To any of you who have not downloaded the latest Fractint developer version with the new "passes=o" option for plotting cumulative orbit sets, I heartily recommend you do so! Subsets of these orbit sets taken along closed curves of the domain space can generate curves that are different in visual character from the parent form, and many of them appear to limn curves that are truly fractal, ie. have noninteger Hausdorff dimension. Watching them form, almost all of them seem to be formed by a small number of "packets" of a large number of cogenerated points that move around the screen like pens. I have managed to render some of these with the domain points on the main cardioid only-- some of the Julia orbit sets so formed are particularly nice. [As an aside, in the following there is the option to do the iteration in the negative direction, which is controlled by the parameters in p2 and p3, ie. instead of z->z^2 + C to do z->+-sqrt(z-C). But since -sqrt(z-C) is just as valid an inverse as sqrt(z-C), there is the conundrum of which one to use at each iteration. For a sequence of n iterations there would be 2^n possible inverse sequences. Obviously there is no way to specify them all, so I just let the user choose a modulo arithmetic driven sequence of which branch to choose each iteration.] ---Hiram /* PARS OF A JULIA ACCUMULATED ORBIT SET FROM THE MAIN CARDIOID WITH C AT [1] INTERSECT [1,1/4] AND A PERTURBED MANDELBROT SET */ RingOfFour { ;HiramBerry; Julia orbit set with the iteration curve on the M-set ; main cardioid, with the C-value set at the tangent pt ; of the main cardioid [1] with the positive first order ; period four bud [1,1/4]. reset=2004 type=formula formulafile=orbit.frm formulaname=Jorbitcardioid1 passes=o center-mag=-0.0130781/0.491833/3.277684/1/90/3.88578058618804789e-016 params=0.25/0.5/0/0/0/0 float=y outside=summ periodicity=0 cyclerange=1/1 orbitdelay=100; colors=@grad0054.map colors=000X3r<40>v16w15x14x13y12z00<42>ze0zf0zg0zh0zi0<2>zj0yj1yj1xk1xk2\ <71>WyUWyUVzV<43>VxtVxuVxuVxvUww<23>syytyyuyy<3>zzz000000 } PerturbedCardioidO { ;HiramBerry; reset=2004 type=formula formulafile=orbit.frm formulaname=Morbitcardioid1 passes=o center-mag=5.78e-006/-7.005e-006/1.103748 params=0.65/0.1/0/0/0/0 float=y outside=summ periodicity=0 cyclerange=1/1 colors=000NrYX3q<39>v16w15x14x13y12z00<42>ze0zf0zg0zh0zi0<2>zj0yj1yj1xk1\ xk2<71>WyUWyUVzV<43>VxtVxuVxuVxvUww<23>syytyyuyy<3>zzz000000 } /* FRMS TO RENDER MANDELBROT AND JULIA ACCUMULATED ORBIT SETS ALONG THE MAIN CARDIOID*/ Morbitcardioid1{ ;HiramBerry; passes=o,orbitdelay=nonzero. Domain points drawn from the ; main cardioid for Mandelbrot iteration. No point density normalization. ; p1=z_perturb,p2=(iteration dir(0=z->z^2+C,nonzero=cycle period(posint)), ; zerofill(0<=int<=cycle)),p3=(start(posint),skip(posint)),where the iteration ; chooses from {z->sqrt(z-C),z->-sqrt(z-C)} , based on the cycling. Eg. to evenly ; alternate the iteration types use p2=(2,1),p3=(1,1). IF(initialized==0) initialized=1 cols=real(scrnmax),rows=imag(scrnmax),pts=cols*rows rctx2=pts*2 density=pi/pts cyc=real(p2),zfill=imag(p2),start=real(p3)-1,skip=imag(p3) bailout=4 ENDIF disti=imag(scrnpix)*cols+real(scrnpix) z=p1 theta=density*(2*disti+1) ct=cos(theta) r=(1-ct)/2 C=0.25+r*(ct+flip(sin(theta))) IF(cyc!=0) pos=start ENDIF : IF(cyc==0) z=sqr(z)+C ELSE pos=pos-floor(pos/cyc)*cyc ; modulo normalize IF(pos<zfill) ; choose iteration type z=sqrt(z-C) ELSE z=-sqrt(z-C) ENDIF pos=pos+skip ; iterate ENDIF |z|<=bailout } Jorbitcardioid1{ ;HiramBerry; passes=o,orbitdelay=nonzero. Domain points drawn from the ; main cardioid for Julia iteration. No point density normalization. ; p1=C,p2=(iteration dir(0=z->z^2+C,nonzero=cycle period(posint)), ; zerofill(0<=int<=cycle)),p3=(start(posint),skip(posint)),where the iteration ; chooses from {z->sqrt(z-C),z->-sqrt(z-C)} , based on the cycling. IF(initialized==0) initialized=1 cols=real(scrnmax),rows=imag(scrnmax),pts=cols*rows rctx2=pts*2 density=pi/pts cyc=real(p2),zfill=imag(p2),start=real(p3)-1,skip=imag(p3) C=p1 bailout=4 ENDIF disti=imag(scrnpix)*cols+real(scrnpix) theta=density*(2*disti+1) ct=cos(theta) r=(1-ct)/2 z=0.25+r*(ct+flip(sin(theta))) IF(cyc!=0) pos=start ENDIF : IF(cyc==0) z=sqr(z)+C ELSE pos=pos-floor(pos/cyc)*cyc ; modulo normalize IF(pos<zfill) ; choose iteration type z=sqrt(z-C) ELSE z=-sqrt(z-C) ENDIF pos=pos+skip ; iterate ENDIF |z|<=bailout }