The snow is long gone from La Julia, but Dr J has been watching the FractoLympics and caught a close up of a very rare 6 cornered show flake! The conditions were just right at the 25,600 iteration mark to form this beautiful flake, Figure 1. The FractoGolf tournament was soggy after last night's heavy rain storm.
These flakes (with Siegel disks) only show up when the Julia set parameter is right on the edge of a cycloid component. I have put an earlier post of mine about Siegel disks in the new iFAQ. For example, in the Mandel-power sets,
1) z := zn + c,
we have a simple equation for the boundary
2) c = x-xn,
where
3) x = e(i*theta)/[n1/(n-1)]
By a little manipulation we get these more efficient equations for your FractoViewer.
4) c = w-enu
where
5) u = theta + (log u)/(1-n)
and theta is an irrational angle like -A+sqrt(B), A and B being integers.
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Figure 1. A six sided snow flake.
Stay warm,
Jay
j6-siegel-blue2 { ; (c) Jay Hill, 1998
; A six pointed snow flake
; p1 contains two parts of the angle
; example p1=-3/8 means angle=-3+sqrt(8)
reset=1960 type=formula formulafile=fotn.frm formulaname=colorit-6js
center-mag=0/0/0.85 params=-3/8 float=y maxiter=25600 inside=bof60
outside=128 periodicity=0
colors=0Wz<60>013012000<191>000
}
frm:Colorit-nJS { ; n corner Siegel disk (c) Jay Hill, 1998
n=(real(p2)); z:=z^n+c
; angle=real(p1)+sqrt(imag(p1))
; angle= log(Julia parameter)/pi
; when real(p1)=0, Julia is on edge of Period 1 component
A=i*real(p1)+sqrt(-imag(p1))
u=A+log(n)/(1-n)
w=exp(u), c=w-exp(n*u), z=pixel:
z=z^n + c
|z| <=4
}
Colorit-3JS { ; three corner Siegel disk (c) Jay Hill, 1998
; angle=real(p1)+sqrt(imag(p1))
; angle= log(Julia parameter)/pi
; when real(p1)=0, Julia is on edge of Period 1 component
A=i*real(p1)+sqrt(-imag(p1))
w=exp(A-.5*log(3)), c=w*(1-sqr(w)), z=pixel:
z=z*sqr(z) + c
|z| <=4
}
frm:Colorit-4JS { ; four corner Siegel disk (c) Jay Hill, 1998
; angle=real(p1)+sqrt(imag(p1))
; angle= log(Julia parameter)/pi
; when real(p1)=0, Julia is on edge of Period 1 component
A=i*real(p1)+sqrt(-imag(p1))
w=exp(A-log(4)/3), c=w-sqr(sqr(w)), z=pixel:
z=sqr(sqr(z)) + c
|z| <=4
}
frm:Colorit-5JS { ; five corner Siegel disk (c) Jay Hill, 1998
; angle=real(p1)+sqrt(imag(p1))
; angle= log(Julia parameter)/pi
; when real(p1)=0, Julia is on edge of Period 1 component
A=i*real(p1)+sqrt(-imag(p1))
w=exp(A-log(5)/4), c=w-w*sqr(sqr(w)), z=pixel:
z=z*sqr(sqr(z)) + c
|z| <=4
}
frm:Colorit-6JS { ; six corner Siegel disk(c) Jay Hill, 1998
; angle=real(p1)+sqrt(imag(p1))
; angle= log(Julia parameter)/pi
; when real(p1)=0, Julia is on edge of Period 1 component
A=i*real(p1)+sqrt(-imag(p1))
w=exp(A-log(6)/5), c=w-sqr(w*sqr(w)), z=pixel:
z=sqr(z*sqr(z)) + c
|z| <=4
}
frm:Colorit-8JS { ; eight corner Siegel disk (c) Jay Hill, 1998
; angle=real(p1)+sqrt(imag(p1))
; angle= log(Julia parameter)/pi
; when real(p1)=0, Julia is on edge of Period 1 component
A=i*real(p1)+sqrt(-imag(p1))
w=exp(A-log(8)/7), c=w-sqr(sqr(sqr(w))), z=pixel:
z=sqr(sqr(sqr(z))) + c
|z| <=4
}
