{VERSION 5 0 "SGI MIPS UNIX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Headi ng 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 48 "Maple Summer Workshop 2004 : Programming Tutorial" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Michael Monagan." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "These examples illu strate recursive programs which assemble PLOT's. The help page for wh at kind of objects you can have in a PLOT and PLOT3D object is in ?plo t, structure.\nOur examples use CURVES and POLYGONS." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "The Brocolli fractal" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 301 "The Brocolli fractal is formed by constructing a \"house\" with r oof height alpha, and then making two houses (recursively) on top of t he two sides of the roof. See picture below. This program constructs the sequence of points going over the fractal from left to right and \+ puts them in a CURVES(...). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 414 "broccoli := proc(n::posint, alpha)\n PLOT( CURVES( [[0,0],br oc([0,0],[1,0],n,alpha),[1,0]] ), \n AXESSTYLE(NONE), SCALING (CONSTRAINED), THICKNESS(2) );\nend:\n\nbroc := proc(p,q,n,alpha,c) lo cal u,v,r,s,t;\n if n=0 then RETURN() fi; \n u := q-p;\n v := [-u[2 ],u[1]]; # rotate u clockwise by Pi/2\n r := p+v;\n s := q+v;\n t : = p+0.5*u+(1.0+alpha)*v;\n r, broc(r,t,n-1,alpha), t, broc(t,s,n-1,al pha), s;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "broccoli( 1,0.2);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6#7'7$\"\"!F(7$F(\" \"\"7$$\"\"&!\"\"$\"#7F.7$F*F*7$F*F(-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6 #%,CONSTRAINEDG-%*THICKNESSG6#\"\"#" 2 462 402 402 2 0 1 10 2 2 9 0 1 1 1.000000 45.000000 45.000000 10030 10061 10056 10072 0 0 0 20540 0 12010 0 0 0 0 0 0 0 1 2 0 0 0 285 82 0 0 0 0 0 0 0 "Curve 1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "lprint(broccoli(1,0.2));" }} {PARA 6 "" 1 "" {TEXT -1 107 "PLOT(CURVES([[0, 0], [0, 1], [.5, 1.2], \+ [1, 1], [1, 0]]),AXESSTYLE(NONE),SCALING(CONSTRAINED),THICKNESS(2))" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "broccoli(2,0.2);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6#7-7$\"\"!F(7$F(\"\"\"7$$!\"#!\"\"$ \"#:F.7$$F*F-$\"$q\"F-7$$\"\"$F.$\"# " 0 "" {MPLTEXT 1 0 24 "lprint(broccoli(2,0.2));" }} {PARA 6 "" 1 "" {TEXT -1 181 "PLOT(CURVES([[0, 0], [0, 1], [-.2, 1.5], [.1e-1, 1.70], [.3, 1.7], [.5, 1.2], [.7, 1.7], [.99, 1.70], [1.2, 1. 5], [1, 1], [1, 0]]),AXESSTYLE(NONE),SCALING(CONSTRAINED),THICKNESS(2) )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "broccoli(8,0.2);" }} {PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6#7[[n7$\"\"!F(7$F(\"\"\"7$$!\" #!\"\"$\"#:F.7$$!#SF-$\"$r\"F-7$$!$U&!\"$$\"%vKqFK$\"(row\"FK7$$!'fZoFK$\"(5Gy\"FK7$$!'>\"4(FK$\"(**Ry\"FK7$$!(o` @(FQ$\"):sz(FQ$\")b#Q\"=FQ7$$!'IzqFK$\"(f$3=FK7$$!&d$oFE$\"'<2=FE7$$!