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+'0p}@(>:A%,J WHEN THE PROGRAM IS RUN THE SCREENWILL BLANK TO SPEED THE CALCULATIONSPmj(b Eq}VEN WITH THIS THE TOTAL TIME TO CALCULATE AND SAVE THE DATA IS BETWEEN 10 AND 20 ȠϠՠҠӠm(Zkk(b WHILE THE PROGRAMr} IS RUNNING PRESSANY FUNCTION KEY (, or ) TO DISPLAY THE dMM(ECURRENT NUMBER OF ITERATIONS, HORIZONTs}AL COUNT AND VERTICAL COUNT.n(((( PRESS TO CONTINUEx F:B2y,@A ((>:A%,(;@t},;@,85(-INSERT DISK WITH AT LEAST 250 FREE SECTORS8(99(0INPUT FILE NAME. DO NOT USE DEVICE OR EXTENDu}ER. FF(>THE PROGRAM WILL SUPPLY A DEVICE NAME D. AND A .DAT EXTENDER;(ENTER NAME OF DATA FILE7((MAX 8 CHARACTv}ERS);R67@<@,.D:,67@,.R67B:,%@:A%,FACORNER ,BCORNER ARE THE BOTTOM LEFT CORNER OF x}THE AREA TO BE PLOTTED%(ENTER ACORNER,BCORNER %,
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B`:@,?AVA QAW@0eAP%@B6 6(This article originally appeared in the March 1986 issue of The Australian Atari Gazette, the newsletter of M.A.C.E., P.O. B$}ox 340, Rosanna, Victoria, Australia 3084.)ADVENTURES WITH THE MANDELBROT SETUSING AN ATARI 800or a slow road to insanit$}y...... by Rita PlukssGirolama Cardano was an astrologer and physician to kings, a compulsive gambler who spent his life on$} the verge of bankruptcy and prison as well as on the brink of atheism and heresy. He was also a scientific writer from whos$}e quill books flowed in fountains.Over 300 years later he has been responsible for my near compulsive behaviour and for gra$}ve doubts being cast upon my sanity. Because of his astounding concept of negative numbers, and fictitious or sophisticated $}quantities, my family have placed a sign on the computer room door - quarantine area - and refuse to come in close contact wi$}th me. All they hear through the crack in the door, or when I emerge for sustenance is "why is it not there?", "where is it?$}". They just shake their heads and hope that there is a power failure and then I will return to human form and remember all $}those tasks that now remain undone.What Cardano put forward is today called an "imaginary" number. This is the square root$} of a negative number. What? you say. Has she finally flipped her lid? We all agree that it is impossible to have a square$} root of a negative number, but, just suppose, if it could be imagined and we combined this imaginary number with a real numb$}er eg 1+2/-2, then, the result would be known as a "complex" number.In fact these "complex" numbers have all sorts of appli$}cations in areas such as atomic physics, engineering etc. But who can understand such highly abstracted concepts? All I'm i$}nterested in is using them to generate pretty pictures.A "complex" number is one that is composed of an ordinary number plu$}s some multiple of the imaginary unit (the square root of minus one). eg 2+5/-1. The customary way of writing /-1 in such n$}umbers is i, and any complex number as a+bi. In our example a is 2, b is 5 and i is the square root of minus one. So we can$} write it as 2+5i.Ordinary numbers can all be thought of as lying along a single straight line without any gaps in it - a c$}ontinuum. But a typical complex number a+bi has no place on the line of ordinary numbers. So what can you do with it? Wher$}e does it fit?Gauss proposed that the complex number could be thought of as labelling a point on a two dimensional plane; w$}here a would be the horizontal distance and b the vertical distance and that the full a+bi could determine the position of a %}point on the plane just as the x and y of a Cartesian coordinate couplet determines the point on a graph.So where does all %}this lead to? It leads to the Mandelbrot Set. The Mandelbrot set is named for Benoit B. Mandelbrot, a research fellow at th%}e IBM Thomas J. Watson Research Centre in Yorktown Heights New York. He developed the field of fractal geometry (the mathema%}tical study of forms having a fractional dimension). This work was carried on by John H. Hubbard, a mathematician at Cornell%} University and Hubbard was one of the first people to make computer generated images of the Mandelbrot set. According to Hu%}bbard it is "the most complicated object in mathematics".If you wish to gain a more mathematical insight into the Mandelbro%}t set, then you will need to have some understanding of two mathematical concepts - iterative procedures and complex numbers.%} The first is fairly easy (just like FOR-NEXT loops) but the second is a bit more abstruse. The articles on the Mandelbrot %}set in both Scientific American (August 1985) and Your Computer (January 1986) will give you sufficient background of both co% }ncepts, but better still, consult with a senior high school maths student.To reiterate, a complex number takes the form of %
