The Sierpinski Curve E.J. Turl 2

Data Compression (Letter) Larry Cobb 2

A Bit More Detail, Please Graham Johnstone 4

Basic Landscape José E. Murciano 7

The Volterra-Lotka System E.J. Turl 8

The Bifurcation Diagram and Chebyshev Polynomials John Topham 10

Fractal Shareware on the Acorn Archimedes Phil Edmonds 15

x5-1 Newton Plot (image) Dr Ian Entwistle 16

Two Mandelbrot Images Cade Roux 17

Mandelbrot With Moving Colours José E. Murciano 19

Editorial and Announcements John de Rivaz 20

Fractal Report is published by Reeves Telecommunications Laboratories Ltd.,

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

Volume 3 no 14 First published April 1991. ISSN applied .


The article pile is now very low, and I would be grateful for some contributions for the next issue, preferably before the last week of April.

RTL offers Low Cost Video Standard Conversion

By the time this appears, RTL will be offering low cost digital video conversions on VHS from any tv standard to any other. Thus if you see a cassette you want that is only available in an alien standard, then send it to us and we'll send it back to you in your local standard. Please note that this is not a copying service - you get one cassette back! (Your original is erased and re-cycled.) You can't break the copyright laws by selling the original and keeping a standard-converted copy.

Anyone wishing to use this service, please write to us detailing what you want converted and we'll quote a price.

If you are a provider of specialist low volume videos, then we would be pleased to negotiate a drop-ship arrangement to despatch your products to the rest of the world, converted to run at the addressee's location.


Reader's Hall of Fame

What? No entries this time! Readers are reminded that if you get an article on fractals published in any mainstream magazine, please try and mention Fractal Report. Tell us, and we'll give your article a mention in this spot. To ensure that your mention of Fractal Report isn't edited out, include it as rem statements in a listing! These are usually reproduced photographically, and are therefore no so easily edited out.

Amygdala Issue 22

This issue had more of the flavour of Fractal Report than previous issues of Amygdala that I have seen. This is probably because the main article was one from a Fractal Report's contributor - Dr Ian Entwistle. He discussed Julia sets where the maximum count is relatively small, and there is an unusual loop termination condition. This is when an iterate escapes from a square rather than the more usual circle. As usual he gives the essential lines of the program and also expands the iterations into real and imaginary parts for readers - although he then goes on to say that a "deliberate mistake" gives prettier pictures!

The second article was a description of a lecture given to various university departments on the Mandelbrot Set. The rest of the newsletter was snippets of news. The second issue of Merz, their ad sheet of $30 rectangles, included many repeats, but readers of Fractal Report may be interested in an Amygdala spin off The Cellular Automatist. This has just started with no 1, and it costs $25 per 10 issues (USA), $30 (Rest of North America), $45 (rest of world.) [Box 219, San Cristobal, New Mexico 87564, USA: Visa and MasterCard accepted.]

A Chaos Application

Mr V.A. Crawley of Hailsham sent us a cutting from The Daily Telegraph concerning an invention made by the paper's science correspondent. He suggests a new version of the "one time pad" where a sequence of numbers is used to code and decode a secret message. A disadvantage of this is that users must keep large volumes or computer files containing lists of numbers to encipher their messages. Mr Robert Matthews suggests that a fractal formula is iterated to produce a string of numbers, the formula and starting point being the key to the code.

Of course, the RND generator in any computer behaves in a similar manner. Fed the same seed, it generates the same string of numbers! (The system may vary between versions of the high level language using it - look at your manual to find out how to get the same sequence every time.)

Nevertheless, Mr Brian Winkel, editor of Cryptologica is quoted in The Daily Telegraph as being enthusiastic about Mr Matthews' idea.

Rec Suggests Answer to Problem

Recreational and Educational Computing, an American newsletter produced by Dr Mike Ecker 909 Violet Terrace Clarks Summit PA18411 USA, mentions the Chudnovsky algorithm in its issue 5,8. This seems to be the answer to my query as to how one can work out the digits of from starting at a particular digit without going back to the beginning. However the item just mentions the name and the fact that two readers independently wrote in and asked about it, and goes on to ask other readers to detail the algorithm. So if anyone out there knows about the Chudnovsky algorithm, please let us all in on the secret! It would be nice if Fractal Report could beat Rec to publishing details.

Also in this issue of Rec the phenomena of Fibonacci music was discussed, with a GW Basic program to play it. "Wallpaper for the Mind" was also covered(!), but to a lesser extent to the coverage already given in Fractal Report. Anyone interested in Rec is referred back to their advertisement in Fractal Report 12, page 17.

P.oem - 12 hours of PC video from a CD ROM

Mr P. Moon sent us some details of Michael F. Barnsley's fractal video compression system for the PC. The full kit costs about �000 and consists of a hard wired card plus software. However clients can send their films on video cassettes and have them encoded. They can then play them on their PCs using a video player program costing about �0. Single frames can also be compressed, thus complex pictures can be included in programs using trivial amounts of memory. The program to reproduce single frames (Targa formatted images or photographs) costs � and the cost for encoding a single frame is �, or � for ten, or �5 for 50. An intriguing feature of the system is that when images are magnified, detail not present in the original is shown. Of course, the detail shown may not be present in the original scene.

