{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "word" -1 257 "" 0 0 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 } {PSTYLE "Problem" -1 256 1 {CSTYLE "" -1 -1 "" 0 0 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 3 4 0 0 0 0 -1 0 }{PSTYLE "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 257 "" 0 "" {TEXT -1 51 "Lesson 18: Application of Series: Geometric Curv es" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 19 "The Snowflake Curve" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 6 " The " }{TEXT 257 15 "Snowflake Cu rve" }{TEXT -1 171 " was initially described by Koch as an affirmat ive solution to the problem of whether there is a continuous curve tha t has no tangent line at any point on the curve. " }}{PARA 0 "" 0 " " {TEXT -1 785 "It is defined as the limit of the sequence of curves g enerated by the procedure snowflake given below. Type in the words d efined below and then enter snow(4) to get a feel for what the snowfla ke curve looks like. The first word performs a basic operation on any segment, which we think of as a list of its endpoints. It replaces t he segment of length d with a sequence of four segments of length d/3 \+ obtained in the following way: Start at the left endpoint of the segm ent and go d/3 of the way along the segment, turn left 60 degrees and \+ go the same distance, turn right 60 degrees and go the same distance, \+ then turn left 60 degrees and proceed d/3 units along the original seg ment to the right endpoint. Note that the resulting path has three po ints where there is no tangent." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "basic := proc(p1,p2)\nlocal dx,dy, p3,p4,p5;\ndx := (p2[1]-p1 [1])/3.;\ndy := (p2[2]-p1[2])/3.;\np3 := [p1[1]+dx,p1[2]+dy];\np4 := [ p1[1]+2*dx,p1[2]+2*dy];\np5 := [p1[1]+1.5*dx-sqrt(3.)/2*dy,\n \+ p1[2]+1.5*dy+sqrt(3.)/2.*dx];\np3,p5,p4,p2;\nend:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 108 "So for example, if we feed the points [0,0], [1,0 ] into basic, we get out the following sequence of numbers:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " basic([0,0],[1,0]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&7$$\"+LLLLL!#5\"\"!7$$\"+++++]F&$\"+Z8 v')GF&7$$\"+mmmmmF&F'7$\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 430 "Notice the left endpoint of the original segment is missing from \+ this list. That is for programming reasons which become clear in the \+ definition of the word flake below. The word flake takes a list of po ints, which represents a sequence of line segments laid end to end, an d returns a list representing 4 times as many line segments, where eac h segment in the original list has been replaced by the 4 segments ret urned by basic." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "flake : = proc(fl)\n local i,curve;\n curve := fl[1];\n for i from 1 to nops(fl)-1 do\n curve := curve ,basic(fl[i],fl[i+ 1] ) ;\n od\n end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "N ow the starting point for the snowflake curve is the equilateral trian gle." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "curve := [[0,0],[1/ 2,1/2*sqrt(3)],[1,0],[0,0]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&cur veG7&7$\"\"!F'7$#\"\"\"\"\"#,$*$-%%sqrtG6#\"\"$F*F)7$F*F'F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "To draw the second stage of the sn owflake curve," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot([fla ke(curve)],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7/7$\"\"!F(7$$\"1+++nmmm;!#;$\"1za_ Y8v')GF,7$$!1T.F`Pspx!#E$\"1WUl%p-Nx&F,7$$\"1+++MLLLLF,$\"1e40$p-Nx&F, 7$$\"1+++++++]F,$\"1'QWy.a-m)F,7$$\"1+++mmmmmF,$\"12*=8p-Nx&F,7$$\"1(p x+++++\"!#:$\"1U,>!p-Nx&F,7$$\"1+++KLLL$)F,$\"1GMzW8v')GF,7$$\"\"\"F(F (7$$\"1+++nmmmmF,F(7$F;$!1+++Z8v')GF,7$F6F(F'-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$%!GF]o-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$%(D EFAULTGFeo" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Now we can define the \+ nth stage of the snowflake curve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "snowflake := proc(n)\n local i,curve,ti;\ncurve := [ [0,0],[1/2,1/2*sqrt(3)],[1,0],[0,0]];\nfor i from 2 to n do curve := \+ [flake(curve)] od;\nti := cat(n,`th stage of the snowflake curve`); pl ot(curve,color=black,style = LINE,\naxes = NONE,scaling = CONSTRAINED, title = ti)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "snowfl ake(4);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CU RVESG6#7]w7$\"\"!F(7$$\"1+++_=&=&=!#<$\"1@Xv&\\,v?$F,7$$!1q`\"*p;kW>!# E$\"1\"yJE*H+:kF,7$$\"1+++/Pq.PF,$\"1U!4:*H+:kF,7$$\"1+++cbbbbF,$\"1jN E([/Di*F,7$$\"14uy.Pq.PF,$\"1#39$)f+IG\"!#;7$$!1n5[%>#=D9F2$\"1^J:)f+I G\"FD7$$\"1=[d^=&=&=F,$\"13=!zu]Pg\"FD7$$!1mgl\"yB)pBF2$\"1M&*e(*3]C>F D7$$\"1mF%3q.Pq$F,$\"1bSS(*3]C>FD7$$\"1mF%=bbbb&F,$\"1-j!o/^_C#FD7$$\" 1f`0+uS2uF,$\"1cd_'*3]C>FD7$$\"1+++66666FD$\"1-j5(*3]C>FD7$$\"1+++'H'H 'H\"FD$\"1W`LY5DXAFD7$$\"1Ob+66666FD$\"1\"*er'>,gc#FD7$$\"1+++#[\"[\"[ \"FD$\"1PkH(>,gc#FD7$$\"1+++nmmm;FD$\"1za_Y8v')GFD7$$\"1Ob+#[\"[\"[\"F D$\"1Eg!p\\,v?$FD7$$\"1ny$36666\"FD$\"1zaU'\\,v?$FD7$$\"1s5,'H'H'H\"FD $\"1?XbX;DGNFD7$$\"18)*=66666FD$\"1o]$ez,!\\QFD7$$\"1Lbo.uS2uF,$\"1@X: &z,!