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G}JB|,#P#DE 1 HI BDEHHII 1 B 1 ,^ 1 70,0La- B V,#PH},^ 1 70 0L#L!-* 1P* 1 y0Yj383}mm ݭI}}`8}``|* ? ɛ,`|:-)| / 1L!`DESTINATION CANT BE DOJ}S.SYS0 0H{ 24Δ 28/L!/) 2 Π 2 0 ξK}hAΞB,0 J 1 BDEHI,HÝDE 1HIHIDELSAVE-GIVE L}FILE,START,END(,INIT,RUN)O S0 1`BDEPHI V` S0H 1 L!M}0 0 1L~0`PLEASE TYPE 1 LETTER,0`hhL! 70 1L0L<1 ,;ɛ7,"ɛ:ݦ1ݥN}A"D|ݤD|ȩ:|ȩ|ɛ,,(/+.ީ1 1,ɛ`轤{NAMEO} TOO LONG B VL!` L1I H1EΝDL1|mDiE` V0`8d/8 i:222 1 LP}!ERROR- 144ɛ+,' 20*.. өr2 1``2TOO MANY DIGITSINVALID HEXAQ}DECIMAL PARAMETER800 0 8 00`,0'D800 H,ɛh`2L1NEED D1 THRU D8uR} ECIMAL PARAMETER800 0 8 00`,0'D800 H,ɛh`2L1NEED D1 THRU D8uEiͩkΩ͙kCop j`j {j`Hi͝Νh`T}L"UNEM:D"NURͩkΩ͙kCop j`j {j`Hi͝Νh`X  DKEBLWIDTERFILEA@ V}88MENU by Les Ellingham, PAGE 6 August 1986. W +@/6-F:A`,%AV$F:Aa,C%@@W% W}@@##@@K:6;@ ,;@,66. -oAA9 X}'A @9A@RKA@]AAoAR@2:(@>:A%,:(@ Y} page  presents<##(@ MANDLEBROT SETSA33(@" courtesy of MACE AustraliaF6(@ Z}(@6(@ 1. mandlezoomK)(@)(@ 2. mandplotP((@((@ 3. datachkU(( [}@((@ 4. coldumpZ)(@)(@ 5. loadscrn_.(@.(@ 6. documentationcb \}b(Y Please ensure that you read the documentation before using the programs.d.)@. @I)! ]}@TAnE6-&@HEA0A@APA`ApA%D:MANDLCAL.BAS%D:MANDPLOT. ^}BAS% D:DATACHK.BAS% D:COLDUMP.BAS%D:LOADSCRN.BASJ(>:A%,--@@6@9- _}F(@J 'AAe'AAe,(@,(@ 1. instructions*(@*(@ `} 2. background/(@/(@ 3. more background.(@.(@ 4. return to menu.)@ a}. @I)!@RA56-&@H5A A"A$@E A6. D:DOM.DOC AP b} !6. D:MANDLBT.DOC! AP 6. D:KELLETT.DOC&V@@6@@V@@ c}K:0 A@: A-@D)@N3!&@*"@2&(@)'3 A0 d}X(@>:,b"AU' A0l v"F:@,!@!" AP Ap(ԠΠϠ e})@(>:A%,$ F:A,A6A09F:A,"A6#@.@9@ f}V(0(%END OF MANUAL - PRESS TO RE-RUND(ANY KEY TO EXITVAdAU8F:B2y,"@#@ g}.@8 AP-F:Ad,AU*AdAU-% A(>:A%, A`(((( h} (S)creen or (P)rinter)@ !!"@)"AA !!"@)"AA  Ap* i}26-@8/@@E:2$4-(>:A%,-( Width 1) 40 columns> ( 2) 80 columns j}H<)@6-&@H< @)!@A@R46-$@@1@@P:4$\B A` k}0(ˠҠĠԠBAdAUf)@@p APD8:MENU4$\B A` l67 MANFILEHVIACNBCNSSCGAVGABAABCOUNm}  ""n} PART 1 - MANDLCAL77refer to Scientific American August 1985;;MACE subset of Mandelbrot set o} by DICK KELLETT(MACE MARCH 19862##*****************************<##*****************************F +'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 %, A0-@@,( e-@@y}e(K SIDE IS THE DISTANCE TO BE COVERED INA POSITIVE GOING HORIZONTAL DIRECTIONoo(g THE DISTANCE COVERED IN A POSITIVE z}GOING VERTICAL DIRECTION IS APPROX. 0.6 THE HORIZONTAL DISTANCE( ENTER SIDE C6-F:AY,$AY{}6AUC(>:A%,2 6-'6-'@2ADJUST FOR PIXEL SHAPE"$$MAIN NUMBER CRUNCHING ROUTINES,-|}-6 6-$%@ 6-$%JK6-6-'6-36-?6-K6-}}T 6-%&6-%@$^6-%@h 6-$6-$6-$r::DISPLAY CURRENT STATE WHEN A FUNCTION KEY IS PR~}ESSED|"F:B2y,@" AHHBLANK SCREEN WHEN FUNCTION KEY IS RELEASED TO SPEEDUP CALCULATIONS*F:B2y},"@*AY *+%,@A@*@6-  *@AUEO}Ly@AY$+@6-@@L(@^-@@y(@ }4-@@"(@4AU@ ApDISPLAY CURRENT STATE=@w!AY3-}@@=(COUNTD-@@#( HOR. POS.5-@0@D( VERT. POS.B-@B((}U-@@( 0-@@9( K-@4@U( $} D:MANDCAL.BASU-@@( 0-@@9( K-@4@U( $)*AHPOVPOFILEDATAe^m } ˛AACHECKS THE ITERATION COUNT FOR MANDELZOOM (MANDLCAL)by Dick Kellett(2MACE MARCH 1}986<***************F***************P= +(>:A%,=9A,;@,;@,Zhh(`ENTER NAME }OF DATA FILE. DO NOT USE DEVICE OR EXT. PROGRAM PROVIDES A D: DEVICE AND A DAT EXTd A`nR67@<@},.D:967@:A%,$-@A'6}-+&@,$@'6-5!@06-&@5+6-%@5 A -(*8, OO(GPRE}SS FOR GRAPH. VERT AXIS IS SCALED TO A SQUARE LAW FOR COUNTS F:B2y,"@AP A0O +@}6-30@=@OAWA/(' AAW}//('1-----------------50----------------1007-@A#"@7,AY&M:8,,"/$@}AY&M:8,,, 6 A D:DATACHK.BAS<@#"@7,AY&M:8,,"/$@I9:-CABLDFILECCCCCOSDLISCHANGTOBHLTIM} } "" - PART 2 MANDPLOT33refer to Scientific American August 1985EEThis reads th}e data file from MANDLCAL and plots the screen(277MACE subset of Mandelbrot set by DICK KELLETT<MACE - M}ARCH 1986F*****************P*****************ZH +>;@,;@,;@,;@,H AP}dV(>:A%,R(=ENTER FILE NAME OF DATA FILE.DO NOT USE DEVICE NAME OR EXT.VnE67@<@,.D:,67@},.E67B:,%@,..DATx A (>:A%,NN(FENTER TO DISPLAY PICTURE WITH USER SELECTED COUNTS FOR C}OLOR CHANGE::(2ENTER TO DISPLAY PICTURE WITH COLOR CHANGE MOD 3MM(EENTER TO PLOT THE COUNTS LOWER THAN THE FIRST }CHANGE POINT IN MOD 3%% @)!@A ! B!"@APp(>:A%,p([THE COUNTS FO}R CHANGE ARE BETWEEN 1 AND 100. ENTER THE CHANGE NUMBERS IN ASCENDING ORDER( A-@@(( _-}@@=(#ENTER COUNT FOR FIRST COLOR CHANGE A_ @)!AA( A-@@(( } Z-@@>($ENTER COUNT FOR SECOND COLOR CHANGE BZ)!AA( A0-@@ }(( Y-@@=(#ENTER COUNT FOR THIRD COLOR CHANGE AY)!AA0 AP A}+@+@@G -A6@4-AYA)@G6-"e"@})"@e6-@$+ ,%@$++,*+ ,,%@$++,*+ ,,%@$+,,N!"@)+"@*+ ,,N6}-+'@&P:'@,,$@%@6"A6-@B @'@&@-,BTOP HALF }OF SCREENJA!@@),&@ABOTTOM HALF OF SCREENT   ^I@&@"@IRET}URN POINTER TO TOP HALF OF SCREENh F:B2y,"@@r F:B2y,"@A@| A`Р}ӠΛb+@%@76-F:A`,%F:Aa,$AVK%@@xb6-F:@,%@6-%@}%A*F:,"@6@F:,"@y@x @@$̠Π}k(>:A%,(k(SOnce design is complete, press SELECT to save to disk. The default name is PICTURE]AdAU}](CPress START to plot a new picture without saving current screenS(((Enter Filename to save designO("DO NOT U}SE DEVICE NAME OR EXTENDERS2 6-B:,"(6. D:PICTURE2 AW67@<@,.D:3-@%}@S67<,.7&@<&@,W ##67%@<%@,..PIC$!!---------------------------}ŠϠˠŠ+@+@@&;%6-F:@,%AV$F:@,36-%Av;6-&0z}6-P:'AV,&6-&+$AV,8AP@MARF:@,bASF:@,nAVzAW:*6}-?:C:hhhLV,<@,*@DX-@@4-@6?0P2@T X N} @D:MANDPLOT.BASDX-@@4-@6?0P2@T X N"ABTIMPR1PR2HDFNCIOVDLREGMIREPASCASLPRPRPICSTM} } W +AA1A@$4(E( ΠWAH@'('( GRAPHICS 7+ TO PX80 P}RINTER,(,(!by Jerry White & Fernando Herrera(S(*(Altered to PX80 and print each S($register separately by Dick }Kellett.-kF(>WHEN PICTURE IS LOADED PRESS TO PRINT REGISTRATION LINEk( PRESS TO COMMENCE PRINTING2+(+(} PLEASE WAIT. SCREEN DUMP LOADING3-@A < A^***SCREEN DUMP****_*BY JERRY WHITE &*`}*FERNANDO HERREA *a* ANTIC MAGAZINE *b* JANUARY, 1984 *c******************dРiRR;A},;A,;@,;@,;@B,;@3,;Ap,n A+)@@;AR@}MAA`_A@${(ENTER DEV:FILENAME.EXTx BԠӛ7 #A!-@}@%"367<,.>:,7 27,76,2,3?6.>:,*67A,.>:,967@,.?6.}67@},.>:,367Ap,.>:,K67@,.7Ap,d67@<@,.7}67Ap<Ap,.7167@}<@,.167@<@,.ˠϠΠś7 #A@!-@@B%"367<,.>:,7 LL}104,104,104,10,10,10,10,170,104,104,157,66,3,104,157,69,3,104,157,68,3KK104,157,73,3,104,157,72,3,32,86,228,169,0,133,21}3,189,67,3,133,212,96٠Ԡś7 #A!-@@3%"367<,.>:,7 ==104,173,48,2,1}33,203,173,49,2,133,204,160,1,200,177,203"@@201,15,240,4,201,79,208,4,233,1,145,203,192,200,208,237,96,Ҡ}Λ-p A(>:A%,,-@@p(?WHEN PICTURE IS LOADED PRESS TO PRINT REGISTRATION MARKS.}(PRESS TO PRINT/-@@0b(#(ENTER REGISTER TO PRINT 9(0. DEF COL ORANGEN(1. DEF COL GREEN}b(2. DEF COL BLUE1**("3.BACKGROUND (REG 4).DEF COL BLACK2[*("4. PRINT ALL REG IN SHADES OF GREY9( INPUT REG=[} )!@A5#$@%A06%-A1AF"!% I%%ҠǮԠ̠ś}J))0,0,0,0,15,15,15,15,0,0,0,0,0,0,0,0S**ҠǮԠ̠ԠΛT))0,0,0,0,0,0,0,0,15,15,15,15,0,0,0,0]}##ҠǮԠ̠ś^))0,0,0,0,0,0,0,0,0,0,0,0,15,15,15,15g))ĠǮԠ̠˛h))15,1}5,15,15,0,0,0,0,0,0,0,0,0,0,0,0q&&Ԡ̠ǮΠӠƠٛr**0,0,0,0,1,0,4,0,3,0,12,0,15,15,15,15!!Р}à٠ԛ; +@$/6-F:@,%F:@,$AV;6-?:C:,,Q@+@@Q6}-?:C:,<@<@<<Av,ԠΠӛ F:B2y,"@A F:B2y,"@A} A A 0@.@@P:: @w(@>:@',>:@d, } COLOR REGISTER (@q"N(@1+- - - - - - - - - - - - +b-@}@m(@q q"@N(@1+ - - - - - - - - - - - - +b-@@m(}@q q"@N(@1+ - - - - - - - - - - - -+b-@@m(@q }q"@N(@1+ - - - - - - - - - - - +b-@@m(@q q"@}N(@1+--------ALL REGISTERS--SHADES OF GREY----------+b-@@m(@q  F:B2y,"@}A F:B2y,"@A F:B2y,@A РҠҠӛ РҠ}Ҡӛ@.@@P:(@>:@',3>:@,>:@',L>:@,>:@},>:@',>:@V, 6-C:,6-C:,ΠϠҠЛb-@9/6-A$@@%%D}6-?:A6<<<,Q(@^(@b W(@(@:(@>:@',>:@d,E@WAd}AUz +AA1A@$4(\(#PRESS TO ENTER NEXT REGISTERz(PRESS FOR }NEW PIC.  F:B2y,"@A* F:B2y,"@A4 A@ AxΠ}Рś/ #A !-A6A0%"+/ 33104,104,133,204,104,133,203,104,141,192,6,104//141,191,6},104,141,194,6,104,141,193,6,169111,133,207,169,191,133,208,160,0,177,203,14100190,6,165,207,240,28,169,0,133,207,173},193,,6,24,105,4,141,193,6,133,205,173,194,6++105,0,141,194,6,133,206,76,95,6,169,1..133,207,173,191,6,24,105,4,1}41,191,6,133..205,173,192,6,105,0,141,192,6,133,206,32))137,6,32,146,6,32,160,6,32,137,6,32//169,6,32,160,6,165,2}07,240,177,56,165,20322233,40,176,2,198,204,133,203,198,208,165,208--201,255,208,151,96,173,190,6,41,3,10,1000170},96,160,0,189,195,6,145,205,232,200,192,,4,208,245,96,173,190,6,74,74,141,190,6++96,160,0,189,195,6,10,10,10,10,24,11}3"44205,145,205,232,200,192,4,208,238,96,0,0,0,0,0,$U-@A32@@U2}^ +AA1A@$C-@@^(ҠԠ7-@}@7(CHECK PRINTER AND PRESS  F:B2y,@A @ B A D:COLDUMP.BAS@xab"xu AUFILENAMSCRENSAVBELABTIMSCRENLOAMODMODENAMENAMDLISISDLISSCHANGPEECREGCRE }  }@SCRENLOD.LSTREF COMPUTE OCT 85 P10MY VERSION... THERE }ISA SAVE PROG TO GO WITH THIS.... SCREEN LOAD ROUTINE SETUP +?;@,9@ ,'6-A }36-A?6-Bp-A6AB( " 2 <104,104,104,170,76,86,228Zd n p }@+s"@ Ax  B`+(+( }WHEN YOU HAVE FINISHED VIEWING %%(PRESS }THE KEY TO LOAD (ANOTHER PICTURE. ((WHICH GRAPHICS MODE ((7 8 9 10 11 15"@ }6-@#"@6-@$"$,ԠРҮΛ1/-@+68,-F:A%,/ 6K+@ }%@76-F:A`,%F:Aa,$AVK%@@x;,-@(A%8,, @6-%@ }%A*F:,"@6@JF:,"@y@xT @@$GET NAME ROUTINE }((((WHAT FILE DO YOU WISH TO SEE?((TYPE D:FILENAME.PIC 0$v>>WAIT FOR Ԡٛ> }>B2y@>,F:B2y,"@"+, @> B`0hBrBSCREEN LOAD ROUTINE|BB  }@@BcARF:@,-ASF:@,?AVA QAW@0cAP@ }B6-?:A6<@,B@B$D:LOADSCRN.BASAA A +$,  }!!GET FILE NAME FOR RETREVIAL-(-("WHICH FILE WOULD YOU LIKE TO VIEW?((TYPE D:FILENAME.PIC &$> } WAIT UNTIL SELECT PRESSED>B2y@>&F:B2y@> B` hBrBSCREENLOAD ROUTINE|BB }@BeARF:@,-ASF:@,?AVA QAW@0eAP%@B6 }-?:A6<@,B@B 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 blMANDLCALBASb DATACHK 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 ,