Obviously these prices indicate a corporate market, but if these techniques could be developed to run in real time, there seems no reason why they shouldn't be used to allow real time video recording or transmission on audio media. With a mass market, the equipment could have a domestic price tag. Who would have guessed a few years ago that a multi-standard VCR and tv standards convertor would become a domestic item?

Machine Code Integer Mandelbrot for QL

C.G.H. Services have introduced an integer Mandelbrot program for the QL, written by Kenneth Murray. It costs �80 by post from C.G.H. Services, Cwm Gwen Hall, Pencader, Dyfed, Cymru SA39 9HA. Please state media required. The program uses 32 bit integer arithmetic, and performs the usual zooms using a menu system. There are provisions for altering the iteration limit, saving the screen and printing it on 9,600 baud Epson printers. Coordinates can be entered by a moveable box or numerically. The program uses the QL's "true" windows and multitasking features. They sent me a copy for evaluation, but unfortunately my usual apalling problems with time made it impossible for me to review it as throughy as it deserves. I can say that it gave quick (for the QL) 8 colour displays. Within the constraints of time, I attempted to get the SuperQboard to print the screen, with "par_use ser" but my NECP2200 just printed a series of line feeds.

They also offer 20 programs for the QL, together with two newsletters, QL Leisure Review and QL Technical Review, and a QL shareware service.

PC Mandelbrot Animation Program from Australia

Not content with sending us Cell Block H and Neighbours, we are now getting computer programs from those enterprising people on the other side of the world. A shareware example is from the Computer Gallery, PO Box 179, Maylands, WA6051, Australia. It is another Mandelbrot display program called The Mandelbrot Viewer and its animation facilities make it stand out from some others. It runs on all common video standards, and comes with a 25 page manual on disk. Registration is $Australian 20, in Australian currency on an Australian bank only. This entitles the owner to receive the latest version and go on a mailing list. I have sent copies to Frachaos and PC Star (both of whom have previously been mentioned in Fractal Report) and shareware enquiries can therefore be directed to either.

Fractint 15.1 Now Available

José Murciano kindly sent me a copy of Fractint 15.1 all the way from Spain. There are a number of new features, vastly improved presentation, a slight speed-up, and a manual of over 100 pages. I have sent copies to the aforementioned shareware dealers, to whom enquiries may be directed.

Mandelbrot With Moving Colours

by José E. Murciano


DECLARE SUB ShiftPalette ()

DECLARE SUB WindowVals (WL%, WR%, WT%, WB%)

DECLARE SUB ScreenTest (EM%, CR%, VL%, VR%, VT%, VB%)



DIM PaletteArray(15)

FOR I = 0 TO 15: PaletteArray(I) = I: NEXT I

WindowVals WLeft, WRight, WTop, WBottom

ScreenTest EgaMode, ColorRange, VLeft, VRight, VTop, VBottom

VIEW (VLeft, VTop)-(VRight, VBottom), 0, ColorRange

WINDOW (WLeft, WTop)-(WRight, WBottom)

LOCATE 24, 10: PRINT "Pulse una telca para salir.";

XLength = VRight - VLeft

YLength = VBottom - VTop

ColorWidth = MAXLOOP \ ColorRange

FOR Y = 0 TO YLength

LogicY = PMAP(Y, 3)

PSET (WLeft, LogicY)

OldColor = 0

FOR X = 0 TO XLength

LogicX = PMAP(X, 2)

MandelX& = LogicX

MandelY& = LogicY


RealNum& = MandelX& * MandelX&

ImagNum& = MandelY& * MandelY&

IF (RealNum& + ImagNum&) >= MAXSIZE THEN EXIT FOR

MandelY& = (MandelX& * MandelY&) \ 250 + LogicY

MandelX& = (RealNum& - ImagNum&) \ 500 + LogicX


PColor = I \ ColorWidth

IF PColor <> OldColor THEN

LINE -(LogicX, LogicY), (ColorRange - OldColor)

OldColor = PColor




LINE -(LogicX, LogicY), (ColorRange - OldColor)

IF EgaMode THEN ShiftPalette



IF EgaMode THEN ShiftPalette






EgaMode = FALSE


SUB ScreenTest (EM, CR, VL, VR, VT, VB) STATIC






VL = 110: VR = 529

VT = 5: VB = 179

CR = 15



VL = 55: VR = 264

VT = 5: VB = 179

CR = 3



SUB ShiftPalette STATIC

SHARED PaletteArray(), ColorRange

FOR I = 1 TO ColorRange

PaletteArray(I) = (PaletteArray(I) MOD ColorRange) + 1


PALETTE USING PaletteArray(0)


SUB WindowVals (WL, WR, WT, WB) STATIC


PRINT "Este programa hace la representacion grafica completa"

PRINT "del Conjunto de Mandelbrot. Por defecto la ventana es"

PRINT "de (-1000,625) hasta (250,-625). Un Zoon de una parte"

PRINT "de la figura. Se efectua entrando nuevas coordenadas."


