Posters. NESUG '96 Proceedings 530. John J. Cohen Advanced Data Concepts ABSTRACT DISCUSSION THE MANDELBROT SET? OUR FIRST TASK

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1 PLAYNG WTH THE MANDELBROT SET John J. Cohen Advanced Data Concepts ABSTRACT Fractals generally and the Mandelbrot Set in particular have offered a rich area for research and exploration for students of mathematics and graphics. A delightful result has been a plethora of dissertations and publications brimming with beautiful graphs. For SAS programmers, in contrast, these have provided us unwished for opportunities for testing the limits of our patience, our programming skills, and the hardware platforms on which we run. All too frequently the unhappy consequence of our dabbling has been headaches, programs which run for hours (or days!), and empty toner cartridges. There is no reason to let others have all the fun, nor to wonder about just what kind of arrangement made it possible for their plots to print. We will offer here the details of a successful experience in producing pictures, secure your first born, and list programs worth copying (assuming proper citation, of course). THE MANDELBROT SET? strikingly beautiful result. The Mandelbrot Set, one of the most popular fractals first suggested by Benoit Mandelbrot in 1979, represents a particular challenge. t will test the capacity of any platform we attempt it on, but will also be particularly rewarding to reproduce. DSCUSSON Our task is two-fold. We must identify an algorithm which will, using generally available technology, allow us to generate the enormous amount of data required to represent the Mandelbrot Set. Secondly, we must display the data in a manner that will convince the viewer that we have replicated (a portion of) the Mandelbrot Set and that it is, in fact, a simple design replicated many-fold. deally, the result will also be aesthetically pleasing. ("Hey, can use that as my screen saver?") We will attempt this task using a combination of base SAS, the SAS Macro language, and SAS/GRAP~. under Windows for Workgroups. The hardware platform consists of a Dell 466ME with 12MB of RAM and an HP LaserJet 4S. OUR FRST TASK Our first task will be to identify an algorithm and create a program which will, using the technology at hand, leave us a graph which is an acceptable replication of the Mandelbrot set. The first attempt, a portion of the entire set, was printed using the WNPRTM printer driver. n testing, it took an average of 25 minutes to generate and plot the Mandelbrot dataset, and another minutes to print the resulting approximately 4,620K print file. A sample is shown just below, and another view appears in the appendices. NTRODUCTON For our purposes, a central premise underlying the study of fractals is that many objects in nature, even if they are not actually composed in this manner, can be described quite accurately by the identification of a simple pattern. This pattern is then replicated hundreds (thousands? millions?) of times using both random and non-random functions (chaotic) to create a complex and often NESUG '96 Proceedings 530

2 OUR NEXT TASK Our next task will be to explore some of the interesting modifications we can make to our initial plot attempt. Here we will reprint the same graph in color. We will use the seven colors available to the Tektronix Phaser 220i printer. (n fact, here we will use the WNPRTG printer driver to offer a flavor of what the color poster will be like.) We will calculate approximate (1) septiles and then create an annotate dataset. n this dataset we will use a PROC FORMAT to associate colors with data points based on where their Y values fall within the septiles. Our output is displayed immediately below, while the program is listed in the appendices. As the entire dataset being graphed is NOT the original, but rather an annotate dataset, we will use PROC GANNO (and thus lose our titles). GRAPlnNG N COLOR (SORT OF) THE PLOT PROGRAM GOPTONS ROTATE=LANDSCAPE CBACK=WHTE CTEXT=BLACK TARGETDEVCE=WNPRTM; SYMBOL C=BLACK V=DOT H=.l CELL PROC GPLOT DATA=MANDEL; PLOT Y * x / NOAXS; TTLEl COLOR=BLACK FONT=SWSSB HEGHT=2 JUSTFY=C '--- The Mandelbrot Curve Set ---'; F YOU WANT TO TRY TinS YOURSELF Copy these programs with our blessing (giving proper citation, of course). Modify them incrementally. Consider testing with a smaller grain value (say, 100) in the algorithm program. The resulting smaller dataset will plot faster. Consider saving versions of the Mandelbrot Set datasets you create to permanent files when playing with the plot programs. Similarly, consider saving final graphs to permanent graphics catalogs. f available, use more powerful hardware. (A colleague's 586-chip machine under Windows95 executed these programs 50% faster and allowed true background printing of the graphs, thus not tying up the machine.) Finally, if you are sending print output to network printers, choosing off hours will make everyone happier. (Trust me on this.) REFERENCES WHAT NEXT? Desired next steps include capturing a better representation of the entire Mandelbrot Set and subsetting and displaying portions of this Set in ways that satisfyingly convince even the casual observer that the whole is simply a replication over many iterations of an original simple pattern. (Our effort in 1995 was somewhat unsatisfying even if technically correct.) Finally, we must display for you, the gentle viewer, a real color graph, and possibly devise a more interesting way of assigning color values to plotted points. Retrieving our titles will be a trivial, but vital coda. The algorithm comes from Tom Frenkel, S Department, Columbia Presbyterian Medical Center. Also, many thanks to Gerry Pauline, Department of Academic Computing, Pace University. H. Peitgen, H. Jiirgens, and D. Saupe, The Mandelbrot Set: Ordering the Julia Sets, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, New York, N.Y., 1992, pp SAS nstitute nc., SAS/GRAP~.User's Guide, Release 6.03 Edition, SAS nstitute nc., Cary, NC, 1988,549pp. SAS is a registered trademark of SAS nstitute nc. in the USA and other countries. indicates USA registration. Dell, HP, and Tektronix are registered trademarks of their respective companies. Windows and Windows95 are registered trademarks of the Microsoft Corporation. 531 NESUG '96 Proceedings