}C=A+Bi where both A and B are real numbers (eg -5, 6, 8.996 etc) and i is defined as the square root of -1. The complex numbe%}r C consists of a real part (A) and an imaginary part (Bi). We can now describe the iterative procedure that generates the Ma%}ndelbrot set. Start with the expression Z=Z*Z+C where both Z and C are complex numbers and allow Z to vary. Set Z=0 to give %
}us Z=0+C=C (C is now equal to Z); substituting this value back into the equation for the next iteration will give Z=C*C+C, th%}en Z=(C*C+C)(C*C+C)+C and so on. The value of Z fluctuates with successive iterations because we have introduced the value i*%}i=-1 into our calculations. The Mandelbrot set is the set of all complex numbers for which the size of Z*Z+C remains finite %}even after an indefinitely large number of iterations and is situated at the centre of a vast two dimensional plane. The bou%}ndary of the set is a fractal (and you thought I'd given up on fractals)! But what a fractal! It must be the ultimate.Our%} interest lies on these edges of the set where Z falls outside the Mandelbrot set i.e. where Z does reach the value of 2 and%} will go off to infinity. The fractal area!The number 2 is the crucial factor in working with the Mandelbrot set. "A stra%}ight forward result in the theory of complex number iterations guarantees that the iterations will drive Z to infinity IF AND%} ONLY IF at some stage Z reaches the value of 2 or greater." But, actually there are very few points where the value of 2 wi%}ll not be reached after a small number of iterations. This situation becomes rarer as the iteration count increases.The Ma%}ndelbrot set itself is the black (static) area within the plane, where Z does not reach the value of 2. (That is, black on th%}e screen, white on the printouts.) It is shaped like a squat, wart covered figure eight lying on its side.The coloured and%} patterned areas around the set are where Z has reached the value of 2 and the colour of the pixel should tell you how many i%}terations it took to reach that value. This is the area that is of greatest interest, the areas that surround the set. Thes%}e areas are a myriad of colours and patterns, no two areas being identical, yet repeating themselves over and over again. Th%}ere are riots of organic looking tendrils and circular sweeps, whirls and whorls, and the colour! You can look at the full s%}et, look at the edges, or you can zoom in as with a microscope and go deeper and deeper, the patterns and colours just keep h%}appening.The program that searches for these values has been called MANDELZOOM (Program listing 1). As the name implies it%} investigates the Mandelbrot set and also allows you to zoom into the set for closer investigation of any part.With the Ata% }ri 800 we do have a couple of minor problems; no double precision therefore we have to limit our zooming, and the resolution %!}we can actually achieve, once again limiting our final screen output. But with these limitations recognised we can produce a%"}cceptable displays within those constraints.Our first programs produced immediate displays - if anyone is interested in the%#}se let me know and I will pass them on to you. The final versions (we think they are the final versions - but we have though%$}t that before!) of the programs that follow this article were put together by Dick. But there have been a small core of enth%%}used people (or nuts?) working diligently through the various stages to come up with the final version. This subset of the M%&}andelbrot set have worked and bolstered each other and they are Chris Ryan, Dick Kellett, Ron Collis, and of course, myself. %'} There have been a couple of enthusiastic late comers, Gary Fyfe and Ian Conner (both from Geelong). The main problem with i%(}mmediate screen plotting was that we could not reallocate where the colour changes should take place. Not having the memory %)}available to read the data into an array, Dick had a stroke of genius - put it to disk as a data file, then when you are read%*}y to plot, read it back from the disk! Brilliant I said! Only one problem, each data file is 246 sectors long! Oh well, wh%+}at else are disks for?It still amazes me that with only 48K of memory (less BASIC of course), an 8 bit ATARI computer and a%,} TV screen that we could produce screens that are quite acceptable considering that what you see elsewhere have been produced%-} on computers much more powerful than ours.BRIEF EXPLANATION OF THE 4 PROGRAMS. See Dick's article (3. More background) on%.