PRINT "Pulse <ENTER> para la ventana por defecto. Pulse otra"

PRINT "tecla para entrar la nuevas coordenadas de la ventana: ";

LOCATE , , 1

Resp$ = INPUT$(1)

IF Resp$ <> CHR$(13) THEN


INPUT "X coordenada superior izquierda: ", WL


INPUT "X coordenada inferior derecha: ", WR


PRINT "Coordenada derecha debe de ser mayor que c.izquierda."



INPUT "Y coordenada superior izquierda: ", WT


INPUT "Y coordenada inferior derecha: ", WB


PRINT "Coordenada inferior debe ser mayor que c.superior."




WL = -1000

WR = 250

WT = 625

WB = -625



Fractal Report Issue 14 Page 19

Fractal Shareware for the Acorn Archimedes

by Phil Edmonds

I have recently obtained some excellent fractal shareware to run on the Acorn Archimedes. This consists of three programs on Careware disk no 9, available for Norwich Computer Services (see ads in Acorn magazines) for around �00. The three programs are supplied in a compressed archived format and must be uncompressed using a utility on the same disk. Once uncompressed, they are run as applications in the normal way, although only two of the applications run in the desktop environment.

The first application is called (imaginatively!) Fractal; it is a fractal pattern generator with operates in the background via a small window. It installs itself on the icon bar and once started by double-clicking on the icon produces fractal-like patterns in a random manner, including some that are similar to Martin's Mappings. It is possible to specify one's own parameters to produce new patterns via a menu system and to save the patterns produced to disk.

The second application is an extremely fast Mandelbrot/Julia set generator called SPEM. This program does not multitask via the desktop, but operates vis its own mouse/menu system. The usual Mandelbrot beetle is generated in about 15 seconds, and while more complex zooms take longer the program is reasonably fast. The program will operate to 64 digit precision, and it is possible to select the number of iterations used up to a maximum of 999. Zooming in is accomplished simply by selecting an area using the mouse and restarting calculation, and the results are displayed as they are calculated. The colours used may be selected by cycling through the 256 colour "palette". By simply holding one of the mouse buttons down amazing pseudo-random colour effects may be obtained. Parameters used can be saved to disk.

The third application is a 3D Mandelbrot/Julia generator/ray tracer. The application multitasks from the desktop and all functions are accessed via a menu system.

The program produces 16 grey scale ray-traced renderings of fractal scenes which may be saved as standard sprite files. Scenes are described by text files created in !Edit or similar. Parsing and ray-tracing are started simply by dragging the icon of the description file on to the application window. The simple command language used in the description files allows one to specify which type of scene is required (Mandelbrot/Julia), the complex number coordinates used, position and intensity of lamps, whether or not shadows are to be produced and much more besides. In common with other ray-tracers, image production takes a fairly long time, but one can watch the image appear in a window on the screen as it is calculated. It is possible to produce some quite stunning images with this program and the disk is worth obtaining for this alone. Fractal report Issue 14 page 15

Letter from Mr Larry Cobb:

I was interested to read Mike Parker's article about file compression in issue 13. Run length encoding (RLE) is a useful technique for reducing the size of graphics (and fractal images). It's great advantage is the relative simplicity of implementation but, as file sizes grow, other more powerful techniques should not be overlooked.

A logical extension of RLE, which recognises patterns data when they appear sequentially, is to build up a table of patterns and assign a reduced binary string to each so that this code can be substituted, no matter where the patterns occur in a graphics file. Such an algorithm is the Lempel-Ziv Welch, which has been used to great advantage by CompuServe in its Graphics Interchange Format (GIF), used for the transmission and storage of graphics. Some fractal investigation programs, not least DRAGONS and FRACTINT, and other more general graphics programs, such as Autodesk Animator, have adopted this standard for obvious reasons. It offers an increased compression of about two over RLE and, being a widely supported standard, it allows the interchange of files between computers, even between different types of machine. The GIF specification is freely available for study and use, without licence fees, from CompuServe International, who own the trademark.

It's interesting to speculate on where file compression can go from here. Can a more powerful standard be developed? There is clearly a need for better compression performance, particularly with the introduction of graphics cards like the IBM XGA. With 24 bits per pixel, these cards can give truly photographic results but where will we find room for the resulting super mega byte files? There is great interest at the moment in so-called "lossy" compression systems. In these a trade-off is sort between loss of picture quality (tiny in some cases) and improved compression performance.

The Joint Photographic Experts Group (JPEG) has developed a standard based of discrete cosine transforms - a method of resampling the data in proportion to the detail included in each area of the image. I can report that the results are very good for photographs but, so far, my tests have proved disappointing for fractal images.

Fractals themselves have been put forward as an alternative compression method - using the matching of fractal patterns to graphics images and then sending the resulting fractal equations.

These lossy systems are clearly a forward but they move the complexity from the transmission and storage media to the coding and decoding hardware. After all, I can "send" you a fractal with very little data:

zn+1 = zn + c

but no one has seen the end of that equation yet!

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