ISSN 0964-5640
And Innovative Computer Applications
Letters 3
Two Algorithmic Composition Programmes Dr Gabriel Landini 4
Fractal Music II - Driving the MPU401 Interface Dr Gabriel Landini 6
Poetry and Computers Gerald England 8
A Simple Model of a Gravitational Lens John Topham 11
Fractal Models John Sharp 14
Applying the Epsilon Cross Method to the Mandelbrot Set Warut Roonguthai 16
A New Type of Fractal Procedure: Chase Functions Roger Bagula 18
Dimensional Iteration In The Mandelbrot Set Roger Bagula 19
On Good Books About Fractals Roger Bagula 20
Fractal Report is published by Reeves Telecommunications Laboratories,
West Towan House, Porthtowan, Truro, Cornwall TR4 8AX, United Kingdom.
Volume 6 no 31 First published January 1994 ISSN 0964-5640.
In the absence of any suggested title that stirred my imagination, I decided to keep the title Fractal Report and add a sub-tile And Innovative Computer Applications.
Many regular readers will prefer an emphasis on fractals anyway, but are nevertheless still likely to be interested in the other ideas introduced.
John Sharp's idea of models from cut a folded paper is something that I hope other writers can build upon. At the moment his models were drawn on a computer by hand (as opposed to the computer calculating where to put the lines). However if this can be extended to the computer producing the cuts and folds, then the idea of 3D computer generated fractal objects first mentioned at the beginning of Fractal Report will have at last appeared.
There was a good response to my call for articles, with the result that there are still some in the file. However more are needed for the next issue, so please get your thinking caps on for the deadline of the end of February.
Newsletter in Polish
Dr Mieczyslaw Szyszkowicz has written to say that he edits a newsletter in Polish PIB Punkt-i-Bit. He is also on the editorial board of an international journal Machine Graphics and Vision which is established by the Polish Academy of Science.
He would be interested in papers on fractals and chaos. (address: 110B, Twyford Street Ottawa K1V 0V7 Canada)
Nonlinear Nonsense
This is the eye catching title of Dick Oliver's free newsletter covering Fractal Modelling, Chaos Science, Artificial Life and in fact all creative graphics. As may be expected it features products he sells from his company Cedar Software, whose new address is RR 1 Box 4495 Wolcott VT05680 USA. However the newsletter is informal in style, and the articles read like magazine articles, not catalogue descriptions. In addition to his own material, shareware and public domain software is reviewed and offered at low prices.
The main article of the Summer 1993 issue is Graphics Gurus Reveal Their Secrets. Another article discusses 3D Fractals and Cedar's latest program Polyray.
The Fall issue for 1993 features the Windows version of Fractal Vision. This is available both in disk and CD versions. The CD version comes with masses of examples and additional shareware graphics programs. The CD version costs only $29 dollars, or $44 if you send for an instruction and user manual published by Sams. (The disk version costs $39 with book.) Overseas orders are sent airmail postage at cost, and paying by credit card is the preferred method. Otherwise send for a quotation.
The book includes a chapter on programming for Windows in C++, but apart from this all the other information in the book is on the CD.
Also in the fall issue is a "how to" with Povray listing from the Sams book Tips of the Graphics Gurus to make a folded carpet thing from a Sierpinski Gasket, and an offer of brightly coloured fractal ties for $24 each, T-shirts $15 and art prints $15, $30 matted.
The newsletter is free to anyone who asks for it, anywhere in the world, so you have nothing to lose by getting on the mailing list, and maybe plenty to gain if you find some of the programs are for you. Mr Oliver says that Europeans and Australians seem very good about ordering something, so it is worthwhile sending free newsletters there. (I must sat that it may have something to do with the low prices when translated into pounds etc.)
Physics Software
The catalogue of the Academic Software Library contains some new and interesting titles which are very much in keeping with the intended readership of this magazine. The organisation publishes computer programs designed to increase understanding of physics, and the intended audience are educators and students.
However the catalogue makes interesting reading for individuals, and with program prices from between $25 and $100 the cost is not unreasonable. Each program comes on disk and is accompanied with a well printed paperback book explaining how to use the program to further understand the topic covered. The books include sections for lecturers and students.
Topics covered include quantum mechanics, relativity, thermodynamics, chaos and fractals, magnetic systems etc.
For your free copy of the catalogue, write to The Academic Software Library North Carolina State University PO Box 8202 Raleigh NC27690-0739 USA.
More Translight
Roger Bagula has sent more copies of his donation supported newsletter Fractal Translight Newsletter. The issues dated April and May 1994 were received via time warp on 5 November 1993, so they will be reviewed in this issue of Fractal Report which will appear in February 1994.
One thing I was pleased to note was that he had reduced his "minimum donation" for people living outside the USA to $50/yr. These two issues contained the usual mixtures of programs, art and comment. The programs were in BASIC for the Amiga, but translation to a PC BASIC is trivial and can be done whilst typing them into the computer. There was another musical score in the April 1994 issue, entitled Bass Blue. The reverse block printing of the score made it very difficult to read, but it seemed to contain four tracks of 13 bars, for two pianos, jazz guitar and electric bass. (I am no musician, so I can't comment further.) Also there was a short story, a Western, The Rescue of Dreadful Gulch. The small print would suggest that Mr Bagula might well make a killing by selling these magazines to opticians to put in their waiting rooms! anyone attempting to read it will soon be convinced he needs glasses.
As usual there were many new fractal iterations, all the more interesting because one could clearly see natural patterns in them. An example is Dimensional Iteration of a Game Value One 2 by 2 Matrix for Real Domain. The resulting printout resembles a bubble forming on a surface. Parts of Creutz's Potts Spin Z2 in Z'=Zs+C looked much like an insect. Mr Bagula has also experimented with doing two iterations at once.
The cover art, from fractal designs, is improving, and now the "price" for non US residents is down to a more reasonable $50 Fractal Translight Newsletter is well worth considering.
R.L. Bagula 11759 Waterhill Road Lakeside CA92040, USA. (Which address he calls "The Home of Outlaw Science".)
For those of you with Amiga computers, Mr Bagula has a range of disks with programs and images on sale. These are very reasonably priced, and I suggest that you write and ask for the current list. The programs are all short, but he says he has written far more than he has ever printed, and the time saved typing them in would be phenomenal. Someone with both an Amiga, a PC and plenty of time may be interested in a collaborational venture to convert them onto a PC disk in a PC supported BASIC.
Incidentally, I have noticed Amigas are now very cheap on the second hand market, presumably because most people want PCs these days because of the software. So anyone wanting a computer good for fractals with a good display and willing to buy second hand could do a lot worse than spend a couple of hundred on one of these. Obviously try and get the vendor to show it to you working before parting with your cash.
REC 55-56
Recreational and Educational Computing nos 55-56 arrived here recently (November). The editorial contained an item on the difficulties with time and running a newsletter, and in particular with people who write in abusive letters because they have not had their work accepted. Dr Ecker points out that he is not a psychiatrist and therefore it would be improper for him to comment too deeply on people's motivations. However he finds that his time spent on the newsletter has by and large been well received. This is even so when he has been unable to publish work he describes as brilliant and creative. Obviously he has to ration his time, so if someone writes twice week he may not get every letter answered by return.
I find similar problems. Last week another of my publications received a letter of abuse because a subscription order the customer had posted on Friday had failed to materialise by the following Wednesday. In fact it crossed with the letter of abuse: various delays in the post had accumulated.
Anyway, REC 55-56 contains several fractal items, including a reprint from Fractal Report 28, José Murciano's Interset. Dr Ecker typed the program in, altering it for his version of BASIC. Incidentally, he asks what "INSTAT" means. This is TurboBasic's shortcut for "If any key has been pressed" and is a very useful break point when experimenting with programs. It is much faster than inkey$, input$ etc., because it doesn't have to say which key is pressed. His comments add value to the original listing
Also in the fractal domain was an article entitled A New Fractal Idea and Cyclic Chaos. The latter was from an article that was to have appeared in the now defunct programming magazine Algorithm.
Computers and Graphics
Dr Clifford Pickover welcomes interesting, well-written articles for the Chaos and Graphics section of the international journal Computers and Graphics (Pergamon Press). He edits this section which appears in each issue of the journal. Topics include the mathematical, scientific, and artistic application of fractals, chaos, and related subjects. The journal is peer-reviewed. Colour is published where appropriate. For more information, contact [email protected] or send papers to Cliff Pickover IBM Watson Research Center, Yorktown Heights New York 10598 USA.
Chaos and Weather Forecasting
An article in The Financial Times of 18 December said that the Meterological Office was coming to terms with the fact that chaos theory makes it impossible to forecast weather for any length of time more than a day or so. In fact they are using chaos theory in computer programs to improve long term forecasting, but this is not for the public or even small business. For �5,000 per year you can get monthly forecasts based upon running several chaotic computer models and taking a weighted average of the results. Even then three in ten of these highly expensive forecasts can be wrong.
From Mr Mike Kirk
Why so few subscribers to Fractal Report? I do not think that interest in the subject is fading. If I look at the shelves in Ellipse, Geneva's computer bookshop, there is a fair number of French and English titles classified under fractals and chaos. Similarly, Orell Füssli in Zürich and Staufacher in Bern can offer a good choice of German and English books. Perhaps this is the reason why Fractal Report is not doing so well. There is now so much other material available. What else was there to satisfy the curiosity of the home computer user when Dewdney's article on the Mandelbrot Set appeared in August 1985 edition of The Scientific American? Not much, and then Fractal Report arrived and provided a useful service. Dare I suggest that the service is now complete, that you should stop Fractal Report (leaving us with memories of an interesting magazine) and not just let your journal fizzle out?
It would seem that (???) Report is going to evolve into something like Scientific American's Computer Recreations with the addition of listings. They were always the pages to which I looked at first so that can't be too bad. I am curious to see where you are going and I have renewed my subscription for another year.
From Mr Roger Castle-Smith
I noted from your editorial in Fractal Report 29 that the future of Fractal Report must be in serious question if there is not a greater flow of articles; this dismays me. The generation of FRACTALS was certainly a craze at one time and I greatly fear that this craze, like so many others in the past, is running out of steam. There have been many articles in Fractal Report which were interesting from the purely mathematical point of view but I suspect that a large proportion of your readership is just interested in the resulting "art" and too much of it becomes a bore after a time. I certainly subscribe to the idea that the only way to give Fractal Report a sporting chance of surviving is, as others have suggested, to widen its scope. The question therefore arises "what should that scope encompass ?". I feel sure that one of the main attractions of Fractal Report is its slant on practical programs with which one can experiment and this underlying theme should continue. I suggest that the following should be considered seriously for inclusion in the title and scope of your publication:
Recreational graphics
Recreational computing (although perhaps covered elsewhere)
Programming hints and tips
I would be prepared to help start the ball rolling with an article on Transera's HTB (High Tech Basic) which will run on a DOS PC. It is almost totally compatible with Hewlett Packard's Rocky Mountain Basic. This I could follow up, or perhaps precede, with an article on the use of the GOTO statement which causes so much irritation in the hands of the expert
My final point concerns cost. The last increase was not inconsiderable although I feel sure justifiable under the circumstances. But my own view is that a further substantial increase would be bound to frighten off many which might regrettably have to include myself.
From Mr John Sharp
Thanks for the latest Fractal Report. Sorry I have not written earlier, but this year has been very full, since I was teaching at a summer school for two weeks in addition to my normal commitments.
I have lots of articles I could write, but not the time, so I can't offer more than a short offering at this stage. It's not a program, but it is fractals. I will try to do something with a program at Xmas (as I usually do) in the hope that you will continue in some form or another, taking the December FR as a signal.
I have been feeling that Fractals are past their peak (or have reached a plateau) for some time now. I ran a course on a Saturday in June in London, using Fractint. Only 5 people enrolled. Usually when I run Saturday classes I fill them with 18 or 19.
I think to extend your scope to mathematical patterns, ray tracing and the general area that Pickover covers would open up both the potential for articles and the audience. Jon Horner's letter was interesting since he is suffering from the same problem, although he is not admitting it as openly as you. His original idea was to have the other things besides fractals, but he is having trouble keeping the database both up-to-date and with come up with new data.
