Editorial John de Rivaz 2
Pictures from Complex Polynomials using Newton's Algorithm John C. Topham 3
A Three Dimensional Julia Plot with Z-Trajectories John de Rivaz 10
Space Filling Iterative Patterns Dr Hugh Daglish 11
Ikeda Map Dr John Corbit 14
Coloured IFS Tilings Dr Uwe Quasthoff 15
Intersecting Universes Michael Kirsch 18
We have had a number of announcements from readers. Mr Bob Harris, of Holly Oakes, Mill Lane, Lymington, Hants, SO41 8LN has some Commodore 64 Mandelbrot and associated programs. If you send him a blank 5.25" disk and return postage and packaging, then he will fill it for you. If you send him two disks, then he will send some 3D wire frame programs and files as well. His offer is in the interests of encouraging more Public Domain C64 programs in this area.
Mr P.J. Tewkesbury, of 26, Gathorp Road, Northern Moor, Manchester M23 0AS has an Amstrad CPC6128 and offers a Hisoft Pascal Mandelbrot generator. A blank 3" disk and �1 gets you this program. A full machine code version is in preparation.
Mr N.W. Fenwick of 164a, High Street, Swanage, Dorset BH19 2PE will send a collection of Amiga A500 Public Domain software for �2.50 including disk, postage and packing. He has also experimented with single lens reflex VDU photography, and recommends using a tripod with cable or timer release in pitch darkness. A shutter speed less than 1/50 sec, say 1/15 sec on 100 ASA film produces a good print. He sent us a print that shows off the Amiga's excellent capabilities on the Mandelbrot Set.
Ms Heather Whitby, of The Association of Maths Teachers, 7, Shaftesbury Street, Derby DE3 8YB, has mentioned her organisation's sets of "Logo Trees" (�3) and "Mandelbrot" postcards (�1.50) delivered. The former includes a program. Although only in monochrome, you get good value for each set, and they can make interesting and attention getting means of communication. "Logo Trees" also has a frieze of printouts and some Logo listings. The Association has an open challenge to produce realistic willows, cypresses, poplars, and oaks.
Welcome to the second volume, and many thanks to all those who renewed their subscriptions. Below I mention some American contemporaries. This will serve two purposes. It will provide readers who wish to expand their horizons with sources for further information, and there is the hope that the editors of these journals will publish similar reviews giving their readers the same opportunity. The subscription arrangements for Fractal Report in America (only) require a check for $23 made payable to "J. de Rivaz". This brings by airmail six issues backdated to the start of the current volume. $46 brings volume 1 as well, together with issues 0 & -1.
Some people said they like reviews of conferences etc., although they are few and far between. I would direct these people to an excellent and polished review newsletter called The Dynamics Newsletter published by Ralph Abraham at the Aerial Press, PO Box 1360, Santa Cruz, California 95061-1360, U.S.A. I know it's American, but the coverage is worldwide and if you get out and about to go to conferences, then you'll not mind raising the few dollars that a subscription requires. It an excellent but struggling publication, so if you like this sort of thing then it needs your support. Write to Mr Abraham for a sample issue and subscription details. He would also be interested in conference announcements or reviews of past conferences, for publication in The Dynamics Newsletter.
At this juncture I can't fail to mention the other American newsletter that is aimed at more experienced fractal enthusiasts, Amygdala. It spreads its net wide as to article type, but it does come up with some real gems at times. Many of you have bought copies of the QL machine code Mandelbrot and Julia programs. The core of these originated from an article in Amygdala describing an Amiga program. Mr Richard V. Robinson's fractal music cassette came from another Amygdala article. (See issue 6 editorial for information.) Although these gems will appear in Fractal Report, probably in more accessible form eventually, there will be a long time delay. Therefore if you want to be in the forefront a subscription to Amygdala would not come amiss. Address: Box 219, San Cristobal, New Mexico 87564, USA.
