Fractals on the Apple II Series J.R. Parkes 4
A Brief Note on Time John de Rivaz 5
Equation Stability F.E. Alexander 6
Nothing But Zooms Amygdala 8
Bifurcation Diagrams - a Detailed Look John C. Topham 9
Program To Generate Land Surfaces Philip Hickin 16
Amiga Fractal Announcement Cade Roux 18
Editorial John de Rivaz 19
Announcements John de Rivaz 20
Fractal Report - The Video
I have had a few letters expressing interest in this project. However whether I myself am going to find the time to make any videos I don't know. Undoubtedly some of you will, and indeed one of you already has. This will certainly be a long term project, but if the results are pleasing enough could prove very worthwhile. Larry Cobb once pointed out that if you add together all the computing power of readers' computers you'd get something approaching a supercomputer, so theoretically it would seem that we could put together something of value given the time.
Oxford University student Cade Roux in a letter published in this issue points out that single frame video recording may not be as easy as it sounds, but I do know that time lapse camcorders exist, and it may be possible to connect an external signal into one of those. Also time lapse monochrome video recorders exist for surveillance purposes, and sometimes appear on the surplus market for about �80. They use 0.5" reel tapes, and a tape that plays in one hour can be recorded over 48 hours, giving a substantial speed up. I am only sorry I never snapped up the one I saw advertised, because I haven't seen one since. However they do exist!
Fractal Music
After an initial dearth, we seem to have come across a few examples. This time we are offering a cassette and compact disk, and an order form is enclosed. The disks will be ordered from America as required, so unless you are really desperate for a CD, I would recommend the cassette. Surface mail shipments form America are notoriously slow as those of you waiting for Nothing But Zooms videos will know. The audio cassettes should be available as soon as initial demand is known, as they are produced by our local firm who are very quick and efficient. However the fact that the musicians have taken the trouble to make a CD, which is very expensive, indicate that they feel sure of their product.
We also know of a local pop group, with both male and female lead singers and various instrumentalists, who have adopted the Mandelbrot Set as their icon. However they have no computer and aren't practical fractal enthusiasts, although some of their instrumental music does appear to have the "same but different" fractal quality. Karen feels that although the music is good Fractal Report readers won't be interested, but I may offer a cassette at a later date if no further computer generated music appears. Certainly if your preference is for rock music you may well prefer their cassettes to our previous offerings. (Do not confuse this with the above offer - the pop cassette isn't on offer this time.)
Call For Articles
At the moment our article file is almost empty, so new articles are sought for issue 10. To remind contributors, the ideal size is two A4 pages camera ready, including some lines of BASIC that describe the basic process being discussed. Please bear in mind economy of space. Past events have revealed that mistakes can emerge even during paste-ups, so the only way of being sure it goes in exactly as you want it is to send it in a suitable form! There is no need for menus and frills in BASIC . In addition if contributors are using PCs, the text on a 5.25" (only) disk would be appreciated as it would enable me to laser print it.
A 3.5" debâcle
A number of people have sent me 3.5" disks. In order to accommodate readers who didn't read my previous editorial about not my having a 3.5" drive and who continue to send these disks I recently bought a drive which broke my controller. In view of the aggravation of this, I have decided not to install a 3.5" drive, until I have more time, and therefore cannot accept these disks. Many thanks to all those who kindly send me programs on these disks, but I am afraid that I cannot use them.
New Newsletter
Mr Andy Lunness has sent us issue 1 of his newsletter Chaos and Complex Cartography. Its motto is Eadem mutata resurgo - "I shall arise the same though changed". (More suited to a cryonics society, I should have thought! Tempus Hostilis Est would be a good motto for Fractal Report, judging from the number of letters I receive and send detailing ambitious projects abandoned through lack of time.) UK subscriptions to 6 issues of Chaos and Complex Cartography cost �7, and the overseas cost is �11. (Cheques to "C&CC") They hope to get six issues out each year, but ask readers to be patient if this time schedule is not adhered to. Subscriptions last for six issues regardless of when they appear. He says that he doesn't aim to rival Fractal Report but to compliment it with articles on the more theoretical side of chaos and fractals. Computer programs as such won't be published, but algorithms will be given. These are relatively easy to transfer into computer programs if you have a structured BASIC like the QL's Superbasic, or the PC's Turbobasic. Those of your familiar with The Science of Fractal Images will know the format these algorithms take. Of course they have the advantage that they are not machine specific, but the disadvantage that there is no absolute check that they work!
Issue 1 has three main articles. The first is an introductory article by Mr Lunness which introduces the reader to the terminology and presents some simple but interesting examples. The second is a review of James Gleik's Chaos - the Making of a New Science and the third an article on drawing fractals from Newton's Method for solving cubic equations. Not mentioned in the contents list there is also a tutorial on the mathematics of complex numbers, and a mathematical proof of the symmetry of the Mandelbrot Set. The newsletter is the same size as Fractal Report, and all the articles are retyped so as to be in the same typeface. This is nice to look at, but runs the risk of causing extra errors to creep in, particularly when mathematical equations are given. Also it is highly consumptive of editorial time that may be better applied elsewhere.
At the printing of the first issue Mr Lunness only had ten supporters, so those of you wanting a more theoretical and mathematical approach may well find his newsletter worth encouraging as a compliment to your Fractal Report subscription. I can't see it lasting very long unless it is supported better than this.
With regards to his article on de Moivré's theorem, he says that it will work for any positive or negative integer. He also states "It is also valid for fractional powers. However the result maps itself onto a Reimann surface, not the complex plane. Rendering this surface onto a plane is a complicated affair which I cannot do, hence the discontinuities. It is also possible to have a complex
power. However the maths is very complicated and the basis for a future article when I can simplify it a bit."
Fractint
Last issue I lamented the problems at Personal Computer World with Fractint, a PC shareware fractal program. Mr Adam Case has very generously offered to copy his shareware Fractint 9.1 free to anyone who sends him a blank 5.25" or 3.5" PC disk formatted to low density (ie 360k 5.25" or 720K 3.5") together with return postage. Send disks and postage to Mr Adam Case, 109, Ferry Road, Hullbridge, Hockley, Essex, SS5 6EL. A similar offer has been received from Jo Gedrych, of Rose low, Whimple Road, Broadclyst, Devon EX5 3BX, only he will fill two 3.5" 720K disks with (presumably a longer version of) the program and documentation, or he will supply disks at cost. A number of people also offered to provide the program for a copying fee, but it seemed pointless printing this in view of the above.
I have now tried Fractint, and can assure readers that it is well worth getting. The reviews it has been getting in magazines like Personal Computer World are well justified. However you just have to use it as given, it is an .exe file written in a mixture of C and assembler. Although it covers many areas, Mandelbrots, Julias, Lambdas, Newtons, planets and landscapes to name but few, there will always be room for the new programs covered in Fractal Report. For example, in version 9.1, higher powered Mandelbrots and Julias were not covered.
Major New Fractal Book
Dr Clifford Pickover has come up with another of his very generous offers to Fractal Report readers. He has written a colour illustrated book totalling no less than 400 pages entitled Computers, Pattern, Chaos, and Beauty, packed with recipes and algorithms for producing images. Martin Gardner the Scientific American computer columnist, said that Dr Pickover has produced a truly stunning survey of the manifold consequences of Chaos and Fractals. Paul Hoffman of Discover Magazine says "Dr Pickover takes the reader on a stimulating odyssey through the world of computer graphics, a world that surprisingly involves the Shroud of Turin, snowflakes, and the genes that cause Cancer."
Dr Pickover has offered to answer any queries implementing the programs in the book, but he asks that queries are sent to Fractal Report and then sent on to him in a bundle. The reason given is that he doesn't want lots of correspondence arriving at his workplace. I know that we have published his address in his announcements before, so please do enter into the spirit of this. We will probably mail enquiries on a monthly basis, so do not expect any quick response, and it may well be that Dr Pickover will want to analyze the commonest queries so that he can formulate a general reply for an appendix for a future printing of the book. So I would ask readers to be patient.
The book will be available in the UK from Alan Sutton Publishing, Phoenix Mill, Far Thrupp, Stroud, Glos GL5 2BU, for a special pre-publication price to Fractal Report readers of �21.50 delivered. This includes �1.50 postage and compares favourably with a publication price of �25. The books measures 246mm by 172mm and has 400 pages with 200 black and white illustrations and 8 pages of colour illustrations. However, to obtain this special price, please mention Fractal Report and also please post your order to arrive BEFORE the publication date of 28 June 1990. Those ordering after publication from booksellers may like to quote the ISBN which is 0 862 99 792 5.
US readers are asked to write to Mr Garrett Kiely, St. Martin's Press, 175, Fifth Avenue, New York, NY10010, USA. He will accept credit card orders, and the book costs about $35 with carriage. (At the time of writing the exact price with carriage is not known).
I also understand that Fractal Report will be mentioned in this book. This will give an enormous boost to our readership both in the UK and abroad. It will result in an increasing range of material being offered as articles, and should benefit everyone concerned.
As far as I am aware, this book is the first book that is similar in content to Fractal Report inasmuch as practical information is given. It is a must for anyone interested in getting these patterns to appear on their own computer screens as opposed to reading what other people have done, without a clue as to how to do it themselves.
A Canadian Fractal Enthusiast Comments
Mr Jon McLaren offers a �25 reward to anyone who can give him the code (for his PC with 8087) that will generate the image on page 57 of the December 1986 Scientific American. He also offers source code in 8087 microsoft assembler and BASIC that computes Mandelbrot and Julia sets using only the 8087 registers, and a Landscape program that runs on EGA or VGA that he believes is superior to the one published in Fractal Report. He expressed his support of Fractal Report authors who concisely explain the maths used and then gives executable code that shows how the maths is converted into BASIC. He cites particularly Ikeda Map and Coloured IFS Tilings as being good examples of this. He hopes that we will get plenty of articles from post graduate students, and that we will be printing code that enables people to get true 3D images of fractal objects. He suspects that it would be no big deal for such a student to supply us with a "light source z-buffer algorithm". Mr McLaren is certainly keen. He tells me he drove 140 miles just for a copy of Generation and Display of Geometric Fractals in 3-D by Alan Norton, only to be disappointed by the lack of code or enough detailed information to enable him to write it. He also told me how he spent on 'phone calls more than the price for which Dynamical Systems and Fractals is marketed in order to obtain a copy. His address is 4384 Bennett Road, Burlington, Ontario, Canada L7L 1Y7.
Certainly I do think it would be useful to have a program module into which one can input x, y, z, and colour which will draw an image illuminated from point x1,y1,z1 with colour(s) c1...cn and be viewed from point vx,vy,vz and have hidden line removal. The module should have universal application, unlike the Mandelbrot and Julia routines that I and others have been working on which rely on plotting particular points first. My suspicions are that such a program will require huge areas of RAM and be particularly unwieldy on PC systems. Hopefully there is some way around this. Such a module could be implemented for a number of different machines and be used by Fractal Report authors and readers for different fractals.
Fractal Novel
A science fiction novel based on fractals will be appearing at Christmas from Gollancz and Bantam Books. It features the Titanic and the Mandelbrot Set. I cannot wait to find out the connection!
I will publish further details as soon as I am given the go-ahead to do so, but as there is a worst case three month delay between receipt of information and publication in Fractal Report many of you who are science fiction enthusiasts may find out before I publish anyway. The main purpose of this note is to save you all wasting stamps writing to tell me about it!
Fractal Software Services
A series of Public Domain and Shareware fractal related software disks for the Amiga. I have a very large collection of PD/Shareware fractal software, which I am currently arranging onto disks. This is just a fraction of the material I have.
Mandelpicts:(John Shortle) A sideshow of Mandelbrot 3.00 images.
Mandelbrots:(Michael Todorovic) A sideshow of MandelVroom images.
Mandel_Gallery 1 & 2 (Under construction)(Cade Roux) The Mandelbrot Gallery, a two disk set, the first containing a sideshow of eleven interlaced images, the second containing thirteen sample images demonstrating each program in the review contained on the disks.
FractalSample1:MandelMountains V1.1 (Mathias Ortmann), EHBPlasma 1.f(Roger Uizun), Chaos (Richie Bielak), FracGen 1.23(Doug Houck) Mandelmountains uses the CPM method to produce extremely smooth 3-D plots of the Set. EHBPlasma produces plasma clouds in 64 colours. Chaos produces Hénnon Maps. FracGen is a versatile fractal generator. This disk has a wide variety of programs since it is the first disk.
FractalSample2: Mandelbrot Set Explorer (Thomas Wilcox) This is the original generator for the Amiga. It has very good palette control, which makes for spectacular pictures.
FractalSample 3 & 4:(Under construction): MandelVroom 2.0 (Kevin Clague) These two disks contain everything you need for a full MandelVroom system, source and lots of images for you to play around with. MandelVroom is the most polished Amiga Mandelbrot program in existence, and supports all kinds of features (88020/881, all modes except 4096 colour mode, multitasking, zooming, panning.)
FractalSample5:(Under construction) CPM (Lars Clausen), DEM (Lars Clausen) MandelShow 1.1 (Nic Shulver), Morell CPM is a good 3-D generator, which has shadows. DEM produces excellent black and white images. MandelShow 1.1 is a standard generator with basic features. This disk will contain a lot more material.
Ordering:
(Cheques/checks payable to Cade Roux)
The disks will all be �2.50/$4.00 each (postage included).
Free services:
1. For a blank disk and postage to the amount of original mailing (UK/US stamps, outside of UK or US no postage required), I will send the requested disk.
2. For a disk containing freely distributable PD or shareware (hereinafter called full disks) I will customise any software I have onto a disk. Please write for a �2/$3 catalogue disk or free printout of the software lists (SASE required).
3. For two full disks, I will send three disks, two of which can be customised.
4. Fractal report readers: FractalSample1 special offer for �2/$3.
Interested in a disk based newsletter? I am starting an Amiga fractal newsletter/magazine of images and programs as well as articles. This will be PD, so all contributions should be allowed to be distributed as part of the newsletter. Anything remotely related to Amiga fractal experimentation is welcome as are reviews of relevant commercial packages, animated sequences and anything fractal. Current plans are for a three disk set at �5/$8, or less if possible. All Feedback to me,
Cade Roux
UK USA
Gonville & Caius
College, Box 3
Cambridge COMSCEUR
CB2 1TA FPO NY 09553-2000
by John de Rivaz
As so many of the letters I send and receive contain some comment about lack of time for various fractal projects, I thought I'd include a brief note of how a future technology could revolutionise life by control of time. If you think that the very idea of the control of time is the ravings of a lunatic, then I suggest that you read New Scientist 28 April 1990 pages 57-61 Wormholes, Time Travel and Quantum Gravity by Ian Redmount. This suggests theoretical grounds for considering that our universe supports time manipulation. My concept, first published in The Immortalist in December 1989, appears below:
It has been said that if anything can be imagined one day it will be done. Something that has been imagined by fantasists and science fiction writers is the time destructor, but most rationalists would regard the concept as nonsense.
A time destructor would be a device that enables the user and everything within his area of influence to move through time slower than the universe as a whole, thus he would be able to get more done than those around him. In a way, tools are time destructors, for example a man with a mechanical digger can move far more dirt than one with a pick and shovel, who can move more than one using only his bare hands. However a true time destructor would be far more than a simple tool. To someone using it, the rest of the universe would appear frozen whilst he caught up with whatever he was doing. H.G. Wells wrote of The New Accelerator, being a drug that speeded people up, and a recent Twilight Zone story featured a mechanical device.
But is this idea basically impossible? It depends on whether time has one or more dimensions. Some say that there are an infinite number of dimensions (why stop at 4?) If this is the case, then maybe what we regard as time are all those dimensions from four up. If time has more than one dimension, then a time destructor would be possible, as when activated the user would simply be creating more time for himself and the selected task in another dimension.
A more serious difficulty is the energy consideration. It may require very large inputs of energy to operate a time destructor. All ideas for time travel devices, such as the one reported recently in The Immortalist*, require the movement of astronomical objects through space - hardly projects for the average individual to use in his own home. However these concepts do show that maybe it could be done one way, and if it can be done one way them there is always a chance it can be done another.
A difficulty that has been touched upon by science fiction writers is that anyone using a time destructor would age according to their body time, not the universe's time. Therefore to a bystander although they would appear super efficient and unflustered, they would age quickly. Of course the complete absence of stress (practically all stress is time-related, or at any rate could be relieved by a time destructor) would lengthen body-time lifespan measurably, but nevertheless if they used the device a lot they would age quickly to an observer. However this difficulty is more apparent to our level of technology, as one capable of making a time destructor would most likely have conquered ageing long before.
In a society where time destructors were commonplace, individuals would live in their own universes, connecting to the common time-stream only when they needed further input from others, or to present the results of whatever task they were performing.
This would be in complete contrast to the present world, where devices like the telephone and cell-phone allow others to intrude upon our lives, regardless of whatever we are struggling with at that moment. An interesting corollary to people living in "the time age" would be that if you wanted to contact someone and he was in his own "time-bubble" you wouldn't have to wait for him to come out of it, as he would leave it at the same instant he entered it, according to the main time path. As long as you were on the main time path yourself, then you can contact your friend.
*Similar to proposals in New Scientist.