I looked at REC and was not impressed with the content for the money. Fractal Report is more detailed and meaty. I also had Algorithm for a year, but it had a simple outlook with too few good ideas. There is obviously a place for beginners, but it seemed to go over the top to pander to them in a noddy way much of the time.
I do hope you can keep going and I look forward to seeing what the last Fractal Report contains before it is reincarnated as "XXXXX Report". After all, if you believe in Longevity, then you ought to believe in it as far as FR goes too.
Keeping my fingers crossed for a continuation.
Additional notes on driving sound card from a BASIC by John de Rivaz:
When using the LAPC-1 channel one is &H91 not &H90 as in the article. On the LAPC-1 channel 0 is reserved for keyboard input. The Midi Monitor program serves no purpose unless a music keyboard is connected - it is intended to display on screen the result of using the keyboard.
Dr Landini also send a printout from a bulletin board concerning connecting the keyboard to the LAPC-1. It ran as follows (edited)-
The Roland LAPC-1 is advertised as a MT32 and a MPU401 on a card. I bought one figuring it was cheaper than getting the two units separately. I didn't have an immediate use for the MPU401, so when I read in the "manual" that I would need an extra connector, the MCP-1, to use the MPU401, I didn't buy one, and figured it was an inexpensive adaptor of some sort. After picking up an old Midi keyboard, I checked the price and it's $150 from Roland! As you can buy an MPU401 cheaper than that, I was quite dismayed, and investigated DIY wiring. Of the five Midi pins, 2 and 5 carry the serial Midi data, and 3 connects the shield. The LAPC-1 has a DB-15 joystick style connector. An hour of experimentation revealed the following
4
x x x x x x x
x x x x x x x x
2
This results in a fully functional link between my keyboard and the MPU401. I am sure thus is not blessed by Roland, and I have no idea whether or not it can harm the LAPC-1, but it does work. And it beats dropping another $150. - Michael Colicos
In correspondence, Dr Landini expressed the opinion that the $150 box contains an optical isolator chip. I would advise readers to take care, and if you are not sure what you are doing don't try it. Fractal Report cannot be held responsible for any loss or damage caused by experiments with hardware.
by Gerald England
There have been attempts to produce computer-generated poetry for over 30 years - most of these have been done to investigate problems in programming rather than been serious literary endeavours. There are programs about that purport to generate poetry. Most rely on a pre-determined set of adjectives, nouns, verbs and adverbs randomly applied. The results are pretty dreadful! Fact is computers can't write poetry, only poets can.
What a computer can do is manipulate data very fast. I've been experimenting using a program called Babble. What Babble does is to analyse a piece of text, split it into its constituent parts, phrases, sentence construction. It can then reconstitute the text in a different order but maintaining to a large degree the basic sentence structure of the original. It can mix together several pieces of text. The texts can be poetry (usually my own), a portion of the Maastricht Treaty, Hamlet's soliloquy, absolutely anything. By mixing these texts it produces something new that is an odd mix from whatever is input. The recorded output from the program is 95% nonsense. The value lies in the other 5% that comes up with intriguing comparisons, juxtapositioning of ideas and the like. From the useful bits, it is then up to the poet to manipulate, edit and rewrite to create something original. Some texts give better results than others and it is a matter of experiment to see what works and what doesn't. These poems, developed using Babble and other computer programs that can manipulate text in meaningful ways, are not computer-generated poetry, but computer-aided compositions. The poet is still responsible for the final work.
It is necessary to emphasise that at least 50% of the texts used are my own original work to start with. The other texts used allow one to bounce one's own writing off something else. I'm only halfway to understanding all this myself. Ultimately the poem has to stand or fall on whether it has anything of its own to say, irrespective of its mode of composition.
I'm not the only poet employing such methods. For the record it's called Post-modernist deconstructionalism but that's merely a label and not all that convenient.
Magazines of interest to anyone producing work of this nature include Transmog (Fiscus Strangulensis, Rt6 Box 138, Charleston, WV 25311, USA), Dada Tennis (Bill Paulauskas, Dreamstate Press, Box 10, Woodhaven, NY 11421, USA) and Lost & Found Times (John M.Bennett, 137 Leland Ave, Columbus, OH 43214, USA).
Babble is Shareware. The registered version costs $25 and is obtainable from Korenthal Associates Inc, 511 Ave of the Americas #400, New York, NY 10011, USA Download Babble
Applying the Epsilon Cross Method
to the Mandelbrot Set
by Warut Roonguthai
Dr Pickover devised the epsilon cross technique in order to reveal both the internal and external structures of some fractal sets. In this article, I will show how to apply this method to the classical Mandelbrot set via a Fractint formula.
In the epsilon cross method, each time we iterate the function of z, we not just test for the boundedness criterion but we also check whether z comes close to the real or imaginary axis. Precisely, we test if z runs into the cross-shaped region defined as below:
Re{z} < or Im{z} <
where is a small positive real number. If so, we will assign the initial point with a colour different from those used inside and outside of the fractal.
This technique usually reveals the dendritic structures in both portions of the set. The method was also explained in [1] and [2] by Pickover himself. Although the technique has been implemented in Fractint as an "inside" option since version 17, the value of epsilon is fixed at 0.01. Moreover, Fractint distinguishs between the points whose orbits pass near the real axis and the points whose orbits pass near the imaginary axis by assigning different colours.
For my method, you can control the epsilon value and the points whose orbits run into the "epsilon cross" will be coloured with the "outside" colour. By this way, the method will reveal only the internal structure and it works best in 2 colours (i.e. B&W).
Figure 1:
The formula below shows how I have applied the method to the Mandelbrot set:
EpsilonCross (XAXIS) {
z = pixel:
z = z*z + pixel,
|z| <= 4 && abs(real(z)) > p1 && abs(imag(z)) > p1
}
The real part of the predefined variable p1 is the epsilon value which you can control with the fractal type parameter params=xxx. The more the magnification, the less the value of epsilon should be used.
As you can see, it is just as easy to apply this technique to other fractals via Fractint formulae.
Here, in the case of the Mandelbrot set, we can use the LastSqr function to speed up (in theory) as shown in the formula below:
EpsilonCross (XAXIS) {
z2 = sqr(pixel):
z = z2 + pixel
z2 = sqr(z),
LastSqr <= 4 && abs(real(z)) > p1 && abs(imag(z)) > p1}
But, in practice, LastSqr does slow down the calculation a little. This also happens to other formulae using LastSqr (try it for yourself). I am not sure whether this is a bug or whether I use the function incorrectly.
I have reported this problem to Mark Peterson who wrote the formula interpreter, but he has not understood my point yet.
The pictures are output samples. They were computed with floating-point math to ensure the accuracy. However, integer math may work as well. The video mode is monochrome VGA. In generating both images, I used passes=1 inside=1 outside=0.
Figure 1 is centred at (-0.0315, 0.7915) and has a magnification of 60. The size of epsilon used is 0.0005 (i.e. params=0.0005).
Figure 2 is centred at (0.224, 0.440) and has a magnification of 4.8. But it was rotated by 90 degrees. Here params=0.001 and maxiter=60. Other parameters not mentioned have values equal to the defaults. Note that Figure 2 is the same view as Figure 27.5 on page 174 of [2] and the picture on page 274 of [3]. But, I think, Figure 27.5 of [2] was misplaced. The picture that was described by the caption may be the one on page 400.
References:
1. Clifford A. Pickover, Inside the Mandelbrot Set, Algorithm 1.1, pp. 9-12, November/December 1989.
2. Clifford A. Pickover, Computers and the Imagination, pp. 169-174, Alan Sutton, Gloucestershire, 1991.
3. Clifford A. Pickover, Computers, Pattern, Chaos and Beauty, Alan Sutton, Gloucestershire, 1990.
Figure 2:
by R. L. Bagula, 2 March 1993
The action movies of our generation seem to have one common feature: the chase! When one thinks of the point function z' generated in a Mandelbrot procedure one sees a path to a saturation limit. What if we create a second function on the single starting point c0 and information about distance and angle such that iteration stops when the two functions get close or diverge singly. I created one such function as a fractal boundary function using the Mandelbrot set as the fractal being chased. It works even if it isn't all that pretty (see front conver). There would seem to be a whole class of functions of this type. The basic program (for the Amiga):
PRINT " input screen partition" :INPUT m
PRINT "input save file name" :INPUT filenames$
SCREEN 1,320,400,4,3
WINDOW 1,"fractal screen",,31,1
pi=3.1415927#
m1=m*385/311
FOR I =-2.75 TO .75 STEP 3.5/(m1+.5)
d=d+1
FOR j= -1.75 TO 1.75 STEP 3.5/(m+.5)
c= c+1
ca=I: cb=j: cc =SQR(ca^2+cb^2) :r=10*(1+cc/(4*LOG(4)))
a=.75:b=1.75
FOR k = 1 TO 150
IF k = 1 THEN
x0=ca
xx=ca
y0=cb
yy=cb
x2=a
y2=b
r2=SQR(a^2+b62)
END IF
r1=r
x10=xx :y10=yy
xx=x10^2-y10^2+ca
yy=2*x10*y10+cb
d1=SQR((xx-x2)^2+(yy-y2)^2)
IF d1<=3.5/(m1+.5) THEN GOTO 5
x2=d1^2*COS(ATN(yy/xx)) +a
y2=d1^2*SIN(ATN(yy/xx)) +b
r2= SQR(x2^2 + y2^2 )
x0=xx:y0=yy
r = SQR(xx^2 + yy^2 )
IF ABS(r-r1)<1 THEN w=-1
IF ABS(r-r1)> 1 THEN w=1
IF r>10 OR ABS(r-r1)<10^-2 THEN GOTO 5
IF r2>10 OR d1<10^-2 THEN GOTO 5
NEXT k
5 REM
IF r=0 THEN r=1
IF xx=0 THEN xx=1
kk=k+ABS(k*LOG(1+cc/(r^2))+w*ATN(xx/yy))
IF kk >5*10^3 THEN kk=5*10^3
cr = 1+kk MOD 15
LINE(311*c/m,385*d/m1)-(311*(c+1)/m,385*(d+1)/m1),cr,bf
IF c>=m+1 THEN c=0
NEXT j NEXT I
GOSUB 100
WINDOW CLOSE 1: SCREEN CLOSE 1 :END
END
Dimensional Iteration In The Mandelbrot Set
by R.L. Bagula, 6 Oct 1993
In not getting my work published by my peers and having to do it myself, I have run into those who imply that I am some sort of "madman" and "fake". All my discoveries and hard work have gained me no respect or help from these men. Today, I have made a fundamental advance in both Mandelbrot and Julia mechanics: the concept of iteration of Hausdorff dimension. I'm sure there are those who will still say I shouldn't be published because I can't spell and have a "bad attitude", but this should at least put to rest these other claims. So what is my new procedure:
1) if k=0 then s'=2
2) if k=>1 then s'=log(s+2*k)/log(s)
3) z'=r^2*exp(i*atan2(y/x)*s')+c
This procedure is just the first of a long line of possible new fractals. This idea came from my ergodic gauge experiments with dimensional spectrums and my superdimensional discovery. My superdimensionals of SO(3) and SO(4) although not very successful showed that fractal dimensional iterations of loglinear functions were possible in dimension space. This morning I had the idea of combining the iteration in ordinary space with the iteration in dimensional space and have been trying it in a number of ergodic gauge mappings. The idea of chaining the angle but not the radius with the dimensional iteration works.
For me the concept of dimensional iteration at the same time as ordinary variable iteration is a "breakthrough" level discovery in fractal science. Let's look at Moran fractals of a ratio:
4) n*(1/r)^s=1
5) log(n)-s*log(r)=0 or s=log(n)/log(r)
This "mad-genius","fake" and "bad attitude" scientist suggests that a new sort of fractal can exist where n=f(s),r=g(s) so that (5) becomes an iteration instead of a set of constants:
6) s'=log(f(s))/log(g(s))
Since n is in Iterated Functional Systems the number of attractor points and 1/r is like a probability ratio, we are seeing both attractor point creation and destruction but also changes in probability and entropy as well. You can as well very the radius with the angle:
7) if k=0 then s'=2
8) if k=>1 then s'=log(s+2*k)/log(s)
9) z'=z^(s')+c
It makes me very sad that I have had to suffer this abuse by my peers in the fractal world. At any time they could, as I have done many times, have swallowed their pride and helped the science advance, but they instead have stuck to the "personalities" issue and my "mental state". I will say that accomplished working procedures that demonstrate totally new concepts and fractal types should have made them see that even were I totally "mad" and a "fake" (who by the way can't spell),I was advancing fractal and scientific knowledge. This set of procedures demonstrates a new higher level of fractals that I have today discovered. R.L.BAGULA 6 OCT 1993 CO ALL RIGHTS RESERVED
by Roger L. Bagula, 7 Oct 1993
As many of you who read The Fractal Translight Newsletter regularly know, I have some favourite books that I consult in my work. In this article I will try to give you an idea of these and some other books I have run into.
1) Fractal Geometry: Mathematical Foundations And Applications, by Kenneth Falconer, John Wiley And Sons, New York, 1990
I got my copy from Media Magic. I've found it a good place to start if you have already working I.F.S., Mandelbrot and Julia programs. He doesn't hit on heavy set theory as if it were the whole world of "fractals". His work on normal random noise in Hausdorff dimension should be read. On K. Falconer: he doesn't answer letters.
2) Measure, Topology, And Fractal Geometry by Gerald A. Edgar, Springer-Verlag, New York, 1990
Again I got my copy from Media Magic. I like Dr. Edgar's work. He is a very respectable traditional mathematician and he places fractal theory well. It was in his book I learned of Dr. Mauldin's multifractal work and Besicovitch-Ursell functions, but his book is not for the faint in heart mathematically speaking! Even though he uses LOGO in his text much of his "proof" stuff is pretty hard for me to follow at times. He almost ignores Mandelbrot Julia techniques. Dr.Edgar answers letters.
3) Fractals, Chaos, Power Laws: Minutes From An Infinite Paradise by Manfred Schroeder, 1991, $33
I got a section of this book as a Xerox from S.P. King on multifractals and it had very good "meat" in it. It can be found in Media Magic's catalogue. As I am not familiar with the whole book I can only say that I think it is probably well worth the money (and if I could afford it I'd buy it!).
4) Mazes For The Mind: Computers And The Unexpected by Clifford Pickover, St. Martins Press, New York.
I am including this because Mal Lichtenstein got a lot of good "code" by translating stuff from this book. I have myself written to St.Martins Press and got no answer: (St. Martins Press 175 Fifth Avenue New York NY 10010 USA)
The three volume set of The Journal of Chaos and Graphics (ed C. Pickover) published also by these guys is a big rip-off in my book. Dr. Pickover isn't totally bad for a Babylonian and actually answers letters but I don't find a lot of originality or integrity in his work, just good library work. There is "meat" here!
5) Fractal Image Compression by Michael F. Barnsley, $39.95
This contains "C" code for image compression using Dr. Barnsley's I.F.S. reduction to linear transforms of bit map Data. (Iterated Systems Inc. 355, Pennbrooke Trace Dunluth, GA 30136 (info from #26 Fractal Report)
On Dr. Barnsley: he doesn't answer letters from people like me (you?). His system has a math fault in being "shear" in the matrices, but it is a landmark in fractals. I like Dr.Mauldin's digraph method better. So there you have some ideas for books for your library to get you "into" the fractal thought mode. R. L. BAGULA 7 OCT 1993 CO ALL RIGHTS RESERVED
Editorial Note - These opinions are those of Mr Bagula, and do not necessarily reflect those of the publisher. Readers are invited to write in with comments on the mathematics that Mr Bagula promotes, and whether they think Dr Barnsley's image compression system is mathematically flawed by being "shear" in the matrices. Some of these articles may have appeared in The Fractal Translight Newsletter, for details see Announcements.
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