{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Helvetica" 0 0 0 0 128 1 0 1 0 0 1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Avalon" 1 12 128 128 128 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "PRE" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "OL" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 }0 0 0 0 4 4 3 10 0 0 2 0 -1 0 }{PSTYLE "R3 Font 0" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 128 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 260 1 {CSTYLE "" -1 -1 "Avalon" 1 12 0 128 128 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 261 1 {CSTYLE "" -1 -1 "Avalon" 1 12 0 0 0 1 2 2 2 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 22 "Fractals for Release 4" }} {PARA 0 "" 0 "" {TEXT -1 12 "May 23, 1999" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Original file location: " }}{PARA 0 "" 0 "" {TEXT -1 67 "http://www.math.utsa.edu/mirrors/maple/maplev/f ractals/fraktale.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 96 "------------------------------------------------------- -----------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Credits: " }}{PARA 0 "" 0 "" {TEXT -1 81 "John Oprea, oprea@math.csuohio.edu, implemetented the J procedu re to pass initial" }}{PARA 0 "" 0 "" {TEXT -1 81 "points and the anon ymous functions to the fractal procedures. His procedures also" }} {PARA 0 "" 0 "" {TEXT -1 34 "caused me to add further fractals." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "The escap e-time fractal procedures are based on a Maple V algorithm taken from \+ the book " }}{PARA 0 "" 0 "" {TEXT -1 89 "'Maple V - Programming Guide ' by M.B. Monagan, K.O. Geddes, G. Labahn and S. Vorkoetter, " }} {PARA 0 "" 0 "" {TEXT -1 16 "Springer Verlag." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Remark on the first proce dure 'mandelbrot': " }}{PARA 0 "" 0 "" {TEXT -1 995 "The modification on mandelbrot uses the fact that Maple can color things according to \+ a function. Also, plotting the zero function keeps the graph in a plan e. Increasing the values of 'grid' in the plot3d command results in \+ better resoultion of the fractal. However, this means longer computat ion time. 'style=patchnogrid' draws the surface with shaded rectangul ar patches with no wireframe grid. This is the best 'style' option fo r this routine. The procedure mandelbrot returns the value m. The hig her m the more likely the probability that the point (x, y) is part o f the Mandelbrot set. Certainly m depends of the number of iterations (here: 30). It is possible that a point satisfies the condition abs( z)<2 after n iterations but is outside the range after n+1 iterations . So to make sure that (x, y) is part of the set you would have to do \+ infinite iterations. m is used in the 'color=mandelbrot' option to vi sualize the set: the red area - the Mandelbrot lake - indicates that " }}{PARA 0 "" 0 "" {TEXT -1 24 "it is part of the set. " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "mandelbrot:=proc(x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " local z, c, m;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " c:= evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " z:=evalf(x+y*I) ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " for m from 0 to 30 while ab s(z)<2 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " z:=z^2+c" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot3d(0, -2 .. 0.7, -1.2 .. 1.2, orientation=[-90,0], grid=[250, 250], style=patchnogrid, " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " scaling=constrained, color=mande lbrot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "plot3d(0, -1.3 \+ .. -1.25, 0.35 .. 0.4, orientation=[-90,0], grid=[100, 100], style=pat chnogrid, \n scaling=constrained, color=mandelbrot);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "mandelbrot:=proc(x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " local z, m;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " z:=evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " for m fro m 0 to 25 while abs(z)<2 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " \+ z:=z^2+(x+y*I)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot3d(mandelbrot, \+ -2 .. 0.5, -1.2 .. 1.2, grid=[50, 50]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Julia sets modified by John Oprea, oprea@math.csuohio.edu " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "julia := proc(c, x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " local z, m;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " z := evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " for m f rom 0 to 30 while abs(z) < 3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " z := z^2+c" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "J := proc(d)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " global phonyvar;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 " phonyvar := d;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " (x, y) -> julia(phonyvar, x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 " plot3d(0, -2 .. 2, -1.3 ..1.3, style=patchnogrid, orientation=[-90,0], grid=[50, 50]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "scaling=constrai ned, color=J(-1.25));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "pl ot3d(0, -1.9 .. 1.9, -1.9 ..1.9, style=patchnogrid, orientation=[-90,0 ], grid=[50, 50]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "scaling=constr ained, color=J(-.74543+.11301*I));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot3d(0, -1.9 .. 1.9, -1.2 ..1.2, style=patchnogrid, orientation=[-90,0], grid=[150, 150]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "scaling=constrained, color=J(-.9+.12*I));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "# LAMBDAFN" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "lambdafn_sin:=proc(x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " \+ local z, m, p1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " z:=evalf(x+y* I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " p1:=evalf(1+I*0.4);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " for m from 0 to 100 while abs(z) <4 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " z:=sin(z)*p1" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot3d(0, -4 .. 4, -3 .. 3, \+ orientation=[-90,0], grid=[50, 50], style=patchnogrid, " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "scaling=constrained, color=lambdafn_sin);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plot3d(0, -1.5 .. 1.5, -1.1 \+ .. 1.1, orientation=[-90,0], grid=[250, 250], style=patchnogrid, " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "scaling=constrained, color=lambdafn _sin);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# BIOMORPH2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# May 22, 1996" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "biomorph2:=proc(x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " local z, m, c;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " z:=evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " c:=z; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " for m from 0 to 100 while ab s(z)<4 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " z:=cos(z)*z^2+c " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot3d(0, -2 .. 2, -1.5 .. 1 .5, orientation=[-90,0], grid=[20, 20], style=patchnogrid, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "scaling=constrained, color=biomorph2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "# NEWTON " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "# Visualization of Newton's method to calculate the c omplex solution of z^n-1=0 by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "# the iteration z(j+1) = z(j) - fn(z)/diff(fn(z)) with fn(z) \+ := z^n-1, and n > 2 the polynomial degree" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "# of fn (also real values for n are allowed)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(z^3-1=0, z);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "# The graph shows the starti ng points from which the method does or does not " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 89 "# converge to one of the above roots. The do cumentation of FRACTINT explains the result: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "# \"The coloring of the plot shows the \"basins \+ of attraction\" for each root of the polynomial " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# -- i.e., an initial guess within any area o f a given color would lead you to one of the roots. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "# As you can see, things get [fractal] a long certain radial lines or \"spokes,\" those being the lines" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# between actual roots.\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "# The fractal properties of Netwon's method to c alculate complex roots were first" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# discovered by John Hubbard of Orsay, France." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "newton:=proc(x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " local z, m;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " z:=evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " for m from 0 to 50 while abs(z^3-1) >= 0.001 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " z:=z-(z^3-1)/(3*z^2)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot3d(0, -2 .. 2, -1.5 .. 1.5, orientation=[-90,0], \+ grid=[250, 250], style=patchnogrid, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "scaling=constrained, color=newton);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "cmplxmarkmand:=proc(x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# \+ May 25, 1996" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# The original form ula 'z:=z^2*c^(p-1)+c' was developed by Mark Peterson." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 " local c, m, z;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " c:=evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " z:=evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " for m from 0 to 30 while abs(z)<2 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " z:=z^2*c^0.1+c" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " ln(m)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plo t3d(0, -2 .. 0.7, -1.2 .. 1.2, orientation=[-90,0], grid=[250, 250], s tyle=patchnogrid, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "scaling=const rained, color=cmplxmarkmand);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "lambda:=p roc(x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "# Lambda" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "# May 26, 1996" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "# " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "# Complex version of t he population growth equation x(j+1)=r*x(j)*(1-x(j))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "# developed by P. F. Verhulst in 1845" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "# Lambda is part of the Julia family" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " local c, z, m;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " c:=0.85+I*0.6;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " z:=evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " f or m from 0 to 30 while abs(z)<3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " z:=c*z*(1-z)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "plot3d(0, -1 .5 .. 2.5, -1.5 ..1.5, style=patchnogrid, orientation=[-90,0], grid=[2 5, 25], " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "scaling=constrained, co lor=lambda);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ifs:=proc(imax)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# Iterated Functions Systems" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 82 "# original program 'Chaos-Spiel fuer ein Farnb latt' written by Heinz-Otto Peitgen," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# Hartmut Juergens and Dietmar Saupe in BASIC," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "# published in: 'Bausteine des Chaos - Fraktale' , Springer Verlag/Klett-Cotta, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " # p. 415, written by the authors mentioned above" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "# June 06, 1997" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " local e1, e2, e3, e4, f1, f2, f3, f4, x, y, xn, yn, z, pts;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " pts:=NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " e1:=0.5; e2:=0.57; e3:=0.408; e4:=0.1075;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " f1:=0; f2:=-0.036; f3:=0.0893; f 4:=0.27;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " x:=e1;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 " y:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " to imax do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " z:=rand()/1e 12;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " if z<=0.02 then " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " xn:=e1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " yn:=0.27*y+f1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " elif z<=0.17 then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " xn:=-0.139*x+0.263*y+e2;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 " yn:=0.246*x+0.224*y+f2" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 23 " elif z<=0.3 then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " xn:=0.17*x-0.215*y+e3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " yn:=0.222*x+0.176*y+f3" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " \+ xn:=0.781*x+0.034*y+e4;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " \+ yn:=-0.032*x+0.739*y+f4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " \+ fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " pts:=pts, [xn, yn]; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " x:=xn;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " y:=yn" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " \+ od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " [pts]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 " plot(ifs(12500), axes=NONE, style=point, symbol=POINT, scaling=constra ined);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "julfn_zsqrd:=proc(x, y)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# Based on a Maple V algorithm take n from the book " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# 'Maple V - Pr ogramming Guide' by M.B. Monagan, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "# K.O. Geddes, G. Labahn and S. Vorkoetter, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# Springer Verlag, modified by John Oprea" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# modified by Alexander F. Walz" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# May 25, 1996" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " local c, z, m;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " c:=evalf(-0.5+0.5*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " \+ z:=evalf(x+y*I);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " for m from 0 to 30 while abs(z)<3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " z :=sin(z)+z^2+c" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "en d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "plot3d(0, -1.91 .. 1 .37, -1.24 ..1.21, style=patchnogrid, orientation=[-90,0], grid=[250, \+ 250], scaling=co" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "nstrained, colo r=julfn_zsqrd);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "The following routine can only be used with Maple V Release 4. Taken from the book 'Maple V Release 4' by Michael Kofler, Addison-Wesley. " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 36 "restart: with(plots, complexplot3d):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "p := (z-1)*(z^2+z+5/4);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := unapply(z-p/(diff(p,z)- 0.5*I), z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "complexplot 3d(f@@4, -4-4*I .. 4+4*I, view=-1 .. 2, style=patchnogrid, orientation =[-143, 65], grid=[50, 50]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "mandelbro t_fast:=proc(x, y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# Based on a \+ Maple V algorithm taken from the book " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# 'Maple V - Programming Guide' by M.B. Monagan, " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "# K.O. Geddes, G. Labahn and S. Vorkoetter, \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# Springer Verlag, modified by \+ John Oprea" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# modified by Alexand er F. Walz" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# May 25, 1996" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " local xn, xnold, yn, m;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " xn:=x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " yn:=y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " for m from 0 to 100 while sqrt(evalhf(xn^2+yn^2)) < 2 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " xnold:=xn; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " xn:=evalhf(xn^2-yn^2+x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " yn:=evalhf(2*xnold*yn+y)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " m" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "plot3d(0, -0.845 .. -0.794, 0.172 .. 0.21, orientati on=[-90,0], grid=[250, 250], style=patchnogrid, \nscaling=constrained, color=mandelbrot_fast);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "chaos:=proc(x1, y1, x2, y2, x3, y3, maxiter, seed1, seed2)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# original procedure written by Tom Williams and modi fied by John Oprea" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " local rand i, x, y, sx, sy, ir, i, j;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " ra ndi:=rand(1 .. 3):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " x:=[x1, x2 , x3]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " y:=[y1, y2, y3]:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " sx:=seed1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " sy:=seed2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " \+ ir:=[sx, sy]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " for i from 1 \+ to maxiter do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " j:=randi( ): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " sx:=(sx+x[j])/2.:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " sy:=(sy+y[j])/2.:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " ir:=ir, [sx, sy] " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 " p lot([ir], style=point, symbol=POINT, scaling=constrained);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "chaos(0, 0, .5, 1, 1, 0, 9000, .5, .5);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "chaos(0, 0, -4, 7, 6, 2, 3000, 1, .5);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "alexerror:=proc(x, iter)" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "# written by Alexander F. Walz" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "# May 29, 1996" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# March 02, 1997" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " local n, pts, xn, xnold, yn, y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 " pts:= [0, 0]; # in case pts is not assigned any value after completion of t he for n loop" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " for y from 0 to 1.1 by 0.01 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " xn := x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " yn := y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " for n to iter while evalhf(xn^2+yn^2) < 4 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " xnold:=xn;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " xn := evalhf(xn^2-yn^2+x); " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " yn := evalhf(2*xnold*yn+y) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " if n >= iter then pts := pts, [xn, yn], [xn, -y n] fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 " # as a variant try: \"if n >= iter then pts := pts, [x, y], [-x, -y] fi;\" to compute the Mandelbrot set " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " od; " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " pts " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "PLO T(POINTS(seq(alexerror(i/100, 500), i=-200 .. 70)), SYMBOL(POINT), AXE SSTYLE(NONE));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "orbits:=proc(x, y, iter) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "# procedure to plot the traject ory a point traverses by being iterated " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "# through the formula 'z(j+1)=z(j)^2+c' (also called \+ 'orbits') for the" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# Mandelbrot s et" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "# written by Alexander F. Wal z" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# May 30, 1996" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 " local a, b, xn, xnold, yn, pts;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " pts:=NULL;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " a:=x; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " b:=y ; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " xn:=x;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 9 " yn:=y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " t o iter do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " xnold:=xn; " } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " xn:=evalf(xn^2-yn^2+a);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " yn:=evalf(2*xnold*yn+b);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 " pts:=pts, [xn, yn]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " od; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " [pts]" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# orbits1.gif 'Spiral'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(orbits(-0.6114369519054 9, -0.36571035906672, 500), style=point, symbol=POINT, " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# orbits2.gif " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 83 "plot(orbits(-0.68181818351150, -0.3109515644 6099, 300), style=point, symbol=POINT, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "# orbits3.gif 'Vortex'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(orbits(-0.34555229917169, -0.60430224984884, \+ 300), style=point, symbol=POINT, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "# orbits4.gif" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot(orbits(+0.24877810105681, +0.58083451911807, 115), style= line, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=cons trained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "# orbits5.gif \+ 'Radiation'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(orbit s(-0.33382209390402, -0.60821359232068, 500), style=point, symbol=POIN T, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constra ined);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "# orbits6.gif" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(orbits(-0.259530793875 46, -0.62385896220803, 500), style=point, symbol=POINT, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# orbits7a.gif 'Curved'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(orbits(-0.14222874119 878, -0.64732701703906, 500), style=point, symbol=POINT, " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# orbits7b.gif 'Coil'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot(orbits(-0.1422287411987 8, -0.64732701703906, 300), style=line, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "# orbits8a.gif 'Spiral Galaxies'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot(orbits(-0.5097751729190 4, -0.60039090737700, 500), style=point, symbol=POINT, " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# orbits8b.gif 'US Star'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "plot(orbits(-0.5097751729190 4, -0.60039090737700, 300), style=line, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "axes=framed, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "# LAMBDAFN_FAST" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# Using hardware coprocessor" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# Based on a Maple V algorithm taken from the bo ok " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# 'Maple V - Program ming Guide' by M.B. Monagan, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "# K.O. Geddes, G. Labahn and S. Vorkoetter, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# Springer Verlag, modified by John Oprea" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# modified by Alexander F . Walz" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# May 31, 1996" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "lambdafn_sinfast:=proc(x, y)" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 26 " local xn, xnold, yn, m;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " xn:=x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " yn:= y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " for m from 0 to 100 while \+ evalhf(sqrt(xn^2+yn^2)) < 4 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " \+ xnold:=xn;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " xn:=evalhf (sin(xn)*cosh(yn) - cos(xn)*sinh(yn)*0.4);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " yn:=evalhf(cos(xnold)*sinh(yn) + sin(xnold)*cos h(yn)*0.4)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 " m" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "plot3d(0, -4 .. 4, -3 \+ .. 3, orientation=[-90,0], grid=[250, 250], style=patchnogrid, scaling =constrained, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "color=lambdafn_si nfast);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot3d(0, -1.5 \+ .. 1.5, -1.1 .. 1.1, orientation=[-90,0], grid=[25, 25], style=patchno grid, scaling=constrai" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ned, colo r=lambdafn_sinfast);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 729 "kamtorus := proc(a ngle, step, ende, points)\n# written by Alexander F. Walz\n# Version 1 .0 - May 31, 1996\n# Version 1.1 - May 23, 1999, four times as fast as v 1.0\n local x, xold, y, orbit, pts, ptscounter, n;\n pts := tab le([]);\n ptscounter := 0;\n for orbit from 0 by step to ende do\n x := orbit/3;\n y := x;\n to points while abs(x) < 10^ 38 and abs(y) < 10^38 do \n # to avoid violation of ranges\n \+ xold:=x;\n x := evalhf(x*cos(angle) + (x^2-y)*sin(angle ));\n y := evalhf(xold*sin(angle) - (xold^2-y)*cos(angle));\n \+ ptscounter := ptscounter + 1;\n pts[ptscounter] := [x, y];\n od; # end of to points\n od; # of for orbit\n [seq(pt s[n], n=1 .. ptscounter)]\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "plot(kamtorus(1.3, 0.05, 1.5, 200), style=point, symbol=POINT, axes=none, scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 773 "koch := proc(p: numeric)\n# SNOWFLAKE\n# algorithm taken from 'Al gorithmen fuer Chaos und Fraktale'\n# by Dietmar Herrmann, Addison-Wes ley\n# original program written in BASIC\n# ported to Maple V, modifie d and optimized by Alexander F. Walz\n# Version 1.0 - July 27, 1996\n# Version 1.1 - May 23, 1999, twice as fast as v 1.0\n local m, n, k, l, s, h, x, y, pts, t, i;\n h := 3^(-p);\n pts := table([]): \n \+ x := 0; y := 0;\n for n from 0 to (4^p-1) do\n m := n;\n \+ s := 0;\n for l from 0 to p-1 do\n t := irem(m, 4);\n \+ m := iquo(m, 4);\n s := s+irem((t+1), 3) - 1\n od; \+ # end of for l\n x := evalhf(x+cos(Pi*s/3)*h);\n y := evalhf (y+sin(Pi*s/3)*h);\n pts[n] := [x, y];\n od; # end of for n\n \+ [seq(pts[i], i=0 .. n-1)];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(koch(5), scaling=constrained, style=LINE, axes=N ONE);" }}}}{MARK "0 15 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 }