So that you might comprehend how I derived the formula for d2jaMandelbrot I'm including the Maxima Cas code that I used . file iterdiff2a.wxm . /* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/ /* [ Created with wxMaxima version 11.08.0 ] */ /* [wxMaxima: comment start ] . (c) Copyright 2017 , sciwise@ihug.co.nz Methodology for expressing differential equations in terms of an iterated function. In particular a first order differential equation and the derivative thereof . Note that f(z,c) might be any iterated function . . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] ' Differential equation we're examining . diff(f(z),z) + 3*f(z)*z = sin(z) for an iterated function z = f(z) , g(z) =f(f(z)) diff(g(z),z) + 3*g(z)*f(z) = sin(f(z)) . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ kill(all)$ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ expand(eq1); expand(diff(eq1,z)) ; /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] For the iterated Mandelbrot fractal function . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ f(z,c):=z*z+c$ g(z,c):=f(f(z,c),c)$ f(z,c); g(z,c); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ ed:expand(eq1)$ edp:expand(diff(eq1,z))$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ ed; edp; /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] This may well simplify the construction of fractint formulas for this type of equation . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] z=Pixel c=Pixel : z=z*z+c ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c [wxMaxima: comment end ] */ /* Maxima can't load/batch files which end with a comment! */ "Created with wxMaxima"$ And file iterdiff2a.mac . /* . (c) Copyright 2017 , sciwise@ihug.co.nz Methodology for expressing differential equations in terms of an iterated function. In particular a first order differential equation and the derivative thereof . Note that f(z,c) might be any iterated function . . */ /* ' Differential equation we're examining . diff(f(z),z) + 3*f(z)*z = sin(z) for an iterated function z = f(z) , g(z) =f(f(z)) diff(g(z),z) + 3*g(z)*f(z) = sin(f(z)) . */ kill(all)$ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ expand(eq1); expand(diff(eq1,z)) ; /* For the iterated Mandelbrot fractal function . */ f(z,c):=z*z+c$ g(z,c):=f(f(z,c),c)$ f(z,c); g(z,c); /* */ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ ed:expand(eq1)$ edp:expand(diff(eq1,z))$ /* */ ed; edp; /* This may well simplify the construction of fractint formulas for this type of equation . */ /* z=Pixel c=Pixel : z=z*z+c ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c */
Here are links to a few images using Sciwise's parameter set. The email systems seem to have relatively new procedures added that "clean up" email formatting, making parameter sets fail in Fractint. I've repaired them in order to run his parameter set, but my repaired version of his parameter set pasted below will be again rendered unable to be run by Fractint... So, I've also put the repaired version of his parameter set onto my server: http://tinyurl.com/170119dEQ-pars or: http://www.emarketingiseasy.com/Sciwise/170119dEQ.PAR Viewing the images full size prevents your browser from resampling and degrading the images. I increased Maxiter from 1024 to 2048, and increased periodicity from 1 to 4. Parent fractal: I manually added shades of cyan, magenta, yellow & gray to the original red, green and blue color map to eliminate un-color-mapped pixels in the image. The Fractint Color Editor: <e> turns out to be easy to use if you're only entering a few new colors. http://www.emarketingiseasy.com/Sciwise/S170119Z.gif An anti-aliased zoom into the rotated left figure: http://www.emarketingiseasy.com/Sciwise/S170119A.jpg To see the image that I calculated as the input to the anti-aliasing process for the image above, here's that same image, except 5 times larger in width and height (1.4 MB). It's fun to scroll and pan around inside the 6000 x 4500 pixel image -- although the colors are a little garish: http://www.emarketingiseasy.com/Sciwise/S170119A.gif A zoom into the 90 degree rotated right side of the parent: http://www.emarketingiseasy.com/Sciwise/S170119B.gif A slightly rotated zoom into the upper right area of: S170119A is an interesting mix of lacy and strongly colored areas: http://www.emarketingiseasy.com/Sciwise/S170119C.gif The tiny central yellow island below S170119A's main figure: http://www.emarketingiseasy.com/Sciwise/S170119D.gif And, finally, the top horizontal island above the main figure in S170119A. The central three areas of gray are colored that way because I didn't put colors into the color map for them -- I didn't realize there were areas of the fractal that had that high an iteration count: http://www.emarketingiseasy.com/Sciwise/S170119E.gif In the above image, the right hand "bulls eye" shows the colors, in sequence, loaded into the color map, starting with bright red in color map location one. If your move your eye from the center of the bulls eye down towards "7 o'clock" you'll see the entire color map's color range, and then the gray colors I didn't modify. And that color map -- I added the 2nd and 3rd row entries: http://tinyurl.com/S170119-cmap or: http://www.emarketingiseasy.com/Sciwise/S170119cmap.jpg - Hal Lane ######################## # hallane@earthlink.net ######################## comment { In this instance I've used Maxima Cas to directly calculate the formula for the differential equation , using the Mandelbrot Set as the function. Again I'm using Newton's method in an attempt to find the roots of the Differential equation ; if successful then I might be able to extend this to more complicated differential equations. } test { reset=2004 type=formula formulafile=fractint.frm formulaname=d2jaMandelbrot corners=-2/2/-1.5/1.5 float=y maxiter=1024 inside=bof60 outside=0 logmap=yes colors=@chroma.map } frm:d2jaMandelbrot(XAXIS) {; Edward Montague (c) 2017 c = Pixel z = c : z=z*z+c ed=-sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp=-2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c z = z-ed/edp .0001 < |ed| } test-w-more-colors { ; Added magenta, cyan, ; yellow & gray shades ; of color. ; Fractint Version 2099 Patchlevel 8 reset=2099 type=formula formulafile=170119dEQ.PAR formulaname=d2jamandelbrot passes=1 corners=-2.054024/1.177097/-1.21167/1.21167 float=y maxiter=1024 inside=bof60 outside=0 logmap=yes colors=000z00<3>K000z0<3>0K000z<3>00Kz0z<3>K0K0zz<\ 3>0KKzz0<3>KK0AA0zzz<3>KKKAAAccc<216>ccc } -------- End of Parameter file ------ --- This email has been checked for viruses by Avast antivirus software. https://www.avast.com/antivirus
I wondered what "Maxima Cas" was in Sciwise's post. Wikipedia https://en.wikipedia.org/wiki/Maxima_(software) says: "Maxima is a full-featured CAS (computer algebra system) that specializes in symbolic operations, but it also offers numerical capabilities[1] such as arbitrary-precision arithmetic: integers and rational numbers which can grow to sizes limited only by machine memory, and floating point numbers whose precision can be set arbitrarily large..." - Hal Lane ######################## # hallane@earthlink.net ######################## -----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of sciwise@ihug.co.nz Sent: Thursday, January 19, 2017 8:07 PM To: Fractint and General Fractals Discussion <fractint@mailman.xmission.com> Subject: [Fractint] DiffNewtMax So that you might comprehend how I derived the formula for d2jaMandelbrot I'm including the Maxima Cas code that I used . file iterdiff2a.wxm . /* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/ /* [ Created with wxMaxima version 11.08.0 ] */ /* [wxMaxima: comment start ] . (c) Copyright 2017 , sciwise@ihug.co.nz Methodology for expressing differential equations in terms of an iterated function. In particular a first order differential equation and the derivative thereof . Note that f(z,c) might be any iterated function . . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] ' Differential equation we're examining . diff(f(z),z) + 3*f(z)*z = sin(z) for an iterated function z = f(z) , g(z) =f(f(z)) diff(g(z),z) + 3*g(z)*f(z) = sin(f(z)) . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ kill(all)$ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ expand(eq1); expand(diff(eq1,z)) ; /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] For the iterated Mandelbrot fractal function . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ f(z,c):=z*z+c$ g(z,c):=f(f(z,c),c)$ f(z,c); g(z,c); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ ed:expand(eq1)$ edp:expand(diff(eq1,z))$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ ed; edp; /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] This may well simplify the construction of fractint formulas for this type of equation . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] z=Pixel c=Pixel : z=z*z+c ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c [wxMaxima: comment end ] */ /* Maxima can't load/batch files which end with a comment! */ "Created with wxMaxima"$ And file iterdiff2a.mac . /* . (c) Copyright 2017 , sciwise@ihug.co.nz Methodology for expressing differential equations in terms of an iterated function. In particular a first order differential equation and the derivative thereof . Note that f(z,c) might be any iterated function . . */ /* ' Differential equation we're examining . diff(f(z),z) + 3*f(z)*z = sin(z) for an iterated function z = f(z) , g(z) =f(f(z)) diff(g(z),z) + 3*g(z)*f(z) = sin(f(z)) . */ kill(all)$ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ expand(eq1); expand(diff(eq1,z)) ; /* For the iterated Mandelbrot fractal function . */ f(z,c):=z*z+c$ g(z,c):=f(f(z,c),c)$ f(z,c); g(z,c); /* */ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ ed:expand(eq1)$ edp:expand(diff(eq1,z))$ /* */ ed; edp; /* This may well simplify the construction of fractint formulas for this type of equation . */ /* z=Pixel c=Pixel : z=z*z+c ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c */ _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint --- This email has been checked for viruses by Avast antivirus software. https://www.avast.com/antivirus
participants (2)
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Harold Lane -
sciwise＠ihug.co.nz