\\QFD7$$\"1%evpTFD7$$ !1zMvSnL;WF2$\"1n]B&4-0\\%FD7$$\"1,U1-Pq.PF,$\"18c,'4-0\\%FD7$$\"1^iz` bbbbF,$\"1bY9XAD6[FD7$$\"1kOe0Pq.PF,$\"1,_U&R-?8&FD7$$!1`#[.TMI4'F2$\" 1bY%fR-?8&FD7$$\"1lp!R%=&=&=F,$\"1)ptTa_FX&FD7$$!1T.F`PspxF2$\"1WUl%p- Nx&FD7$$\"1lp!Rp.Pq$F,$\"1RFS$p-Nx&FD7$$\"1op!Rbbbb&F,$\"1KQOWGD%4'FD7 $$\"1gOe0uS2uF,$\"1'G$)Qp-Nx&FD7$$\"1:K.56666FD$\"1#yJIp-Nx&FD7$$\"1?k ?&H'H'H\"FD$\"1B31UGD%4'FD7$$\"1n$e06666\"FD$\"1q8M#*H+:kFD7$$\"1D'z.[ \"[\"[\"FD$\"1k)*3\"*H+:kFD7$$\"1IGbmmmm;FD$\"1e40UJvNnFD7$$\"1!4uB&=& =&=FD$\"1iCg$*H+:kFD7$$\"1B$HBAAAA#FD$\"13Iy%*H+:kFD7$$\"1[`>Qq.P?FD$ \"1nR:XGD%4'FD7$$\"1.M%=AAAA#FD$\"1q89$p-Nx&FD7$$\"1ny$=f#f#f#FD$\"1l) *)>p-Nx&FD7$$\"1(4xyxxxx#FD$\"12*=1%GD%4'FD7$$\"1JB$='H'H'HFD$\"1g$Q3p -Nx&FD7$$\"1+++MLLLLFD$\"1e40$p-Nx&FD7$$\"1+++?&=&=NFD$\"1^?,WGD%4'FD7 $$\"10j'RLLLL$FD$\"1(f#*\\*H+:kFD7$$\"1+++1Pq.PFD$\"1WJ(\\*H+:kFD7$$\" 1+++!*))))))QFD$\"1M,ZUJvNnFD7$$\"1JB$Qq.Pq$FD$\"1I'=7H.l0(FD7$$\"1#*Q qLLLLLFD$\"1$3Q,H.l0(FD7$$\"1hYm<&=&=NFD$\"1Dr'*RMDxtFD7$$\"1(>qOLLLL$ FD$\"1A(z?f.!)p(FD7$$\"1ny$[q.Pq$FD$\"1q-c#f.!)p(FD7$$\"1^#=.*))))))QF D$\"1i8#Qu`(=!)FD7$$\"1Ob+wS2uSFD$\"1:3/$f.!)p(FD7$$\"1+++YWWWWFD$\"16 $*)=f.!)p(FD7$$\"1+++J'H'HYFD$\"1`$=6u`(=!)FD7$$\"1Y>NYWWWWFD$\"1**))H \"*Q]R$)FD7$$\"1+++;[\"[\"[FD$\"1&RZ.*Q]R$)FD7$$\"1+++++++]FD$\"1'QWy. a-m)FD7$$\"1+++%=&=&=&FD$\"1%H$)o)Q]R$)FD7$$\"1%pL]bbbb&FD$\"1YF?&)Q]R $)FD7$$\"1+++o.Pq`FD$\"1,A#ft`(=!)FD7$$\"1+++bbbbbFD$\"13_U)e.!)p(FD7$ $\"1pw;Ef#f#fFD$\"19nP*e.!)p(FD7$$\"13hH6666hFD$\"1fs0SPv=!)FD7$$\"1R` L(H'H'H'FD$\"1=#G.f.!)p(FD7$$\"1.)Htmmmm'FD$\"1@cr(e.!)p(FD7$$\"1K@;\" [\"[\"['FD$\"1v]`PMDxtFD7$$\"1\\$44p-Nx&FD7$$\"1_Y!='H'H'zFD$\"1k)*)3%GD%4'F D7$$\"1T#ekxxxx(FD$\"1e4X\"*H+:kFD7$$\"1;\"*zY\"[\"[\")FD$\"16/(3*H+:k FD7$$\"1&ymHLLLL)FD$\"12*=%RJvNnFD7$$\"1\"e$z<&=&=&)FD$\"1k)*Q!*H+:kFD 7$$\"1L;W))))))))))FD$\"1?$4'*)H+:kFD7$$\"1v.i-Pq.()FD$\"1r(G'RGD%4'FD 7$$\"1u.i()))))))))FD$\"1I(*\\!p-Nx&FD7$$\"1M;We#f#f#*FD$\"1%=>%*o-Nx& FD7$$\"1.$4YWWWW*FD$\"1J(**3%GD%4'FD7$$\"1.$4'H'H'H'*FD$\"1*oq1p-Nx&FD 7$$\"1(px+++++\"!#:$\"1U,>!p-Nx&FD7$$\"1/$4c\"[\"[\")*FD$\"1'f4(RDv_aF D7$$\"1I4;++++5Fdil$\"1b0[\"R-?8&FD7$$\"1M;WI'H'H'*FD$\"1+6'4R-?8&FD7$ $\"1&ym\\WWWW*FD$\"1a0=TAD6[FD7$$\"1zNzH'H'H'*FD$\"17:0#4-0\\%FD7$$\"1 jT/++++5Fdil$\"1vpTFD7$$\"1< Z/++++5Fdil$\"1GMp$z,!\\QFD7$$\"16fiI'H'H'*FD$\"1vRP%z,!\\QFD7$$\"1_Y! 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GFaan-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFA ULTGF]bn" 1 6 0 1 10 0 2 6 1 1 1 1.000000 45.000000 45.000000 0 0 "Cur ve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Here's an animation of \+ the snowflake curve 'growing in place'. 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F^bv7$$\"1+++hw)4K%FaqFjgwFacn7$$\"1+++0e8pCFaqF,7$$FeiwFaqFjgw7$$Fbjw FaqF,F+F6F:-F?6#%A5th~stage~of~the~snowflake~curveGFBFFFJFNFFFJ" 1 2 0 1 10 0 2 9 1 1 1 1.000000 44.000000 46.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 256 "" 0 "" {TEXT -1 52 "Problem: What is the length of \+ the snowflake curve?" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 256 9 "Solution:" }{TEXT -1 88 " To calculat e the length of the snowflake curve we see that the first stage has le ngth " }{XPPEDIT 18 0 "s1 = 3" "6#/%#s1G\"\"$" }{TEXT -1 33 " units; t he 2nd stage has length " }{XPPEDIT 18 0 "s2=s1+s1/3" "6#/%#s2G,&%#s1G \"\"\"*&F&F'\"\"$!\"\"F'" }{TEXT -1 6 " , or " }{XPPEDIT 18 0 "4/3 *s1 " "6#*(\"\"%\"\"\"\"\"$!\"\"%#s1GF%" }{TEXT -1 42 " ; in general the nth stage has length " }{XPPEDIT 18 0 "sn = s(n-1) + 1/3 *s(n-1)" "6 #/%#snG,&-%\"sG6#,&%\"nG\"\"\"F+!\"\"F+*(F+F+\"\"$F,-F'6#,&F*F+F+F,F+F +" }{TEXT -1 6 ", or " }{XPPEDIT 18 0 "(4/3)^(n-1)*s1" "6#*&)*&\"\"% \"\"\"\"\"$!\"\",&%\"nGF'F'F)F'%#s1GF'" }{TEXT -1 120 " . So the leng th of the nth stage goes to infinity as n gets large. This says the s nowflake curve is infinitely long.\n" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 13 "A Spacefiller" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 395 " Da vid Hilbert in 1891 described a continuous curve whose range is the un it square! This was contrary to the intuition of the time, which was \+ that the range of a path had to be 1-dimensional. Here is a Maple wor d that draws an approximation to Hilbert's space filling curve . T he approach to defining the drawing procedure peano (below) is similar to that used to define snowflake (above)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "basic := proc(p1 ,p2,p3)\nlocal dx,dy,p4,p5,p6,p7,p8,p9;\np4 := .5*(p1+p2);\np9 := .5*( p2+p3);\np5 := p4+(p2-p9);\np6 := p2+(p2-p9);\np7 := p2+(p4-p1);\np8 : = p9+(p4-p1);\np4,p5,p6, p2 ,p7,p8,p9, p3 ;\nend:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 138 "peano := proc(fl)\nlocal i,cur;\ncur := [f l[1] ] ;\nfor i from 1 by 2 to nops(fl)-2 do\ncur := [op(cur),basic( \+ fl[i],fl[i+1],fl[i+2] )]\nod;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fl := [[0,0],[1,1],[2 ,0]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#flG7%7$\"\"!F'7$\"\"\"F)7$ \"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "for i from 1 to 4 do fl := peano(fl) od:\nti := cat(4,`th stage of Peano's curve`);\n plot(fl,style=LINE,axes=NONE,title=ti,\nscaling=CONSTRAINED);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tiG%;4th~stage~of~Peano's~curveG" } }{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVESG6$7][ m7$\"\"!F(7$$\"1++++++]i!#F_u7$F.F^y7$F2F`y7$F7F^y7$F 2F[y7$F7Ffx7$F