&vS'FE$\"'# >#=FE7$$!&$>oFE$\"'xS=FE7$$!'!H1(FK$\"(f>%=FK7$$!(yq=(FQ$\"):oP=FQ7$$! )6ILsFW$\"*!\\.V=FW7$$!(`)HsFQ$\")$*4]=FQ7$$!(vc5(FQ$\")oPa=FQ7$$!((4D sFQ$\")L%)f=FQ7$$!)*[;A(FW$\"*q2p'=FW7$$!(K/<(FQ$\")byr=FQ7$$!',^qFK$ \"(>j'=FK7$$!'T2oFK$\"(I^'=FK7$$!'hvpFK$\"(\"z#)=FK7$$!(K]4(FQ$\")vD)) =FQ7$$!)Re\"4(FW$\"*!>K&*=FW7$$!(n./(FQ$\")(*>+>FQ7$$!(X4#pFQ$\")Kt%*= FQ7$$!(B(ppFQ$\")x#p!>FQ7$$!)h]=pFW$\"*]0=\">FW7$$!(yx%oFQ$\")b!=\">FQ 7$$!'+*z'FK$\"(6'**=FK7$$!&3j'FE$\"'&>)=FE7$$!%>iF?$\"&J'=F?7$$!%z`F?$ \"&!f=F?7$$!%ffF?$\"&*>>F?7$$!&3P'FE$\"'vQ>FE7$$!'S9mFK$\"(R*R>FK7$$!( y&QnFQ$\"):mN>FQ7$$!)6![y'FW$\"*!\\,T>FW7$$!(`8y'FQ$\")$z![>FQ7$$!(vrl 'FQ$\")oN_>FQ7$$!((fwnFQ$\")L#y&>FQ7$$!)*[Jx'FW$\"*q()['>FW7$$!(K>s'FQ $\")bwp>FQ7$$!'^-mFK$\"(*Hk>FK7$$!'\"*ejFK$\"(5J'>FK7$$!'6FlFK$\"(r2)> FK7$$!(Klk'FQ$\")vB')>FQ7$$!)R3VmFW$\"*!>I$*>FW7$$!(n=f'FQ$\")(z\")*>F Q7$$!(XCZ'FQ$\")Kr#*>FQ7$$!(B7_'FQ$\")x!\\+#FQ7$$!)h+qkFW$\"*]&y4?FW7$ $!(y#*R'FQ$\")by4?FQ7$$!']]jFK$\"(\"f(*>FK7$$!&B='FE$\"'$*z>FE7$$!&0x& FE$\"'3h>FE7$$!&(QfFE$\"'8.?FE7$$!'!p5'FK$\"(\"z??FK7$$!(AjA'FQ$\")vDE ?FQ7$$!)R(GA'FW$\"*!>KL?FW7$$!(d;<'FQ$\")(*>Q?FQ7$$!(NA0'FQ$\")KtK?FQ7 $$!(855'FQ$\")x#\\/#FQ7$$!)hz\\gFW$\"*]0)\\?FW7$$!(o!zfFQ$\")b!)\\?FQ7 $$!'HIfFK$\"(6w.#FK7$$!'4idFK$\"(]*>?FK7$Fgel$\"(RV/#FK7$$!(o3\"eFQ$\" )N`c?FQ7$$!)6lfdFW$\"*I691#FW7$$!(B*)o&FQ$\")8Th?FQ7$$!(X,k&FQ$\")o@\\ ?FQ7$$!(n8f&FQF[gl7$$!)*Q1_&FWFffl7$$!(A%paFQFafl7$$!&#=bFEF\\fl7$F\\h l$\"'&*>?FE7$$!%]`F?$\"&z(>F?7$$!$x%F9$\"%<>F97$$!$N$F9$\"%_=F97$$!$$R F9$\"%(*>F97$$!%5XF?$\"&z0#F?7$$!&=#\\FE$\"'vw?FE7$$!'Sl^FK$\"(Rz2#FK7 $$!(y&*G&FQ$\"):mt?FQ7$$!)6!eL&FW$\"*!\\,z?FW7$$!(`BL&FQ$\")$zg3#FQ7$$ !(v\"3_FQ$\")oN!4#FQ7$$!((fF`FQ$\")L#e4#FQ7$$!)*[TK&FW$\"*q()G5#FW7$$! 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18 "Serpinski's Gasket" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "The follow ing draws Serpinski's gasket by staring with an equilateral triangle a nd \"cutting out the equilateral triangle in the middle\" by drawing t he four equaliaterateral triangles around it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "serpinski := proc(n::posint)\nPLOT( CURVES([ser p([0,0],[0.5,0.8660254040],[1,0],n)]),\n AXESSTYLE(NONE), SCALING (CONSTRAINED) );\nend:\n\nserp := proc(p,q,r,n) local m1,m2,m3;\n\nif \+ n=0 then RETURN(p,q,r,p) fi;\nm1 := 0.5*(p+q);\nm2 := 0.5*(q+r);\nm3 : = 0.5*(r+p);\nserp(p,m1,m3,n-1), m1, serp(m1,q,m2,n-1), m3, serp(m3,m2 ,r,n-1), p\nend:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "serpi nski(1);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6#717$\"\"!F(7$$\"# D!\"#$\"+?q7IV!#57$$\"\"&!\"\"$F(F(F'F)F)7$F1$\"+SSDg')F/7$$\"#vF,F-F) F0F0F87$\"\"\"F(F0F'-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG " 2 311 311 311 2 0 1 10 0 2 9 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 "Curve 1" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "lprint(serpinski(1));" }} {PARA 6 "" 1 "" {TEXT -1 262 "PLOT(CURVES([[0, 0], [.25, .4330127020], [.5, 0.], [0, 0], [.25, .4330127020], [.25, .4330127020], [.5, .86602 54040], [.75, .4330127020], [.25, .4330127020], [.5, 0.], [.5, 0.], [. 