{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "" -1 256 "" 1 24 128 0 128 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 14 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 }{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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 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 "" 0 257 1 {CSTYLE "" -1 -1 "" 1 18 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 1 18 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 "" 0 259 1 {CSTYLE "" -1 -1 "" 1 18 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 "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 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 "" 0 264 1 {CSTYLE "" -1 -1 "" 1 14 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 "" 0 265 1 {CSTYLE "" -1 -1 "" 1 14 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 "" 0 266 1 {CSTYLE "" -1 -1 "" 1 14 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 "" 0 267 1 {CSTYLE "" -1 -1 "" 1 14 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 "" 0 268 1 {CSTYLE "" -1 -1 "" 1 14 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 "" 0 269 1 {CSTYLE "" -1 -1 "" 1 14 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 256 33 "\301rvore de Pit\341goras bidimensional" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 26 "Yolanda Kioko Saito Furuya" }}{PARA 258 "" 0 "" {TEXT 258 26 "Departamento de Matem\341ti ca" }}{PARA 259 "" 0 "" {TEXT 259 35 "Universidade Federal de S\343o C arlos." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 261 256 "Apresentamos aos estudantes interessados um programa qu e gera o fractal chamado \301rvore de Pit\341goras. Temos aqui a vers \343o bidimensional do fractal. Este programa pode ser executado no Ma ple V Release 4 e possivelmente em vers\365es posteriores, e faz parte do " }{TEXT 260 20 "Hipertexto Pit\341goras" }{TEXT 263 354 ", dispon \355vel na Internet a partir do endere\347o http://www.dm.ufscar.br Ne ste Hipertexto est\341 tamb\351m dispon\355vel um programa an\341logo \+ para gerar a vers\343o tridimensional do fractal. Nossa figura foi ada ptada de um programa proposto por Harm Derksen. A figura de Derksen po de ser acessada na Internet no endere\347o http://www.maplesoft.com/cy bermath/samples.html" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots): with(linalg):" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT 267 42 "Definindo as propor \347\365es do tri\342ngulo base:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "cateto0:= 3: cateto1 := 4: h ip0 := sqrt(cateto0^2+cateto1^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "tri0 := [[cateto0,0], [0,0], [0,cateto1]]:" }}} {EXCHG {PARA 266 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 269 32 "Fun\347\365es e defini\347\365es auxiliares:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "pit0 := [[[ cateto0,-cateto0],[0,-cateto0],[0,0],[cateto0,0]]," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "[[-cateto1,0],[-cateto1,cateto1],[0,cateto1],[0,0]] , tri0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "u := evalm(tri0 [3]-tri0[1]): u2 := norm(u,2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "mov := proc(X,v,P) local lambda, Q, x,y, v2, s;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 " v2 := norm(v,2): lambda := v2/u2;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " s :=solve(\{x*u[1]-y*u[2] = \+ u2*v[1]/v2, y*u[1]+x*u[2] =" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "u2*v [2]/v2\},\{x,y\}); assign(s): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 " \+ Q := convert(evalm(lambda*matrix(2,2,[x,-y, y,x])&*(X-tri0[1])+P ), 'list');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " RETURN(Q);" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " end:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " movpit0 := proc(P,Q) local v; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " \+ v:= evalm(Q-P); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "[map(mov,pit 0[1],v,P),map(mov,pit0[2],v,P),map(mov,pit0[3],v,P)] end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " Ploti := proc(i) local fig00, fig01, fig02, movpit; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " movpit := Pit[i]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " fig00 := polygonplot(movpit[1], color=y ellow):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 " fig01 := polygonplot(m ovpit[2], color=green):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " fig02 \+ := polygonplot(movpit[3], color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " display(\{fig00, fig01, fig02\}, scaling=constrained): " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " end:" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT 271 19 "In\355cio da sequ\352n cia" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Pit[0] := [[0,0], [hip0,0], [hip0,-hip0], [0, -hip0]] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "PitPlot[0] := polygonp lot(Pit[0], scaling=constrained, axes=none," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "color=cyan): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Pit[1] := [op(movpit0(Pit[0][1], Pit[0][2]))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "PitPlot[1] := display( \{PitPlot[0] ,Ploti(1)\}): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "epsilon : =0.3: Pare := false:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Val e[1] := true: Vale[0] := true:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "n := 5: k := 1: # n \351 n\372mero de itera\347\365es e k a or dem inicial" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 146 "O valor de n aci ma pode ser modificado para aumentar ou diminuir o n\372mero de itera \347\365es. Valores altos podem aumentar muito o tempo de computa\347 \343o. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "while k <= n do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " for i from 2^k to 2^(k+1)-1 \+ do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " j1 := iquo(i,2): r1 := irem(i,2): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " if Vale[j1] = \+ false then Vale[i] := false: PitPlot[i] :=" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "PitPlot[i-1]: break " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " elif r1 =0 then P:=Pit[j1][1][1]: Q:=Pit[j1][1][2]: " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " else P:=Pit[j1][2][1]: Q:=Pi t[j1][2][2]: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " Pit[i] :=m ovpit0(P,Q): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " PitPlot[i] := display( \{PitPlot[i-1],Ploti(i)\}):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " if (norm(Q-P,infinity) " 0 "" {MPLTEXT 1 0 20 "Vale[i] := true: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "k:= k+1: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 27 "Desenhan do o \372ltimo est\341gio" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "i: display(PitPlot[i-1]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 34 "A nima\347\343o da sequ\352ncia de itera\347\365es" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a := floor( (i-1)/10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " display([seq (PitPlot[a*k], k=0..10)], insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }