This time I had to send out a "request for papers" to some of our more established authors in an attempt not to fill the newsletter with my own stuff, which is often below the standard set by our other authors. So please keep articles coming for the next issue. Remember that I can now read both sizes of disk and therefore you can send material on the 3�" disks as well, either density, MSDOS format.

I have recently got MSDOS 6.0, and if you haven't already got a defragmentor and compressor it is well worth its price of �65.33. But if you order after 30 June it will cost �100.53. I must congratulate Microsoft in offering it cheaper to start with rather than attempting to recover its development costs from the first few customers. However they make the reduction look less than it really is by quoting prices before carriage and VAT in their advertisements. Their advertising agency must have slipped up here! [Microsoft Upgrade Centre Unit 1 Mill Farm Business Park Millfield Road Hounslow Middlesex TW4 5PY]

Elsewhere in this issue you will see a review that mentions images from text, and also my own attempt at curlicues from text. I would like to stress the immense interest there is in generating images from data that can enable people to interpret it quickly. An interesting project would be to get mapping or iterative plots from text that produce images that somehow give a "fingerprint" of the author. The human genome project produces vast quantities of data that needs some form of visualisation, and therefore any results in this area will have immediate practical interest.

Many Fractal Report readers are interested in the artistic value of the images produced, and therefore even if the images you can get from text aren't particularly useful scientifically they may still be worth reporting on. Who knows, it may lead to a train of thought that could result in a new area of data analysis.

Another area of data analysis that could benefit is the search for extra terrestrial radio signals from intelligent life. Any signal picked up is likely to be very similar to random noise and could easily be rejected as such. But maybe some computational process could convert this data, which appears as strings of (binary) numbers, into patterns that could show up any underlying structure.

In any of these instances, the pattern doesn't have to mean anything in relation to the input, it just has to show that there is something there other than noise. In the case of text, of course, there is a slight difference in that a "signature" of the author would be useful to historians and possibly people with time to spend on copyright issues.

Announcements

REC vol 7 no 8

REC is still reorganising its lapsed publishing schedule. This issue should have appeared last year, and the next issue will be a triple issue that should get up to date. No 8 contains an item on complex powers, ie z^z where z=x+iy. The major article in this issue is on mathemagical black holes, iterations that always end up with the same number regardless of the number you start with. (A sort of one dimensional fractal iteration I suppose.)

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

Collection of Natural Fractals

Mr G.T. Swain kindly sent in a copy of The Telegraph Magazine of 3 April which included an article The Patterns of Chaos. This concerned a book to be published by Cornerhouse Publications in the autumn entitled Fractal Landscapes. Instead of being about patterns drawn by computer, it is a collection of photographs taken on a world tour for the purpose by Bill Hirst. Several examples were given in the article, all in monochrome.

The article mentioned that Dr Mandelbrot was born in Warsaw and said that he brought an entirely new concept into geometry. It described how Mr Hirst "Finally plucked up courage to show the fruits of his work to Dr Mandelbrot himself". After examining the photographs for two hours, Dr Mandelbrot is reported to have said in his Polish accented American voice: "You know, this man really understands fractals."

I might add that I have often looked at the patterns in surf as waves break, the patterns in the sand caused by streams flowing down the beach to the sea, and the mottled patterns sometimes left by the waves on the beach, and seen fractals. There is much at the seaside apart from the coastline and seaweed to interest fractal spotters.

Visions of ...

Dr Pickover has asked me to point out that the title of his book is Visions of The Future NOT Visions of Tomorrow, as reviewed in the last issue. It can be obtained from Science Reviews Ltd., 18 Oaklands Gate Northwood Middlesex HA6 3AA for �25. US readers should contact St Martins Press 175, Fifth Avenue New York NY10010. Its ISBN is 0-905927-09-5.

Iterated Systems Fight Crime

In Fractal Report 26 I reviewed Iterated Systems' fractal compression software demo disk. An article in New Scientist of 8 May revealed that the software is being used to store pictures of criminals, and indeed process them as fractals for comparison. The intention is for police to be able to picture suspects and then compress the images to 5k files when they can be compared in seconds with images on the police national computer. Even if this system is not perfect, it can produce a short list that is much easier for a police man to sort out manually. It is suggested that the images are actually easier to compare in their fractal format.

Letters

From Mr Timothy Harris

Many thanks for including the review of CAL in issue 26 and mentioning V3.60 in the editorial section. Although this will no doubt help the program to become more widely known there are a number of inaccuracies in the review:

CAL stands for Confusion and Light, rather than Chaos and Light as was written.

A fractal may be saved as a bit image at any time - it does not need to be completed. Previously saved images can therefore be loaded and continued. This feature has been present since the very first version of the program and other options, such as the batch completion mode, would not be there if only complete images could be saved. I have now re-phrased some of the help screens to emphasise that it is not necessary to wait for the image to be finished, in case this presented a misleading picture.

The comments on the user interface were very useful and I have included many of the changes suggested. The number of iterations is set separately from the fractal menu and the current Set Position option has been complemented by a centre/mag version. When setting the zoom box L can be used to toggle between large and small movements and A to lock the aspect ratio.

The keys E F V for decreasing the red, green and blue levels were chosen simply because they are immediately to the left of the R G B keys respectively. I have made it so CTRL R G B can be used, although I have personally found the need to use two keys at once in such a situation can make it more difficult - eg when trying to simultaneously increase the red whilst decreasing the green component it would be necessary to pump the CTRL key up and down while alternately pressing R and G. There is also an additional option, selected by pressing ? in the palette editor, which creates a random palette.

The caps-lock state is no longer shown on the status line and the strings for num-lock and insert have been shortened to avoid obscuring any information occurring there.

The new version, V4.0, includes an export to PCX option which should get around the problem of including CAL images in DTP or graphics packages. Furthermore the compiler for user defined formulae has been sped up around threefold, additional functions have been introduced to allow the formulae to include more elaborate internal colouring schemes and L-systems are evaluated in around half the time taken in previous versions.

From Mr John Sharp

Last Year you found out about Dr Pickover's Mazes book being on special offer from St Martin's Press. If you hear of any more let us know. They were very good. I sent my credit card number and received the book at the price quoted without any shipping charges! The book is excellent. There are many new things, plus the usual rehashing of article he has written. I suspect he is becoming a front man. He can't have the time to do so much.

I have had some contact with John Horner. His publication Frac Cetera is good, considering how isolated he is. I am lucky in being able to get to good libraries. He has trouble even seeing New Scientist, although he is using electronic communication well. He has also communicated with the Stone Group and has got a beta test version of Fractint 18.

American dollar subscriptions: There are two ways better than the bank. The first I found was American Express do a dollar check for a fee of �2. I went into their offices in the Haymarket (off Piccadilly Circus) and got one instantly over the counter. Western Union have advertisements all over London saying that they can wire money cheaply, although I haven't investigated how much it does cost.

The last page of issue 26 was interesting and brought to mind another point which I will look at developing, which is: "are the images in our mind rather than discrete entities?" I will see if I can develop the idea further, although the article may not get written just yet.

Fractal Report ought to keep going with the interesting articles which still keep coming.

An open letter to Dr. Meech

from Joyce Haslam

Dr. Gaston Julia was investigating z² + c as an attractor using a program like this one for the set at (P,Q):

10 cx = P: cy = Q

20 x = 0.1: y = 0.1 : REM seed values

30 FOR i=0 TO 6000

40 wx = x - cx: wy = y - cy

50 IF (wx>0) THEN theta = atan(wy / wx)

60 IF (wx<0) THEN theta = 3.14159 + atan(wy / wx)

70 IF (wx=0) THEN theta = 3.14159/2

80 theta = theta / 2

90 r = sqrt(wx * wx + wy * wy)

100 IF RND>0.5 THEN r = sqrt(r) ELSE r = -sqrt(r)

110 x = r * cos(theta): y = r * sin(theta)

120 IF i>10 THEN POINT( -a + (x + b) * c/d , (e - y) * f/g )

130 NEXT

The Science of Fractal Images gives a=5, b=4, c=500, d=8, e=2, f=250, g=4 for the screen display factors.

Since we now have machines instead of log tables, we iterate like this for each pixel of a grid:

100 x = pixel_x : y = pixel_y : r = P : s = Q

110 color = 0 : least = 4.0

120 REPEAT

130 newx = x * x - y * y + r : newy = 2 * x * y + s

140 x = newx : y = newy

150 magnitude = x * x + y * y : color = color + 1

160 IF magnitude < least THEN least = magnitude: index = color

170 UNTIL color > maxit OR magnitude > 4

The Mandelbrot calculation is the same except for the first line:

100 x = 0 : y = 0 : r = pixel_x : s = pixel_y

In prose, each Julia set is for a single c; and each iteration starts by setting z to the pixel value. The centre point (z=0) of each Julia is the point c on the Mandelbrot display. Each iteration in the Mandelbrot set starts by setting z to zero and c to the value of the pixel.

Dr. Mandelbrot's only display option (on a teletype line printer) was to colour points inside the set black and leave points outside the set white. Now we usually show the contours outside the set in colours, with the interior of the set black or white.

Binary decomposition is a nice and easy black & white or three-colour display. Plot the interior of the set in either white or a colour (I usually choose yellow). Outside the set, plot the point in black if the final value of y is positive, but in white if the final value of y is negative. (Try plotting according to the final value of x instead of y.) Four-way decomposition is also easy. If x and y are both positive, the point is in the first quadrant (use colour 1); if they are both negative, the point is in the third quadrant (use colour 3); etc. By using related colours (say 1=dark blue, 2=light blue, 3=dark green, 4=light green) you might see both decompositions in the same plot.

The variables least and index are the key to two of Fractint's more esoteric effects. They keep track of the nearest approach of the point to the origin. For inside = bof60 use colour = 75 * sqrt(least); for inside = bof61 use colour = index. The significance of the nearest approach of the z-trajectory to the origin is discussed in an appendix of the Fractint manual and in Beauty of Fractals, page 62.

Since you are writing programs yourself instead of relying only on Fractint, you can display the points inside the set by a method traditional for the outside (with points outside the set white or a neutral colour) or vice-versa. Outside=bof60 is great fun. It will give a screen full of colours even for dendrites like the Julia at (0, 1); perhaps the authors of Fractint could be persuaded to include it as an option. Some displays will be meaningful and others will not; the interpretation is up to you.

There is no substitute for reading The Big Three: The Fractal Geometry of Nature, The Beauty of Fractals, and The Science of Fractal Images. Your local library should be able to borrow them for you if you are having difficulty finding them for sale. I would also recommend R.T.Stevens Fractal Programming in C to any beginner (if you can read Gruniad prose you can read C). There are important articles in the Scientific American issues of Nov 81, Dec 86, and Aug 90, and the Computer Recreation columns of Aug 85, Sept 86, Dec 86, July 87, Nov 87, Feb 89, July 89, and Jan 91 are about fractals. Some of these latter reappeared in Dewdney's book Armchair Universe. Again your library should be able to help you.

The Julia set has featured in many articles in the back issues of Fractal Report, which are still available. Some articles are more general than their names imply (eg Mandelplot for Amstrad 6128 in Fractal Report 4, The Mandelbrot Explained: A Beginners Guide in Fractal Report 20). The inverse iteration of the Julia set comes up in Fractal Report 8 (animated!) and again in Fractal Report 20. Kate Crennell's article in Fractal Report 11 has display ideas, as do Dr. Jules Verschueren's articles in Fractal Report 12, Fractal Report 15 and Fractal Report 23, and Dr. Ian Entwistle's in Fractal Report 15 and Fractal Report 18. There is an article in Fractal Report 17 about Julia sets and the Z-trajectories of the points in them. YAMS in Fractal Report 6 is a solid guessing algorithm that has saved me hours of time and inches of fingernails.

Happy fractaling!

Coming attraction

Rumour says that the Phoenix set and its Julias will be hard wired into version 18 of Fractint. They give splendid images for inside=bof60 and inside=bof61 for iterations between about 30 and 150. Zero is the only finite attractor, so details emerge and then vanish again as iterations increase.

inside=bof60 inside=bof61

Ushiki's Phoenix maxit=64

Phoenix Julia at -0.395, 0.258 maxit=32

Phoenix Julia at 0.035, 0.640 maxit=32

Joyce Haslam, 112 Keighley Road, Colne, Lancs. BB8 0PH

Quantum Fractals at Least?

by Y. Bozzonetti.

The why?

What is the usefulness of fractals?

Nothing! says the technophobic.

For a mathematical abstract domain unknown twenty years ago, there has nevertheless been some achievements. Today, fractals are the basis of a new art domain using computers. Tomorrow, that may turn into games and adventures in some virtual reality worlds. On a more basic footing, they start to invade the data compression domain, something very useful for computers, television and telecommunications. If sci-fi videophone becomes someday a reality for everyone, it will be for a good part a fractal offspring. This is not mere gadgets, TV is the key to the education door in developing countries, where teacher shortage is a fact of life for all governments. These countries cannot pay for an extensive ground based transmitter network, and individuals can't pay for satellite receivers. A narrow band AM radio transmitter is the only economical solution; only fractal data compression allows us to distribute pictures in this way. In the years to come, fractal based technologies may so contribute to save millions of live in poor regions ... and suppress many related problems in developed nations.

Yes, fractals start to be useful. They are too an element in a mathematical toolbox enabling to devise new scientific knowledge, that is, the technology and economy of tomorrow. Quantum physics is seen by many as an abstruse theory, one of its branches, solid state physics, is nevertheless the basis of all the modern electronics industry. Indeed, transistors, diodes and other integrated circuit components are made from semiconductors tailored by solid state physics rules. Before looking at fractals in that promising domain, it may be worth to look back and see what has been done.

When we need to know how an object reacts to a single force (or a sum of similar forces) we use the Newtonian physics rules. When there are more objects, for example three or four, Newtonian formalism becomes very cumbersome and Largange, Hamilton or Jacobi recipes enter into play. What if very many objects must be worked out at the same time? If the interactions between them is weak, we can use the statistical mechanics rules, this is the so called thermodynamics, with applications in all kinds of thermal engines and well beyond. If interactions are strong, for not very many objects, we have a state called chaos. Chaos looks as disorder at first, nevertheless it can be reversed - it remains deterministic. It is not predictable in its outcome, but remains computable. Doing the computation, that is, the experiment, we find the result. In practice, chaos is very sensitive to the starting conditions. Fractals are one mathematical illustration of chaos. There is no theory up to now for very many objects in strong interaction.

Fractals display an essential property: scale invariance, they look the same at every scale. What if a physical force field becomes fractal? What, if at any large scale force, we add a local one with a significant strength only at short range? The consequences are far reaching, they are what we call quantum mechanics!

Because in most case interesting situations imply interactions between many objects, Newtonian quantum mechanics is very tedious, Lagrangian and Hamiltonian formalisms are the rule here. Jacobian quantum mechanics is technically better, but remains too complicated for most current human minds. In the quantum thermodynamics, all the quantum parts vanish and we are left with classical physics, that is, not scale invariant force fields. Strangely, there seem to be no chaotic behaviours in the quantum domain. There is no room for both, fractal forces and fractal results of their actions. No theory explains that, it is only an experimental fact. Is there an unknown law behind that, or simply our experimental quantum systems remain too simple to display such a complex behaviour?

Modern physics use two quantification systems, and chaos starts when there are many objects in strong interaction. Here, "quantifications" are our objects, two of them may be a far too low value to start a chaos system. Can we find more quantifications? The answer is probably yes. Modern mathematical tools explain a local force field that is, a quantification, as the result of a kind of integral (a fibre bundle) built on a moving coordinate system. Many quantifications may be built in this way with strange properties (this is why we have don't found them before, they are too uncommon to reveal themselves if we don't know where to look at them).

One possibility is the continuous quantification, an infinite set of local forces on all scales and true fractal invariance. Here may stay the key to fractal quantum mechanics. But what will a fractal quantum system look like?

Assume we have something taking some values at different places in space. We can describe that "something" by a function of the space. Because an applied force will displace the "something", it transforms the original function into another. This is a mathematical object called "operator". A function transforms a number into another number, for example a coordinate into the local value of "something"; at the next step, operators transform a function into another one. If the operator takes into account local force components, it is said to be a quantum operator. This is not physics, only a lexicon for usual mathematics quantum slang.

To compute the effects of a quantum operator on a function, we simply multiply the first by the second. Everything works as if functions were defined as the points in a space, the space of functions. As ordinary space, that one allows to project a point along a coordinate system. This one is not made of orthogonal arrows but from so called orthogonal functions. Any function of the space is then reduced to a set of coefficients multiplying the coordinate functions. That looks complicated at first but we have simple shifted from words as length, height... to others as first, second... Legendre or Hermite polynomials.

Ordinary space contains only three dimensions, on the contrary, most functions need an infinite set of coordinate polynomials to be broken into a coefficient set. Such infinite dimensional function spaces are called Hilbert spaces. Now, with function reduced to a set of numbers, we can use on each of them any operators. We have in fact reduced functions to numbers so that operators work in Hilbert spaces as functions in ordinary one. Operator formalism is a powerful tool in both, classical and quantum physics. In most case, classical physics is learned before the mathematical chapters on operators and quantum mechanics. So, most people think quantum physics is a strange domain because it uses operator formalism. There is no physics in that, simply, quantum operators include a local force term with scale defined by the Planck's constant. Classical ones have no such local term, but otherwise work in the same way.

What if we turn now to very many quantifications? For each quantification, there will be a force term, a function element and an operator component to deals with it. To summarize, force fields, functions and operators becomes fractals. When a function is broken into a set of numbers along a Hilbert's space coordinate system, each coordinate can't no more be a Legendre polynomial or something similar, it must be the corresponding fractal. That fractal must expands on a space as large as the domain where quantum continuity duals. If all quantization remain constrained in the domain between zero and two in the complex plane, then so will be the fractals of coordinate functions. Here, zero is the distance from the original or ordinary quantification and two is the maximal distance, in unit of Planck's constant from that point.

All of that is not a physical theory, simply a mathematical model ready to be applied if physical considerations or experiments need the introduction of continuous quantification. My object here is not to produce such a theory, but simply to demonstrate than such a theory may well be the next step in our attempt to understand the world. If such speculations will finally succeed is another question. Our problem here is with fractals. Displays of fractal formulae using Legendre or Hermite polynomials is so a fundamental building block in the process. The next time anybody ask you: "Why are you doing pictures on your screen?" you'll have an answer: To illustrate the scientific paper describing the next physical theory of the world.

The how to.

Here are the first Hermite polynomials:

- H(0) = 1,

- H(1) = 2Z

- H(2) = 4Z^2 - 2

- H(3) = 8Z^3 - 12Z

- H(4) = 16Z^4 - 48Z^2 + 12

- H(5) = 32Z^5 - 160Z^3 + 120Z

To find the following polynomials in the set, we can use the recurent law:

H(n+1) = 2.Z.H(n) - 2.n.H(n-1),

for example:

H(6) = 2.Z.H(5) - 8H(4).

Here are the Fractint formulas for H(2) to H(5).

HER2 {Z=P1,C=PIXEL : Z=4*Z*Z-2*C, |Z|<=4}

HER2A {Z=C=PIXEL : Z=4*Z*Z-2*C, |Z|<=4}

HER2B {Z=PIXEL,C=P1 : Z=4*Z*Z-2*C, |Z|<=4}

HER3 {Z=P1,C=PIXEL : Z=8*Z*Z*Z-12*Z*C, |Z|<=4}

HER3A {Z=C=PIXEL : Z=8*Z*Z*Z-12*Z*C, |Z|<=4}

HER3B {Z=PIXEL,C=P1 : Z=8*Z*Z*Z-12*Z*C, |Z|<=4}

Here, C is included in the product of the second term, it may be put on the first one or simply added:

HERC3 {Z=P1,C=PIXEL : Z=8*Z*Z*Z*C-12*Z, |Z|<=4}

HERC3A {Z=C=PIXEL : Z=8*Z*Z*Z*C-12*Z, |Z|<=4}

HERC3B {Z=PIXEL,C=P1 : Z=8*Z*Z*Z*C-12*Z, |Z|<=4}

HERCC3 {Z=P1,C=PIXEL : Z=8*Z*Z*Z-12*Z+C, |Z|<=4}

HERCC3A {Z=C=PIXEL : Z=8*Z*Z*Z-12*Z+C, |Z|<=4}

HERCC3B {Z=PIXEL,C=P1 : Z=8*Z*Z*Z-12*Z+C, |Z|<=4}

HER4 {Z=P1,C=PIXEL : Z=16*Z^4 - 48*Z^2 + 12*C, |Z|<=4}

HER4A {Z=C=PIXEL : Z=16*Z^4 - 48*Z^2 + 12*C, |Z|<=4}

HER4B {Z=PIXEL,C=P1 : Z=16*Z^4 - 48*Z^2 + 12*C, |Z|<=4}

HER5 {Z=P1,C=PIXEL : Z=(32*Z^5 - 160*Z^3 + 120*Z)/C, |Z|<=4}

HER5A {Z=C=PIXEL : Z=(32*Z^5 - 160*Z^3 + 120*Z)/C, |Z|<=4}

HER5B {Z=PIXEL,C=P1 : Z=(32*Z^5 - 160*Z^3 + 120*Z)/C, |Z|<=4}

Here, the use of C is in fact a free choice, it may be added, entered in one or more product and so one. With 18 formulas to test there is some work for computers, other variants and larger Hermite's polynomials are left as exercise, - a concluding remark found too often in mathematical books.

In a real quantum calculation, fractals must be computed on a sum of polynomials, each with its own coefficient such P1. Who wants to try them?

Fractals at the Seaside

by John de Rivaz

The shortage of articles this month made me realise that there are other ways of getting fractal images besides using a computer. Everyone knows that the coastline is a fractal, so what about something more interesting. Only about a half a mile from my house there is a continuous process generating transient fractal images - the action of rocks on waves approaching the shore. As a wave passes over a rock at just the right depth, a series of vortices form a triangular pattern which changes over a period of several seconds as it fades away. Along the shoreline the pattern is more confused, but the "rock triangles" are very pronounced.

The figure can be made to look more life a fractal by using Photofinish to enhance the image using "equalise" and then setting to black and white.

The first picture on the right is the image after equalisation. The resolution of the video camera and computer imaging system is such that if the whole triangle was shown all the detail would be lost. Below that is the image without equalisation, is more as a photograph. The bottom image is another vortex pattern.

Scope for further experiment would be to take each frame of a video of a dissolving "rock triangle" and somehow get the computer to display it as a 3D view of a solid object. This could be done by taking each frame, putting it to black and white, and then regarding it as a "slice" to be placed on top of the previous one.

It would also be interesting to see if there is a simple mapping formula which produces similar patterns on the computer screen.

Curlicues From Text

by John de Rivaz

Writing in Visions of the Future, in an article entitled Fractals and Genetics in the Future, H. Joel Jeffrey suggested that it may be worthwhile to input text into IFS programs to generate patterns that may give some instant fingerprint of the author. This sounded interesting, but the result on page 134 wasn't particularly stunning.

When I got John Sharp's article on curlicues, I thought that here is a possible pair of ideas that can be merged. An initial venture was simply using the ASCII values of the letters, all shifted to upper case and reduced so that A=1, B=2 etc. It wasn't particularly interesting. Then I tried getting a probability matrix of the letters of the alphabet, which again, as it had so few points, was pretty mundane. However the next step was to take each letter and do a probability of appearance of the letter that follows, producing a curlicue for each letter.

This produced the result as follows for Mr Bozzonetti's article, taking the text up to the first of the formulae:

For the next Longevity Report Mr Bozzonetti wrote an article entitled What is Life concerning the nature of physics, biology thermodynamics and the process we call life. It was fed into the program and produced the following result:

They certainly don't produce a definite signature of the text, but the little squiggles don't look random. Maybe curlicues aren't the way to convert text to a visual image, but maybe one of our readers can think of something better. Here is the listing of the program:

screen 12

pi=4*atn(1)

pi2=2*pi

s=2000

window (0,0)-(4*s,3*s)

' experiment with this value:

a=1: 'length of vector

open "c:\turbobas\programs\boz.txt" for input as #1

dim p(26,26)

let a1=0

while not instat and not eof(1)

let a=asc(ucase$(input$(1,1)))-64

if a>0 and a<27 then let p(a,a1)=p(a,a1)+1:a1=a

wend

for n=1 to 26

let np=n-1

let ylocate=6-(1+int(np/6)):yshift=ylocate*1150-750

let xlocate=1+6*(np/6-int(np/6)):xshift=xlocate*1200

let wd=0:heading=0:x=0:y=0:pset(xshift,yshift),0

for n1=1 to 26

let wd=wd+p(n1,n)

if wd>pi2 then

let wd2=wd/pi2

let wd=pi2*(wd2-int(wd2))

end if

heading=heading+wd

if abs(heading)>pi2 then

let h2=abs(heading)/pi2

heading=pi2*(h2-int(h2))*sgn(heading)

end if

x=x+a*cos(heading)

y=y+a*sin(heading)

line -(x+xshift,y+yshift),15

next n1

locate 32-ylocate*6,xlocate*12: print chr$(n+64);

next n

locate 1,1

print "Press any key to end"

while not instat

wend

end

Magazine review: Fractals

John de Rivaz

Thanks are due to Yvan Bozzonetti for drawing this to my attention. It is a new peer reviewed academic journal published by World Publishing Co. 73, Lynton Mead Totteridge London N20 8DH UK. (USA: 1060 Main Street River Edge NJ07661) Cost per four issues is $123 to individuals (or libraries in developing countries), $211 others.

The flyer and call for papers says that "as readers represent a diversity of disciplines, papers published will have the broadest appeal and can be read and understood be readers of diverse backgrounds." It also said that pictures and schematics will be reproduced in colour.

Although rejecting an exchange subscription, the editor kindly sent me a copy of issue one for review. Unfortunately I cannot repay this kindness by urging every Fractal Report reader to buy it also. This is not because there is anything wrong with Fractals as such, but probably for the same reason that the editor thought an exchange subscription inappropriate. The reason is that Fractals is aimed at a highly academic rather than practical audience.

There were some interesting articles, but no attempt was made to suggest any practical work the readers could do on their own computers, although the authors had obviously done a lot of "keyboard bashing".

I read a couple in detail, one being another attempt to model the stock market using fractal techniques. A number of graphs were given, and the caption under one said An example of market price evolution simulated by Eq. (4) with = 0, N = 100, = 1.0, = 0.4 and = 0.01. B_i(0) are randomly chosen in the range (-1.0,1.0).

When one looks at equation 4, one sees:

B_i(t+1) = B_i(t) + _i(t) + a_i + c_i{P(t)-P(t_prev)}

No mention of , N, , , and ! Sure, a longer equation containing them could be constructed from the text, but then it is not equation four, and readers anxious to get the patterns on their own machines cannot do so easily. This may be more a criticism of learned papers generally than those in Fractals, but it is a problem to readers with limited time and/or limited mathematical ability.

The other paper I read I found of greater interest, and this was entitled Long Range Correlation in Human Writings. I read it after writing the curlicues article elsewhere in this issue. Here, the authors had taken text and replaced each letter and some punctuation marks with binary strings, thus A becomes 00000, b 10000 etc. Case is ignored. The writing is reduced to strings of 0s and 1s, but the authors say other codes are as valid. A random walk program is then applied, with a 0 as a downward step and a 1 as an upward step.

The resulting waveform is then subjected to mathematical processes and various results obtained for different written works fed into the program. The authors concluded that "the explanations and origins of the scaling behaviour is still beyond us".

However inconclusive it may be, this article is one that generates interesting ideas in the mind of the reader. I wonder whether someone will discover a program into which one can feed text and get detailed and complicated fractal patterns depending on the text chosen etc.? I think that the concept of making images from text is an exciting one, artistically, scientifically and philosophically. Lets see more of it in Fractal Report soon!

Fractal Fiction: The Face

by John de Rivaz

Tarvin looked across the ruins of his office. The auctioneer had removed the last of anything saleable the day before, and he had been told to collect his personal belongings and hand in his keys when he collected his cards and redundancy money. He took one last glimpse out of the window - a window he never saw as he sat at work, as it had been hidden by a row of filing cabinets that had now been sold.

The London skyline had already been altered, by the huge steel spheres that had replaced many of the already demolished office buildings. Tarvin wondered what the now deposed Price Charles would have thought of that alteration to the skyline.

He picked up his personal laptop computer, and flicked through the programs on its massive gigabyte memory. Like everyone else, he had the "Linda Set" program, and had wasted many an hour looking for new faces. The fractal set, first discovered by maths student Linda Smith using a simple equation linking common surds, was originally thought to be unremarkable. It produced a swastika like shape with spirals and curls, beaches and bays and islands etc. But with improved personal computing people searched deeply into it had found a four armed spiral with little blobs off it. When magnified, these blobs turned into caricatures of the faces of important people in history.

Tarvin remembered the initial flurry of excitement when this was discovered, and instant fame for Linda Smith. The popular press soon dubbed the set "The Linda Set" as most of them thought the originator of Julia Sets to be female. There was a new boost for religions, and people dug deeper and deeper into the set looking for pictures of saints and religious leaders, and eventually for the face of God. Politicians were there, as well as scientists, explorers and artists. Heroes and villains also had their place, with one exception - no one had found Adolf Hitler.

The Linda Set spawned a new electronic craze - dedicated high speed personal computers designed to facilitate the search deeper and deeper into the spirals of history. Tarvin remembered the day that a live television show networked all the world's most powerful supercomputers to go deeper than ever before into the face spiral. World wide interest was greater than the moonshot or the UK's royal weddings and divorces. The spiral ended with what at first seemed a blob, but when magnification was extended, they found Adolf, right in the middle, with no more to go. It took humanity only a few minutes to reach the conclusion that Hitler and God were one and the same.

The establishment of most countries was based to some extent on religion, and the resulting collapse of law and order might have meant an end to civilisation, except for the sudden rise of an obscure movement that had been struggling for decades with less than 1,000 members.

Tarvin had been as keen as everyone else to sign up for cryonic suspension - freezing on death in the hope of reanimation and cure by future science - as an alternative to becoming a victim of the concentration camp in the sky. He was in a well paid job and could easily set aside the cost to be available upon his death, or "deanimation" as the cryonicists called it.

However, people were not ready to believe that strongly in anything now, and when an alternative proposal to make war on death itself appeared, politicians proposing it had landslide victories. Soon, the drive to efficiency in the war effort became relentless. The people of the world were united as never before against a common foe. It was better to be a floor sweeper in a biology laboratory than to be in the highest paid job in some irrelevant enterprise. If something wasn't part of the war effort, it was just rejected and ignored. Cryonics continued as a long stop insurance, but the main effort was then on stopping the ageing process - searching for the elixir of life. Anything else was a waste of time and effort.

As he left the building, Tarvin took one last souvenir of his past life of riches. He carefully unscrewed a brass plate from the wall:

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