75, .4330127020], [1, 0], [.5, 0.], [0, 0]]),AXESSTYLE(NONE),SCALING(C ONSTRAINED))" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "serpinski(5 );" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6#7c^p7$\"\"!F(7$$\"&Dc\" !\"'$\"+)QHjq#!#67$$\"%DJ!\"&$F(F(F'F)F)7$F1$\"+v(eET&F/7$$\"&vo%F,F-F )F0F0F87$$\"$D'!\"%F4F0F'F5F57$F9$\"+j\"))*=\")F/7$$\"&+D'F,$\"+w(eET& F/F5F?F?7$F<$\"+b<`#3\"!#57$$\"&D\"yF,F@F?FBFBFK7$$\"%v$*F3F6FBF5F;F;7 $FLF-7$FOF4F;FQFQFN7$$\"'v$4\"F,F-FQFRFRFS7$$\"$D\"!\"$F4FRF;F'FGFG7$F L$\"+%pkJN\"FJ7$$\"&]P*F,FHFGFZFZ7$FO$\"+KwzB;FJ7$FTFenFZFgnFgnF]o7$$ \"&+D\"F3FHFgnFGFjnFjn7$FT$\"+r0V%*=FJ7$$\"'+]7F,F[oFjnFaoFao7$FW$\"+5 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7$FhbmF-7$F[cmF4F[bmFfcmFfcm7$F[cmF67$FbcmF-FfcmFgcmFgcmFicm7$FecmF4Fg cmF[bmFhcmFhcm7$FbcmF@7$$\"'+v$*F,FEFhcmF[dmF[dmFdcm7$$\"'DJ&*F,F@F[dm F\\dmF\\dmF_dm7$$\"&vo*F3F6F\\dmFhcmFjcmFjcm7$F`dmF-7$FcdmF4FjcmFedmFe dmFbdm7$$\"'vV)*F,F-FedmFfdmFfdmFgdm7$\"\"\"F(FfdmFjcmF[bmFh\\mFj[lF'- %*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG" 2 483 483 483 2 0 1 10 0 2 9 0 1 1 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20540 0 12010 0 0 0 0 0 0 0 1 0 0 0 0 308 385 0 0 0 0 0 0 0 "Curve 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "The following does this in 3 dimensions using a regular tetrahedron with vertices (1,1,1), (-1,- 1,1), (-1,1,-1), and (1,-1,-1). It draws recursively the four tetrahe dra on the four corners." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 428 "serpinski3d := proc(n)\n PLOT3D(serp3d(n, [1,1,1], [-1,-1,1], [- 1,1,-1], [1,-1,-1]),\n SCALING(CONSTRAINED),COLOR(RGB,1,0,1)); \nend:\n\nserp3d := proc(n, a, b, c, d)\n if n=0 then return POLYGON S([a,b,c], [a,b,d], [a,c,d], [b,c,d]) fi;\n serp3d(n-1, a, (a+b)/2, \+ (a+c)/2, (a+d)/2),\n serp3d(n-1, (a+b)/2, b, (b+c)/2, (b+d)/2),\n \+ serp3d(n-1, (a+c)/2, (b+c)/2, c, (c+d)/2),\n serp3d(n-1, (a+d)/2, (b +d)/2, (c+d)/2, d)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "serpinski3d(1);" }}{PARA 13 "" 1 "" {INLPLOT "6(-%)POLYGONSG6&7%7%\" \"\"F(F(7%\"\"!F*F(7%F*F(F*7%F'F)7%F(F*F*7%F'F+F-7%F)F+F--F$6&7%F)7%! \"\"F4F(7%F4F*F*7%F)F37%F*F4F*7%F)F5F77%F3F5F7-F$6&7%F+F57%F4F(F47%F+F 57%F*F*F47%F+F=F?7%F5F=F?-F$6&7%F-F7F?7%F-F77%F(F4F47%F-F?FF7%F7F?FF-% (SCALINGG6#%,CONSTRAINEDG-%&COLORG6&%$RGBGF(F*F(" 3 359 359 359 2 0 1 10 0 2 6 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -13096 -16395 0 0 0 0 0 0 0 "Curve 1" "Curve 2" "Cur ve 3" "Curve 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "lprint(se rpinski3d(0));" }}{PARA 6 "" 1 "" {TEXT -1 209 "PLOT3D(POLYGONS([[1, 1 , 1], [-1, -1, 1], [-1, 1, -1]],[[1, 1, 1], [-1, -1, 1], [1, -1, -1]], [[1, 1, 1], [-1, 