Nearest to Fractal Report in content is Dr Clifford Pickover's bi-annual newsletter The Journal of Chaos and Graphics. Dr Pickover isn't as strict as I am in aiming for practical information, but many of the contributions are similar to Fractal Report articles. It appears that some early issues were sent out free. Dr Pickover can be contacted at the IBM Watson Laboratory, Yorktown Heights, NY10598, U. S. A. He is also anxious for written contributions to this and other journals.
Although some of you wanted an arrangement to pay for American subscriptions in sterling, we haven't come up with a formula that is economic of money and time to make this possible. For those readers who might want to buy things by mail from America, it is not that hard to get a $US dollar account. Ask your UK bank for an introduction to an American bank where you can open a dollar account by mail. If you keep more than two or three hundred dollars in the account, then there are no banking charges on some accounts. It has not been illegal for British people to have overseas accounts for some time. Mrs Thatcher returned this freedom soon after she came to power. It was denied citizens during the war in most countries, but the UK with its history of non-Libertarian governments was one of the last to restore it.
A Three Dimensional Julia Plot with Z-Trajectories.
by John de Rivaz
First may I apologise to any author who feels that his article has been displaced by this one. However I feel that this illustrates how a few simple lines of BASIC (in this case GW BASIC - I have now got a PC, and my Spectrum has given up the ghost in disgust!) can produce an interesting display, and what's more it's fun to watch it build up.
The 3D Julia pattern follows ideas given by Lewis Siegel in Fractal Report 5 and the Z-Trajectories follow ideas given by Ed Hersom in Fractal Report 1, and others in unpublished works. The Julia plot is built from the program by Liam Proven in Fractal Report 4.
The program produced below has the 3D Julia centred (when fully displayed) at the lower portion of the screen. On the top right had side there is a "sun" consisting of all the Z-trajectories superimposed. On the top left hand side all the z-trajectories for a single line are superimposed, and erased at the end of the line. In addition, pressing the "B" key at any time causes the z trajectory to be blanked at the end of each point's iteration, and pressing "N" returns it to the next line condition.
100 SCREEN 9:A%=350:B%=600:COLOURS=16:KEY OFF
110 XMIN=-1.75:XMAX=2:YMIN=-1:YMAX=2.25
120 P=-.39054:Q=.58769:CMAX%=100
130 DX=(XMAX-XMIN)/(A%-1):DY=(YMAX-YMIN)/(B%-1):C1%=1:M%=200:CLS
140 FOR NY%= 0 TO A%-1
150 FOR NX%= B%-1 TO 0 STEP -1
160 K%=0:C1%=C1%+1:IF C1%=COLOURS THEN C1%=1
170 X=XMIN+NX%*DX:Y=YMIN+NY%*DY
180 K%=K%+1:XN=X*X-Y*Y+P:Y=2*X*Y+Q:X=XN:'label dloop
190 IF (X*X+Y*Y)>M% THEN C%=K% MOD COLOURS :GOTO 230 :'dplot
200 IF K%=CMAX% THEN C%=15:GOTO 230:'dplot
210 PSET (550+X*5,70+Y*5),C1% :PSET(100+X*5,70+Y*5),C1%
220 GOTO 180 :'dloop
230 XX%=NX%+.9659*NY%:YY%=450-100*K%^.2-.2588*NY%: 'dplot
235 IF POINT(XX%,YY%)=0 THEN PSET(XX%,YY%),C%
240 IF INKEY$="b" THEN BLANKING =1:BEEP
250 IF INKEY$="n" THEN BLANKING =0:BEEP
260 IF BLANKING AND C%<>1 OR NX%=B%-1 THEN LOCATE 1,1:FOR J=1 TO 11:PRINT " ":NEXT J
270 NEXT NX%
280 NEXT NY%
This program is given not just as something to type in and run, but something to play with. You can change the Julia coordinates in line 120, for example. Can you find a quicker way to blank out the left hand Z-Trajectories than printing strings of blanks (line 260)?