3 The Algorithm %global grain TERCT; %let grain=500; %let YCENTER=-0.114; The Handelbrot CUrve Set set parameter values for testing results %let TERCT=100; %let XCENTER=-0.772; Data MANDEL(Keep=X Y) Length X Y 8 ; GRAN =&grain; TERCT = &TERCT; XCENTER = &XCENTER; YCENTER = &YCENTER; XRANGE =.01 ; YRANGE = XRANGE * 0.6 XSTART = XCENTER - DeRANGE XEND = XCENTER + (XRANGE YSTART = YCENTER - (YRANGE YEND = YCENTER + (YRANGE STEP = XRANGE / GRAN ; Set Granularity of The Display '" of terations Y-Axis Midpoint Y-Axis Midpoint X-Axis Range Set Aspect Ratio For Display / 2) Starting Value For Cl Loop / 2) Ending Value For Cl Loop / 2) Starting Value For C2 Loop / 2) Ending Value For C2 Loop C1 & C2 Loop ncrement --- (Cl,C2) Xs A Complex Number Whose Value xs Cl + C2 * X --- Do C1 = XSTART To XEND By STEP ; Do C2 = YSTART To YEND By STEP Link TESTPT ; f (PRS = 'Y') Then Do; End Return End X = C1 Y = C2 Output Test The X-Axis values Y-Axis values Value of (Cl,C2) This Point s A 'Prisoner', not flying off into infin- ity despite repeated squar- ing and adding End Do (C2) Loop End Do (Cl) Loop Test A Particular Value of (Cl,C2) *1 TESTPT: Z1 = 0 Z2 = 0 Do N = 1 To TERCT By 1 ; Sqnare Z ADd Add C To Xt SQZ1 = Zl * Zl - Z2 * Z2 ; SQZ2 = 2 * Zl * Z2 SQZl = SQZ1 + Cl SQZ2 = SQZ2 + C2 ; -- This Point xs Definitely Not A Member of The Set -- Return ; f «SQZl * SQZ1) + (SQZ2 * SQZ2) > 156) Then Do; PRS = 'N' Do Not Add To Curve Set Return Return For The Next Point To Test Zl SQZl Z2 = SQZ2 PRS = 'Y' ; Hold This Value And This One Too! End Do (N) Loop Point s OK, Add t To The Curve Set Return For The Next Point To Test NESUG '96 Proceedings 532

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5 Playing With Color 1* Create an Annotate Dataseset using the original dataset *1 1* First use program-to-write-a-program for PROC FORMAT PROC UNVARATE DATA=MANDEL NOPRNT; VAR Y; OUTPUT OUT=ANNOC MN=MNY RANGE=RANGEY; FLENAME TEMP 'C:\COLOR.PGM'; 1* The program is written out to here *1 DATA COLOR; SET ANNOC; FLE TEMP; 'PROC VALUE 1* create color mapping in PROC FORMAT ARRAY MAXRANGE {7} MAXl - MAX7; 1* Calculate Ranges *1 DO = 1 TO 7; MAXRANGE() = MNY + (RANGEY * (/8»; END; FORMAT MAX1 - MAX7 9.6; 1* gdevice PBASER allows 7 colors 'LOW - MAXl '="MAGENTA" 1* Put value MAXl '-' MA -2 MAX2 '-' MAX3 '='CYAN" MAX3 '-' MAX4 MAX4 '-' MAX5 MAX5 '-' MAX6 '="RED" MAX6,- HGH' '="BLACK" i' RUN; %NCLUDE TEMP; RUN; 1*** execute PROC FORMAT ***1 1* Convert Original Dataset nto Annotate Dataset DATA ANNOC; SET MANDEL; 1* initialize annotate variables *1 LENGTH FUNCTON STYLE COLOR $8 TEXT $1; RETAN HSYS '3' XSYS YSYS '2'; 1* create Observations for plot *1 FUNCTON='LABEL'; STYLE='SPECAL'; TEXT='A'; SZE=.l; COLOR=NPUT(PUT(Y,COLOR.),$6.); 1* 1* We lose titles, use DATASYS GOPTONS ROTATE=LANDSCAPE CBACK=WHTE PROC GANNO ANNOTATE=ANNOC DATASYS; plot dataset using PROC GARNO for exact X,Y coordinate values CTEXT=BLACK TARGETDEVCE=WNPRTG; *1 *1 NESUG '96 Proceedings 534