} how each of the programs work.PROGRAM 1: MANDELZOOM - MANDLCALPerforms the calculations on that part of the Mandelbrot s%/}et under investigation and writes it as a data file to disk (15-30 hours), depending on how much of the area is within the ac%0}tual set (black).PROGRAM 2: MANDPLOTReads the data file obtained by program 1 and plots that data to screen. (15-20 minute%1}s)PROGRAM 3: DATACHKScans the data disk, reads the number of iterations required to generate each piece of data (pixel) an%2}d collates this information. This allows you to see at what levels of iteration to make your colour changes when running pro%3}gram 2.PROGRAM 4: COLDUMPThis has nothing to do with the Mandelbrot set, but is a screen dump to dump each colour register%4} separately to the printer (7+ screen). By using this, colour carbon paper and a good eye when you roll your paper back, you%5} too can produce colour printouts as were seen at the February meeting.Have fun and good luck! For more information on the%6} Mandelbrot set refer to Scientific American (August 1985) or if really involved try The Fractal Geometry of Nature by Benoit%7} B.Mandelbrot (W.H.Freeman & Co. New York, 1977) and Fractals - Form, Chance and Dimension by Benoit B.Mandelbrot (W.H.Freema%8}n & Co. San Francisco, 1977). New York, 1977) and Fractals - Form, Chance and Dimension by Benoit B.Mandelbrot (W.H.Freema$ THE MANDELBROT SETThis disk is a double sided one. The front side contains the four programs listed within the Mandelbrot ):}articles in the M.A.C.E newsletter. They allow you to create the Mandelbrot data file, plot the data file to screen, investi);}gate where to make the required colour changes, and finally to dump the screen to printer, colour register by colour register)<} (for colour printouts), or in a variety of grey shades.The flip side contains a slide show (using Fader 2) of graphics 7+ )=}Mandelbrot screensaves. These were generated by Dick and Rita.To generate your own Mandelbrot use Program 1 and refer to F)>}igure 1 in the article (figure not on this disk). This is the complete set, and from here you can select the coordinates to )?}investigate any part of that set. ACORNER (Real coordinate) is the horizontal axis, BCORNER (Imaginary coordinate) is the ve)@}rtical axis. SIDE is the horizontal length of the 'square' you wish to view. (The ratio is 1 horizontal to .64 vertical.) )A}The smaller the SIDE value, the more powerful the zooming function and you find yourself deeper within the set. I have gone )B}to 8 decimal places (and further). The deeper you go, the more precise you need to be with your measurements to get the righ)C}t coordinates to find something of interest, and you do need some luck, otherwise you may find nothing but the blackness of t)D}he set itself.Load in program 1. Insert a disk with 250 free sectors, this will be your data disk. Follow the prompts. T)E}urn off the screen and leave the computer and drive to work for the next 10-30 hours while you do all those other tasks that )F}need doing. When all the computation has finished and the file has been completed, turn on the screen which will show FINISH)G}ED and the name of the file it was saved to.Load in Program 2. Type in the name of the picture file, then the name of the )H}data file to generate the picture. For the first run through select option 2 (MOD 3) and watch the mystery of the selected a)I}rea unfold before your eyes (15 minutes). Save this screen by pressing the SELECT KEY. Use Program 5 (Loadscrn) to retrieve)J} this picture at a later time.Program 3. Follow the prompts. This program shows you at what levels of iterations activity)K} was occurring. This will help in the choice of where to set the colour changes for the best effects. After noting (or dump)L}ing) the information generated by this program use that information to experiment where to place your colour changes in progr)M}am 2. Continue experimenting until you have the effect you desire.Program 4. If you have a PX-80 printer this will dump y)N}our screen to the printer either in shades of grey, or in separate colour registers.Program 5. This is a rough and ready r)O}etrieval program. Follow the prompts (graphics 15 is graphics 7+). I added this as an afterthought, just incase you did not)P} have the means to retrieve the picture files you had created. This small program will actually retrieve any type of saved s)Q}creen except for compressed screens.PROGRAM 1 MANDELZOOM part 1 - MANDLCAL.BASPROGRAM 2 MANDELZOOM part 2 - MANDPLOT.BASP)R}ROGRAM 3 DATACHK.BASPROGRAM 4 COLDUMP.BASPROGRAM 5 LOADSCRN.BASSIDE 2 is an autorun. (Press START to speed up the displ)S}ay of each screen.) Just slip it into the drive, turn on the computer, sit back and watch a slide show of what you can produ)T}ce using the programs on the front side of the disk. on the computer, sit back and watch a slide show of what you can produ(7(This article originally appeared in the March 1986 issue of The Australian Atari Gazette, the newsletter of M.A.C.E., P.O. B-V}ox 340, Rosanna, Victoria, Australia 3084.)MANDELBROT SETS and the ATARIby DICK KELLETTHave you looked at those compute-W}r generated pictures in SCIENTIFIC AMERICAN AUGUST 1985 and YOUR COMPUTER YEAR BOOK JANUARY 1986 and thought you would like-X} to try the same type of graphics? The articles in these magazines suggest setting up arrays of up to 1000 by 1000 and using-Y} 1000 iterations to check if each point is in the MANDELBROT set. This is fine if you have access to a super mini or a mainfr-Z}ame but not much help with only 32K available. While the ATARI cannot match the resolution of the published pictures, GRAPHI-[}CS 7+ (GRAPHICS 15+16 on XL and XE models) will give sufficient resolution and three colours plus background colour to produc-\}e interesting pictures.The pictures are produced by using the equation Z^2+C where Z and C are complex numbers and repeatin-]}g the calculation with the answer replacing Z in the equation. Counting the number of iterations before Z=2 and assigning a -^}colour to this number generates the picture. If the number of iterations exceeds the selected maximum (in our case 100 itera-_}tions) the area (pixel) is in the MANDELBROT set and is plotted in the background colour.After many frustrating hours plott-`}ing screens directly and having a single colour or a small area in one corner, I realized that the number of iterations selec-a}ted to change colours was very important and differed for each picture.My approach to the problem was to use two programs. -b} The first one called MANDLCAL.BAS selects an area according to the formulae in the SCIENTIFIC AMERICAN article, scales it to-c} suit a GRAPHICS 7+ screen and stores the results on disk as a DATA file of 246 sectors. I used one hundred iterations to ap-d}proximate whether or not the point was in the MANDELBROT set.The second program called MANDPLOT.BAS allows you to select th-e}e levels of iteration that colour changes will occur at, and then plot the point in the selected colour. Points in the MANDE-f}LBROT set are always plotted in the background colour.The colour changes may be selected to change at preset counts, plot t-g}hrough the colours in MOD.3 or plot colours below the first change in MOD.3 The same colour is used for the lowest and high-q}Kb%DOS SYSb*)DUP SYSbSAUTORUN SYSbUMENU blMANDLCALBASbDATACHK BASbMANDPLOTBASb-COLDUMP BASbLOADSCRNBASbNMANDLBT DOCb9DOM DOCb'UKELLETT DOCest counts as there is sufficient difference in the position of the points plotted to avoid running the areas together. It a-r}lmost gives the effect of having an extra colour.Both of these programs run very slowly. MANDLCAL takes between ten and tw-s}enty hours to calculate the data file. The worst case involves 100*159*191 separate calculations for an area completely in t-t}he MANDELBROT set. Obviously, an area with a lot of background points takes longer to calculate than one with a lot of colou-u}r.MANDPLOT will set up a graphics 7+ screen and plot the picture from the data file in approximately twenty minutes. Befor-v}e the picture is plotted you will be asked for a file name to save the picture to. If you do not select a name, the picture -w}will be saved using the default name of PICTURE.The pictures are saved as 62 sector files. I prefer to save my pictures wi-x}th individual names and use the DOS copy file option to create a new file called PICTURE, which can then be loaded into ATARI-y} ARTIST (MICRO ILLUSTRATOR) by pressing the CLEAR key after ATARI ARTIST has been loaded. The colours can then be adjusted a-z}nd pattern fills added as required. The picture can then be saved in the normal way and used with FADER 11 as a slide show. -{} The pictures could also be put through Rapid Graphics Converter and be used as a background file for MOVIE MAKER etc.Both -|}these programs will compile with the MMG compiler. MANDLCAL does not show a significant increase in speed and I let the prog-}}ram run overnight and while at work the next day. When compiled MANDPLOT will plot the picture in approximately ten minutes -~}instead of twentyfive minutes.If you try these programs I would suggest that you obtain a copy of the photograph of the ful-}l MANDELBROT set from the SCIENTIFIC AMERICAN article, as it shows the co-ordinates for the set. This will allow you to sele-}ct the starting points (ACORNER, BCORNER) and the length of the side for the area to be plotted (SIDE). If you have trouble -}locating this article see RITA or DICK at the next meeting or come to the screen art group.To get started try the following-} points. The first set of figures should give you the complete MANDELBROT set (a colour version of Fig.1).ACORNER=-2.5 -}BCORNER=-1.25 SIDE=3.5 ACORNER=.2665 BCORNER=-.0049 SIDE=.002ACORNER=-.9 BCORNER=.263 SIDE=.005Happy -}plotting. Lots of patience!CORNER=.2665 BCORNER=-.0049 SIDE=.002ACORNER=-.9 BCORNER=.263 SIDE=.005Happy ,