1, -1], [1, -1, -1]],[[-1, -1, 1], [-1, 1, -1], [1, - 1, -1]]),SCALING(CONSTRAINED),COLOR(RGB,1,0,1))" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "serpinski3d(3);" }}{PARA 13 "" 1 "" {INLPLOT " 6^o-%)POLYGONSG6&7%7%\"\"\"F(F(7%#\"\"$\"\"%F*F(7%F*F(F*7%F'F)7%F(F*F* 7%F'F-F/7%F)F-F/-F$6&7%F)7%#F(\"\"#F6F(7%F6F*F*7%F)F57%F*F6F*7%F)F8F:7 %F5F8F:-F$6&7%F-F87%F6F(F67%F-F87%F*F*F67%F-F@FB7%F8F@FB-F$6&7%F/F:FB7 %F/F:7%F(F6F67%F/FBFI7%F:FBFI-F$6&7%F57%#F(F,FPF(7%FPF6F*7%F5FO7%F6FPF *7%F5FQFS7%FOFQFS-F$6&7%FO7%\"\"!FZF(7%FZFPF*7%FOFY7%FPFZF*7%FOFenFgn7 %FYFenFgn-F$6&7%FQFen7%FZF6F67%FQFen7%FPFPF67%FQF]oF_o7%FenF]oF_o-F$6& 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[m7%FdjlF\\[m7%F(F]tF]t7%FdjlFc[mFj[m7%F\\[mFc[mFj[m-%(SCALINGG6#%,CON STRAINEDG-%&COLORG6&%$RGBGF(FZF(" 3 483 483 483 2 0 1 10 0 2 6 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5 " "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18 " "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "C urve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37 " "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "C urve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56 " "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "C urve 63" "Curve 64" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 286 "One can sh ow that the fractal dimesion of Serpinski's gasket in 3D is 2, i.e. it has area. This one can see if one rotates the visual so that it fill s up a plane as follows. Try increasing the recursion to verify this. Rotate this object so you can see that it really is the fractal." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plots[display3d](%,orientat ion=[-90,180]);" }}{PARA 13 "" 1 "" {INLPLOT "6_o-%)POLYGONSG6'7%7%\" \"\"F(F(7%#\"\"$\"\"%F*F(7%F*F(F*7%F'F)7%F(F*F*7%F'F-F/7%F)F-F/-%&COLO RG6&%$RGBGF(\"\"!F(-F$6'7%F)7%#F(\"\"#F;F(7%F;F*F*7%F)F:7%F*F;F*7%F)F= F?7%F:F=F?F2-F$6'7%F-F=7%F;F(F;7%F-F=7%F*F*F;7%F-FEFG7%F=FEFGF2-F$6'7% F/F?FG7%F/F?7%F(F;F;7%F/FGFN7%F?FGFNF2-F$6'7%F:7%#F(F,FUF(7%FUF;F*7%F: FT7%F;FUF*7%F:FVFX7%FTFVFXF2-F$6'7%FT7%F6F6F(7%F6FUF*7%FTFhn7%FUF6F*7% FTFinF[o7%FhnFinF[oF2-F$6'7%FVFin7%F6F;F;7%FVFin7%FUFUF;7%FVFaoFco7%Fi nFaoFcoF2-F$6'7%FXF[oFco7%FXF[o7%F;F6F;7%FXFcoFjo7%F[oFcoFjoF2-F$6'7%F E7%FUF*F;7%FUF(FU7%FEF`p7%F;F*FU7%FEFapFcp7%F`pFapFcpF2-F$6'7%F`pFao7% F6F*FU7%F`pFao7%FUF;FU7%F`pFipF[q7%FaoFipF[qF2-F$6'7%FapFip7%F6F(F67%F apFip7%FUF*F67%FapFaqFcq7%FipFaqFcqF2-F$6'7%FcpF[qFcq7%FcpF[q7%F;F;F67 %FcpFcqFjq7%F[qFcqFjqF2-F$6'7%FN7%F*FUF;7%F*F;FU7%FNF`r7%F(FUFU7%FNFar Fcr7%F`rFarFcrF2-F$6'7%F`rFjo7%F;FUFU7%F`rFjo7%F*F6FU7%F`rFirF[s7%FjoF irF[sF2-F$6'7%FarFirFjq7%FarFir7%F*FUF67%FarFjqFbs7%FirFjqFbsF2-F$6'7% FcrF[sFbs7%FcrF[s7%F(F6F67%FcrFbsFis7%F[sFbsFisF2-F$6'7%Fhn7%#!\"\"F,F `tF(7%F`tF6F*7%FhnF_t7%F6F`tF*7%FhnFbtFdt7%F_tFbtFdtF2-F$6'7%F_t7%#Fat F " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "4" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }