by as many different authors, each with one or more illustrations and an explanation of the origin of the images.

In an introduction Dr. Pickover states that he had scoured the four corners of the earth in a quest for unusual people and their fascinating patterns. "Some of the patterns," he remarks, "are ultramodern, while others are centuries old." He points out that the line between science and art is a fuzzy one, the two being fraternal philosophies formalized by ancient Greeks, and that today computer graphics is one method through which scientists and artists reunite these philosophies by providing scientific ways to represent natural and artistic objects.

The book covers three different categories of patterns, those derived from nature, those generated mathematically and those drawn by hand.

The first Part of the book, Representing Nature, occupying some 40 pages, is devoted to patterns of natural origin, ranging from a cross-section of part of an apple tree to wave patterns occurring during a chemical reaction. Some of these, surprisingly, look as though they had been mathematically generated, illustrating the highly structured character of natural processes.

The third Part of the book, "Human Art", occupies some 36 pages, and contains some extremely interesting and striking hand-drawn images, derived from many diverse sources - Celtic, Islamic, Persian and Japanese for instance - as well as some modern examples - op art, art deco and others. These do not have the mathematical character of the images in the first Part; they have their own imaginative complexity and ingenuity.

The middle Part, Mathematics and Symmetry, is the major component of the book, occupying some 326 pages, and is the section likely to be of particular interest to readers of Fractal Report. Here we find an unbelievable variety of mathematically generated images. Naturally many of these are fractals - extensions of Mandelbrot and Julia sets, biomorphs and mappings of transcendental complex functions. Particularly noteworthy are the many contributions by Dr. I. D. Entwistle, whose name is well known to readers of Fractal Report. The images submitted by him show outstanding originality and artistry; of their class they are unique. Other contributors worthy of special mention here include Earl F. Glynn and Mieczyszlaw Szyszkowicz for their striking and original images.

Other mathematically generated images include unusual Lissajous figures, tilings, cellular automata and wallpaper-like designs.

Although many contributors provided pseudo-codes of the programs from which their images were derived, many referred simply to one or other commercial software packages that they had used. This is a disappointing feature of the book. Although this was perhaps unavoidable it would have been helpful if an indication could have been given of where the software was obtainable.

Each contribution is accompanied by a list of references to relevant literature and there is also an extensive general bibliography. An interesting observation by the editor is that in 1989 the world's scientific journals published about 1,200 articles with the words "chaos" or "Fractal(s)" in the title.

In addition a glossary of mathematical terms is given for the benefit of non-technical readers.

This beautifully illustrated and informative book can be highly recommended to readers of Fractal Report.


by Jules Verschueren, 23 Mar 1995.

Internet: [email protected]

All solutions are obtained with the Newton method:

Z(n+1) = z(n) - F[z(n)] / F'[z(n)],

with F' being first derivative of F; but some liberties were taken in search of the miniature Mandelbrots.

The Julia/Mandel space bar toggle does unfortunately not work (at least not up till v19.2) and therefore these 2 types were programmed separately. A formula title starting with 'M' means a Mandelbrot type set, 'J' is a Julia type set (most commonly used with the Newton algorithm).

The 'M' is not always present as it often fills the whole plane with 1 colour.

Important: In order to create the nicest Julia sets within Newton's method, write down the coordinates of the corresponding Mandelbrot point and insert it as the Julia constant in the corresponding Julia formula.

You will soon find out that familiar 'Mandel' structures give familiar Julia sets while non-familar ones can give completely unknown Julia sets...

1. The best known Newton formulas:

F(z) = z^p - 1 = 0

F(z) = z(z^p - 1) = z^p - z = 0

- for integer p roots are located at e^(2Pi*ik/p), k=1..p or : real root part at cos(2Pi*k/p) and imaginary part at sin(2Pi*k/p);

- for z(z^p-1) there is the additional superstable root for z = 0;

- there is XYAXIS symmetry whenever p is even;

- negative integers are OK but can faster and better be displayed with an algorithm like described in my article in Fractal Report 15;

- also non-integer values are valid and you can observe some bifurcation when you use p close to an even power, eg. 3.88... From here onwards to the even power, you get an area where a two-cycle is reached (the 'set' colour as no 'root' is attained) at the place where the new root develops (nothing of latter property is mentioned in FRACTINT).

2. One parameter families of polynomials of degree 3:

A. F[Z] = z^3 + (c-1)z - c = 0 (NewtonA) (Beauty of Fractals, pp. 100)

B. F[Z] = z^3 + (c-1)z + 1 = 0 (NewtonB) (Fractals Everywhere, pp.284)

C. F[Z] = z^3 - c*z^2 - z + c = (z-c)(z-1)(z+1) = 0 (NewtonC) (with c only having an imaginary part in JNewtonC, some "biological modelling" (eg. cell nuclei) is simulated: Barnsley, Fractals Everywhere, pp. 285-9).

(Other variants are clearly possible and can give nice results.)

NewtonClassA, B and C use the classical |z|<=4 bailout;

3. Parameter families of polynomials

F(z) = zp + (c-1)z - c = 0,

zp + (c-1)z + 1 = 0,

zp1 + (c-1)zp2 - c = 0,

(z-1)(z-c+L)(z+c+L) = 0.

With exception of this last (previously documented) formula I created the above formulas because they produce these marvellous miniature "Mandelbrot" and Julia sets within the "Newton" environment, something many people have been asking for in eg. Fractal Report. Every single point of this infinite number of small Mandelbrots produces a nice corresponding "Newton carried" Julia set.

4. Newton methods for e^z-1 and cos(z): easy and nice; try some color cycling once you found a good spot.

- Normally the |zold-z| bailout test is used to detect when a root is closely approached; with 'zold'= the previously calculated 'z'.

- Variables: n=nominator (or a power), d=denominator.

- In theory, MNewtonMandA, MNewtonN and MNewtonNM with n=3 (and m=1) should give identical fractals, but they sometimes don't: there must be something odd in the FRACTINT ^ calculations before v19.0 when the initial starting z=0 for the Mandelbrot type sets, but since this error is consistent, the corresponding Julia sets still seem to agree and we can still have fun. The solution to this problem is to use a very small starting perturbation for z (eg. 0.0001).

This error was supposed to be corrected in v19.0 but as before there are still problems with Mandel types starting with z=0: therefore I added P2 to the MNewtonN and P3 to MNewtonNM formulas. Using P3 is unfortunately only possible as of v19.0 and for earlier versions you can just set P3=.0001.

These same problems are still present in v19.2.

- All systems should be run in FLOATING POINT mode !

A typical Frabatch.bat file might look like this (try it!):

Fractint Type=Formula FormulaFile=Newton.Frm FormulaName=JNewtonMandA 
corners=-0.416/0.416/-0.312/0.312 Float=Yes Params=0.32172/1.67341/0/0

JNewton (XAXIS) {; F(z) = z^P1 - 1 = 0, the classical Newton sets
		 ; eg. P1=3             [no Mandelbrot version]
	zold=z, n=z^P1-1, d=P1*z^(P1-1),
	0.001 <= |zold-z|  }  ;|n| bailout also OK 

JNewtonEven (XYAXIS) {; F(z) = z^P1 - 1 = 0, the classical Newton sets
		      ; for even powers, eg. P1=4  [no Mandelbrot
	zold=z, n=z^P1-1, d=P1*z^(P1-1),
	0.001 <= |zold-z|  }  ;|n| bailout also OK 

JNewtonVar (XAXIS) {; F(z)=z^P1 - 1 =0 with n=Real(n), a nice variant
	z=pixel:                                ;[no Mandelbrot
	zold=z, n=Real(z^P1-1), d=P1*z^(P1-1),
	0.001 <= |zold-z|  }  ;|n| also OK  

JNewton_z (XAXIS) {; Newton method for z(z^(p1)-1)=0  [no Mandelbrot
		   ; use any +real value for p1,  
		   ; p2 is relaxation, normally p2=1
	zold=z, zp=z^p1,
	z = z-p2*z*(zp-1)/((p1+1)*zp-1),
	0.0001 <= |z-zold|  }

JNewtonA (XAXIS) {; z^3+(c-1)z-c=0 with c=p2, eg. p2=.5
		  ; p1=Julia constant
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3+p2)/(3*z2+p2-1) + p1,
	0.001 <= |zold-z|  }

MNewtonA (XAXIS) {; z^3+(c-1)z-c=0 with c=p2, eg. p2=.5
		  ; Newton method for Mandelbrot like set
	z=p1:     ; starting perturbation (normally 0)
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3+p2)/(3*z2+p2-1) + pixel,
	0.001 <= |z-zold|  }

JNewtonMandA  {; z^3+(c-1)z-c=0, p1=Julia constant,
	z=pixel, c=p1:
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3+c)/(3*z2+c-1),
	0.001 <= |zold-z|  }

MNewtonMandA (XAXIS) {; z^3+(c-1)z-c=0 (p1 is starting perturbation)
		      ; see BoF pp. 100
	z=p1, c=pixel:
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3+c)/(3*z2+c-1),
	0.001 <= |zold-z|  }

MNewtonMandAp (XAXIS) {; z^3+(c-1)z-c=0 (p1 is starting perturbation)
	z=p1, c=pixel:
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3+c)/(3*z2+c-1) + pixel,
	0.001 <= |zold-z|  }

JNewtonMandAp (XAXIS) {; z^3+(c-1)z-c=0, p1=Julia constant 
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3+c)/(3*z2+c-1) + p1,
	0.001 <= |zold-z|  }

JNewtonClassA {; z^3+(c-1)z-c=0 with c=p2, Newton with "Classical" bailout

		; p1=Julia constant, eg. c=p2=.5, p1=(.928676,.065712)
		; or (.7093765,.0012123): use high res and zoom...
	z2=sqr(z), z3=z2*z,
	z = (z3+z3+p2)/(3*z2+p2-1) + p1,
	|z| <= 4  }

MNewtonClassA (XAXIS) {; z^3+(c-1)z-c=0 with c=p2, p1 is start
		       ; Newton with "Classical" bailout, Mandelbrot
	z=p1:          ; try p2=.5
	z2=sqr(z), z3=z2*z,
	z = (z3+z3+p2)/(3*z2+p2-1) + pixel,
	|z| <= 4  }

JNewtonB  {; z^3+(c-1)z+1=0 with c=p2, p1=Julia constant 
	   ; eg. c=0 and p1=(-1.10172,0.194155) or (-0.58842, 0.106472)
	   ; or c=0.5, p1=(-0.247727,0.0029096)
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3-1)/(3*z2+p2-1) + p1,
	0.001 <= |zold-z|  }

MNewtonB (XAXIS) {; z^3+(c-1)z+1=0 with c=p2 
		  ; Newton method for Mandelbrot like set
	z=p1:     ; starting perturbation
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3-1)/(3*z2+p2-1) + pixel,
	0.0001 <= |zold-z|  }

MNewtonMandB (XAXIS) {; z^3+(c-1)z+1=0 (p1 is starting perturbation)
	z=p1, c=pixel:
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3-1)/(3*z2+c-1),
	0.001 <= |zold-z|  }

MNewtonMandBp (XAXIS) {; z^3+(c-1)z+1=0 (p1 is starting perturbation)
	z=p1, c=pixel:
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3-1)/(3*z2+c-1) + pixel,
	0.001 <= |zold-z|  }

JNewtonClassB  {; z^3+(c-1)z+1=0 with c=p2, Newton with "Classical"
		; p1=Julia constant, eg. c=0, p1=(-.813,.20172) 
		;      or (-.82827,.073353)   or (-.63429,.1130856)
	z2=sqr(z), z3=z2*z,
	z = (z3+z3-1)/(3*z2+p2-1) + p1,
	|z| <= 4  }

MNewtonClassB (XAXIS) {; z^3+(c-1)z+1=0 with c=p2, p1 is start
		       ; "Classical" bailout with Mandelbrot like set
	z2=sqr(z), z3=z2*z,
	z = (z3+z3-1)/(3*z2+p2-1) + pixel,
	|z| <= 4  }

JNewtonC  {; z^3-c*z^2-z+c = (z-c)(z-1)(z+1) = 0, with c=p2
	   ; p1=Julia const, (-.472764,.033645) or (.38177,.11479)
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3-p2*(z2+1))/(3*z2-p2*(z+z)-1) + p1,
	0.001 <= |zold-z|  }

MNewtonC (XAXIS) {; (z-c)(z-1)(z+1)=0, with c=p2
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3-p2*(z2+1))/(3*z2-p2*(z+z)-1) + pixel,
	0.001 <= |zold-z|  }

JNewtonMandC  {; f(z)=(z-ic)(z-1)(z+1), p1=Julia constant
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3-Imag(c)*(z2+1))/(3*z2-Imag(c)*(z+z)-1)+p1,
	0.001 <= |zold-z|  }

MNewtonMandCp (XAXIS) {; (z-ic)(z-1)(z+1)=0, p1 is start perturbation
	z=p1, c=pixel:
	zold=z, z2=sqr(z), z3=z2*z,
	z = (z3+z3-Imag(c)*(z2+1))/(3*z2-Imag(c)*(z+z)-1)+pixel,
	0.001 <= |zold-z|  }

JNewtonClassC  {; (z-c)(z-1)(z+1)=0, c=p2, Newton with "Classical"
		; p1=Julia constant, eg. c=p2=.5, p1=(-.75111,.01939)
		; or (-.708582,.052785) or (.797473,.0044221)
		; for P2 imaginary value only (p1=0) -> biological
	z2=sqr(z), z3=z2*z,
	z = (z3+z3-p2*(z2+1))/(3*z2-p2*(z+z)-1) + p1,
	|z| <= 4  }

MNewtonClassC (XAXIS) {; (z-c)(z-1)(z+1)=0, c=p2, p1 is start
		       ; "Classical" bailout with Mandel type set
	z=p1:          ; try p2=.5
	z2=sqr(z), z3=z2*z,
	z = (z3+z3-p2*(z2+1))/(3*z2-p2*(z+z)-1) + pixel,
	|z| <= 4  }
JulInNewton  {; f(z)=(z-1)(z-c+L)(z+c+L)=0 
	      ; try c=P2=.8738 + .9201 and L=P1=.5 (real)
	z=pixel, L=Real(p1), c2=sqr(p2), L2m1=L+L-1, L2=L*L:
	zold=z, z2=sqr(z), z3=z2*z,
	0.001 <= |zold-z|  }

MandinNewton (XYAXIS) {; f(z)=(z-1)(z-c+L)(z+c+L),L=real(P1)<>0,try
	z=0, L=Real(p1), c2=sqr(pixel), L2m1=L+L-1, L2=L*L:
	zold=z, z2=sqr(z), z3=z2*z,
	0.001 <= |zold-z|  }

MandinNewtonp (XAXIS)  {; f(z)=(z-1)(z-c+L)(z+c+L)=0 with L=real(p1)
	z=0, L=Real(p1), c2=sqr(pixel), L2m1=L+L-1, L2=L*L:
	zold=z, z2=sqr(z), z3=z2*z,
	0.001 <= |zold-z|  }

MNewtonN (XAXIS) {; z^n+z(c-1)-c=0, n=p1, eg. n=5, zoom right side
		  ; normally p2=0; due to parser error try .0001
	z=p2, c=pixel, n=p1, n1=p1-1:
	zold=z, zn1=z^n1, zn=zn1*z,
	0.001 <= |z-zold|  }

JNewtonN  {; z^n+z(c-1)-c=0, n=p1, p2=Julia constant
	   ; eg. n=5, p2=(.53532164,.0046549) - its full of them!
	z=pixel, c=p2, n=p1, n1=p1-1:
	zold=z, zn1=z^n1, zn=zn1*z,
	0.001 <= |z-zold|  }

MNewtonNp (XAXIS) {; z^n+z(c-1)-c=0, n=p1
		   ; normally p2=0, due to parser error try .0001
	z=p2, c=pixel, n=p1, n1=p1-1:
	zold=z, zn1=z^n1, zn=zn1*z,
	z=(n1*zn+c)/(n*zn1+c-1) + pixel,
	0.001 <= |z-zold|  }

JNewtonNp  {; z^n+z(c-1)-c=0, n=p1, p2=Julia constant
	    ; eg. n=3, p2=(-.01465,.63363)
	z=c=pixel, n=p1, n1=p1-1:
	zold=z, zn1=z^n1, zn=zn1*z,
	z=(n1*zn+c)/(n*zn1+c-1) + p2,
	0.001 <= |z-zold|  }

MNewtonNp2 (XAXIS) {; z^n+z(c-1)-c=0, n=p1, c=p2
		    ; normally p3=0, due to parser error try .0001
		    ; p3 doesn't work before v19.0->add with editor
	z=p3, n=p1, n1=p1-1:
	zold=z, zn1=z^n1, zn=zn1*z,
	z=(n1*zn+p2)/(n*zn1+p2-1) + pixel,
	0.001 <= |z-zold|  }

MNewtonNM (XAXIS) {; z^n+z^m(c-1)-c=0, n=p1, m=p2
		   ; normally p3=0, due to parser error try .0001
		   ; p3 doesn't work before v19.0->add with editor
	z=p3, c=pixel, n=p1, n1=p1-1, m=p2, m1=p2-1:
	zold=z, zn1=z^n1, zn=zn1*z, zm1=z^m1, zm=zm1*z,
	0.001 <= |z-zold|  }

JNewtonNM (XAXIS) {; z^n+z^m(c-1)-c=0, n=p1, m=p2 
		   ; p3=Julia constant, doesn't work before v19.0
	z=c=pixel, n=p1, n1=p1-1, m=p2, m1=p2-1:
	zold=z, zn1=z^n1, zn=zn1*z, zm1=z^m1, zm=zm1*z,
	z=(n1*zn+m1*zm*(c-1)+c)/(n*zn1+m*zm1*(c-1)) + p3,
	0.001 <= |z-zold|  }

JNewtonez (XAXIS)  {; Newton f(z)=e^z-1=0, use corners=-8/8/-6/6
	z=pixel:                               ; [no Mandelbrot
	0.001 <= |z-zold|  }

JNewtonCos (XYAXIS) {; Sin, Sinh, Cosh give the same shifted/rotated
by Pi/2
		     ; zoom the main origin structure and cycle the
	z=pixel:                                  ; [no Mandelbrot
	zold=z, s=sin(z), c=cos(z),
	z = z+p1*c/s,
	0.0001 <= |zold-z|  }                     ; |c| also OK 

The Well-Tempered Fractal

by Robert Greenhouse

I would like to announce the availability of version 3.0 of The Well-Tempered Fractal which can be obtained by ftp from Noel Griffin's ftp site SPANKY.TRIUMF.CA. Many thanks to those of you who have sent me feedback on my earlier versions. The READ.ME file is included below:

The Well-Tempered Fractal v3.0 (Copyright 1993-1995) is a composer's Tool for the derivation of musical motifs, phrases and rhythms from the beauty and symmetry of fractals, chaotic attractors and other mathematical functions


The Well-Tempered Fractal is a program which composes musical melodies by mapping fractal images to musical scales. Ten different fractal types may be mapped to twenty-one different musical scales. Twenty symmetry operators may be applied to the image as it is plotted. The output may be heard on an ordinary PC speaker or written to a MIDI event file which can be converted to a standard MIDI file for use with MIDI instruments or sound cards. The aim of the program is to produce fractally generated melodies to be used by composers as the starting point for entire musical compositions.

The following files are included in the archive called WTF30.ZIP

WTF30    EXE     72098 04-09-95   1:08p      The main executable of
WTFTITLE EXE     49718 03-26-95  10:57a      The title screen for WTF
WTF30J   TXT     47042 05-20-95   2:47p      The manual for WTF 3.0
FILE_ID  DIZ       410 05-20-95   4:29p      Archive descriptor file
READ     ME       3041 05-31-95   5:38p      This file
HOP01    GIF      8429 04-11-95   6:52a      .GIF of fractal used to
                                             compose HOP01C.MID
HOP01C   MID     24964 04-11-95   7:06a      A MIDI file of a
                                             composition emerging
                                             from the fractal shown
                                             in HOP01.GIF
HOP01TXT ZIP     13258 05-20-95   2:52p      The raw .TXT file output
                                             HOP01.GIF written by
                                             (ZIPped to save space)
IVAN1B   MID     87940 05-30-95   9:40p      Another MIDI file created
                                             from the Ivanov fractal
KAMTOR1  MID     24700 05-30-95   9:05p      Another MIDI file created
                                             from the KAM Torus
EXAMPLES TXT      3781 05-31-95   5:35p      A description of the
                                             MIDI files
CHEBY    GIF     23330 05-20-95   3:13p      Typical fractals from
DUFFING  GIF      3956 05-20-95   3:17p
HOPALONG GIF      6719 05-20-95   3:25p
IKEDA    GIF     14466 05-20-95   3:25p
IVANOV   GIF      8349 05-20-95   3:25p
JULIA    GIF      3411 05-20-95   3:26p
KAMTORUS GIF     11572 05-20-95   3:26p
MIRA     GIF     10315 05-20-95   3:26p
SEAHORSE GIF      3567 05-20-95   3:26p
SGICON   GIF      7161 05-20-95   3:26p

Both executables must be present for the program to run and the archive must be distributed with all files intact.

The program is an IBMPC/DOS program and requires no additional hardware to create the fractals and play music through the PC speaker. It will in addition produce MIDI event files which may be converted to standard MIDI files using Piet van Oostrum's utilities (MF2T.EXE, T2MF.EXE) which are available by ftp at To play the MIDI files, a sound card or sound module which supports MIDI music is required.

I welcome feedback, comments, suggestions as well as any examples of MIDI files which users create.

Registration is $15: Robert Greenhouse 3401 Hillview Ave Palo Alto California 94304 (USA)

More on gravitational lenses please!

by Yvan Bozzonetti

(Editorial note - I found this on the hard disk and couldn't find it anywhere in Fractal Report 34 onwards. Sorry for the delay [or maybe I put it in a space without recording it in the contents page].

The second paper from John Topham on gravitational lensing (Fractal Report 33) was particularly interesting for me and some other reader as well I assume. As an amateur astronomer, I am a member of a lot of clubs in that domain. One of them is completing a large telescope, one meter in diameter, with a 2048 X 2048 CCD array. One of the first planned challenge for that instrument, that summer, should be to detect Einstein's rings in distant galaxies and quasars.

If this experiment succeeds, there may be more of the same kind in the near future. Anybody with a PC and a modem can try that activity: Software Bisque ( 912 Twelfth Street, Suite A Golden Colorado 80401 USA) sells for $500 or so a package including : THESKY, SKYPRO and an interface to send order to a 24" robotic telescope on Mount Wilson. You choose your object with THESKY and you command both, the telescope and its CCD camera with SKYPRO. Don't bother about your local weather, the hour or anything, you will receive your CCD pictures taken with an ST6 camera in a few minutes via your telephone line.

[Editorial note: the cost for observing time is $300 per half-night session, or $500 per whole night. I should imagine that if you have a worthwhile project, rather than just wanting to play around, this is very good value for money. You need to telephone by voice in advance to arrange a session, and reservations are taken from clubs or private individuals. A free half hour training session can be arranged during daylight hours, and it may be advantageous to have two telephone lines, one for voice and the other for data. More information from: Gil Clark, [email protected], fax 818-793-4570, or write: Telescopes in Education PO Box 24 Mt Wilson CA91023 USA]

There has been in the astronomical literature some studies about exotic objects with uncommon mass distribution, such ring (ring made from a finite cosmic string), disks (spinars) or multipoint objects (a black hole pair for example); An illustration of the possible gravitational lensing effect would be useful in a search on CCD pictures of the sky.

The varying light amplification in a so called microlensing event is another subject to experiment with: When a dim star or a black body passes by our line of sight to a background star there is a light amplification of the background object by the gravitational lensing. If the dim star has some large planetary body around it, there may be a spike in the light curve. What if there is a dust and gas disk? This case may be very common but there is no study of the subject.

In the half a dozen of microlensing candidates found up to now, most of them seem to have a mass near 0.1 solar mass, this is the value predicted for some uncommon form of matter such as boson stars or a five dimensional Kaluza-Klein soliton. Such objects may eventually interact with light through fields other than gravitation. Such fields may fall off faster than the common 1/r2 gravity. How that affect the lensed picture and the light curve?

A fractal window on Mars.

by Yvan Bozzonetti

Having received some CD-ROM from magazines and other sources, I have turned to the only open possibility: To buy a CD player. With that in mind, I have started to look at new advertising for that media.

Virtual Reality Laboratories, Inc. (2341 Ganador Ct., San Luis Obispo, CA 93401) sells a fractal programme named Vistapro, making mountains, trees, landscapes of all kind. There are many supplementary packages from all around the world, but the one I find the most interesting (I have not all of them yet!) is Mars Explorer. It contains a Mars map made from pictures of the Viking mission. This is not the best map on the market, but the simplest to use. With Vistapro, the perspective can be altered so that the landscape seems to be seen from an airplane, not an orbiting satellite. This closeup picture is supplemented by fractal rendering. You can see the land such it is, dry under a yellow sky or look for what could be a distant future of terraformed Mars. Green trees thrive on the border of a lake in the Olympus Mons caldera. White clouds drift in a blue sky...

A friend has tested the system a step further: using pictures of the Moon recovered from another CD-ROM distributed by the Astronomical Society of the Pacific, he has terraformed the moon. Who will do the same with the now published Venusian map?

This tool could well serve to "write" some picture books on the solar system or more strange places. What about terraforming an integrated circuit picture, may be an interesting subject for some advertising or art gallery.

This is an ancient maze pattern, believed to be over 4,000 years old, to be found carved in Rocky Valley, near Tintagel, North Cornwall

Helen Jagger Wood

Images from a three Point Attractor

by Marius-F. Danca

P.O. Box 524 Cluj Romania

In Fractal Report Introductory Issue an article was presented named Images from a three Point Attractor by J.C. Topham. Unfortunately I haven't seen Portraits of Chaos by Ian Stewart in New Scientist, issue no 1689, that inspired J. C. Topham. The system is described by the following recursive equations:


yn+1=(2xn2+2yn2-p)yn+xnyn, n=0,1,... (1)

called "coupled equations" because x depend on y and y of x. To see the attractors of the system we must determinate first the critical points (or stationary points) of the system (1). This points verify the following system:


f2(x,y)=0 (2)



f2(x,y)=(2x2+2y2-p)y+xy (3)

The solutions are:


that becomes for p=1.5 (the value chosen by J.C. Topham):

X1(0,0), X2,3(0.5, 0.866025), X3(0.5 -0.866025)

X4(-0.625, 1.08253), X5(-0.625, -1.08253) (5)

Now it must analyse the kind of the critical points (5). For this reason we get an linear approximation to the system (1) in the neighbourhood of the critical points meaning that we substitute x with a+x' and y with b+y' where (a,b) are one the critical point and (x',y') one vector.1,2,3 Anyhow this linearisation don't influence the next steps. Then we calculate the roots of the characteristic equations (Eingenvalues):




is called the Jacobian, and:

Because of invariancy of the Jacobian under the linearisation that we have discussed, we can calculate

direct from (3). One says that one critical point X is attractor point if all the Eingenvalues of equation (6) for A calculated in X satisfy ||<1. If the point X is and attractor then starting from one point (x0,y0) of the plane with relations like (1) the orbit goes to X.

If at least one of the roots verify ||>1 then the point X is reppeling (if the real part of the Eigenvalues are both positive) or saddle (when one of the Eigenvalues has the real part positive).

If the roots of (6) are ||1 and ||=1 and are simple, than an Hopf bifurcation occurs, transforming the attractor X into so called cycle when the orbit is an closed curve, but this theory is too long to be presented heare. So we begin to analyse the points X12345

1) X1(0,0)

The characteristic equation is:




so X1 is a repelling point

2) X2(0.5, 0.866025) The characteristic equations are:

and the roots are:

X2 being an saddle. Also X3 is an saddle and X45 one reppeling point. So all the critical points of system (1) are unstable. But what was drawn in Fractal Report? The attraction basin of cycles, but not of the critical points calculated above.

In other words when one fixed point isn't attractive then one cycle exits. The criterion for one analytical determination of cycles being to complicated I determined them with one numerical algorithm (Julia) finding four cycles (fig.1) Each of them contains two points:

1) [(-0.433336, 0.1889257), (0.380283, -0.280817)]

2) [(-0.433336, -0.1889257), (0.380283, 0.280817)]

3) [(-0.51722, 0), (0.36534, 0)]

4) [(0.053053, 0.46974), (0.053053, -0.46974)]

that are visited alternatively. It means that if we start with (x0,y0) in (1) from one of the attraction basin of system the orbit goes to corresponding cycle visiting alternatively his two points. Now we can understand why the y position of the Basic code from Fractal Report are +/-0.1889257 and -0.469743.

The horizontal symmetry of the cycles is explained by the y parity of the system after division by y.

How can we determinate the attraction basin ? If the orbit of the (x0,y0) goes to one of the two points of one cycle, than we plot the appropriate point (i,j) in the screen with certain colours (fig.2,3).

(xn,yn) is close enough to the cycle when:


|yn-y'|< n=0,1,2,...

where (x',y') is one of the two points of the cycle and an very small positive number called precision of the algorithm. Now it's clear that we can choose other values for p in (1). To determine the appropriate attraction basin we use the relations (1) but other critical points will appear with or without cycles (e.g. p=3).

In the Basic code the rectangle that contains the attraction basins in the plane is defined by left-down corner and his side ("leftx"="lefty"=-1.2. "side"=2.4). The screen frame is defined by "frame" ("frame"=300 pixels). The precision can be take 0.01 or less. Different values for parameters can give different images (fig. 2,3).

Comparative to Julia's algorithm in this case it isn't necessary to count the iterations. The escape radius hasn't a big importance. That is necessary only to determine the points that are attracted by infinity. So it can be taken bigger than 4 because we know that if the modulus of one complex number z=x+i*y is bigger that 4 than if we iterate it this escapes to infinity. In fact the algorithm has a lot in common with the Newton's method that is (of course) a Julia algorithm.


1. Ray Radheffer, Dan Port, "Differential Equations Theory and Applications, Jones and Bartlett Publishers Boston, 1991,

2. Hao Bai Lin Chaos II, World Scientific, Singapore, 1990,

3. H. G. Schuster, Deterministic Chaos, VCH, Weinheim 1989.

REM Attraction Basin of System
REM Article in Fractal Report, Introductory Issue
REM Marius-F. DANCA July 1995

INPUT "leftx="; leftx: INPUT "lefty="; lefty: REM left-down
INPUT "p="; p: REM Parameter of the System
INPUT "size="; size: REM Dimension of the System in the complex plane
INPUT "frame="; frame: REM Dimension of the System on the screen
escape = 10: REM escape ratio
eps = .001: REM precision
x12 = -.434: y1 = .189: y2 = -.189: REM coordinates of cycles
x34 = .381: y3 = .281: y4 = -.281
x56 = .053: y5 = .47: y6 = -.47
x7 = -.517: y7 = 0
x8 = .365: y8 = 0
stepc = size / frame: REM step in the complex plane
x = leftx: REM (x,y) point in the complex plane
FOR i = 1 TO frame: REM (i,j) point in the screen
	x = x + stepc:
	y = lefty
	FOR j = 1 TO frame
		y = y + stepc
		xx = x: yy = y
100             REM iteration
                 xxx = 2 * xx * xx * xx + 2 * yy * yy * xx - p * xx - .5 * xx * xx
+ .5 * yy * yy
		yy = 2 * xx * xx * yy + 2 * yy * yy * yy - p * yy + xx
* yy
		xx = xxx
		m = xx * xx + yy * yy
		GOSUB 300: REM test of the (x,y)
		IF (m < escape) AND (test = 0) THEN GOTO 100
		IF test = 1 THEN PSET (i, j), col
		NEXT j

300 : REM test subroutine; if test=1 then (x,y) had reached one
IF ABS(xx - x12) < eps AND ABS(y1 - yy) < eps THEN col = 1: test = 1:
IF ABS(xx - x12) < eps AND ABS(y2 - yy) < eps THEN col = 3: test = 1:
IF ABS(xx - x34) < eps AND ABS(y3 - yy) < eps THEN col = 5: test = 1:
IF ABS(xx - x34) < eps AND ABS(y4 - yy) < eps THEN col = 7: test = 1:
IF ABS(xx - x56) < eps AND ABS(y5 - yy) < eps THEN col = 9: test = 1:
IF ABS(xx - x56) < eps AND ABS(y6 - yy) < eps THEN col = 11: test = 1:
IF ABS(xx - x7) < eps AND ABS(y7 - yy) < eps THEN col = 13: test = 1:
IF ABS(xx - x8) < eps AND ABS(y8 - yy) < eps THEN col = 15: test = 1:
test = 0

Fractal 97

"Fractals in the Natural and Applied Sciences"

4th International Working Conference

April 8 - 11, 1997, Denver, Colorado, USA

Call for Papers

Aims and Scope

The conference is intended to provide a forum for the dissemination of the latest research findings in the broad field of fractals. Interdisciplinary submissions are strongly encouraged.

Conference Topics (Amongst Others)

Applications, Diffusion, Disordered systems, Dynamical systems, Fractal surfaces, Growth phenomena, Image analysis, Multifractal formalism, Pattern formation, Phase transitions, Self organization, Turbulence, Visualization

Paper Submission

Four copies of a full paper in English, typed on one side only and limited to 5000 words (11 pages in total), should be sent by post to arrive no later than August 5, 1996. No email or fax submissions will be accepted. [That shows we're in the 20th century -ed!] Submission of a paper implies that the work has not been previously published nor is currently under consideration for publication elsewhere. In order to be considered, the first page should contain the title, authors' names and affiliations, phone and fax numbers, full electronic and postal addresses, an abstract of not more than 200 words, and at most five keywords. All papers will be refereed and those accepted and presented will be published.


The Conference is promoted by the International Federation for Information Processing through its Specialist Group on Fractals and Chaos, SG15, is organized by Kingston University, UK and hosted by University of Denver, USA.

Conference Chairman M. M. Novak (Uk)

Organizing Chairman T. G. Dewey (Usa)

Programme Committee currently being formed


5 August 1996 Deadline for full papers

14 October 1996 Notification of acceptance

8 November 1996 Camera ready copy

8-11 April 1997 Conference


Please submit your paper to:

Miroslav M. Novak

School of Physics




Tel: +44 181 547 2000 +44 181 547 7419

Email: [email protected] Kingston University

Fax: +44 181 547 7562

Fractal 95 conference.

A copy of abstracts of papers accepted for the Fractal 95 Conference can be found at gopher://

The full proceedings have just been published as Fractal Reviews in the Natural and Applied Sciences M. M. Novak (ed.), Chapman & Hall, London, August 1995 ISBN 0-412-71020-X

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