When I was a school boy, many decades ago when libraries had real books, I found at the library an album of snow flakes. It had more than 100 pages of snow flake images, white on black. There must have been 50 on each page, arranged in rows and columns. Every one was different. Some complicated, some simple. All very pretty. The text indicated they were made with a microscope. But you know, you can actually see these shapes without a microscope if you catch a snow flake on a dark mitten and hold it in the sun light just right. The snow flake will constantly change shape, depending on temperature and humidity. I told Dr. J about this and that I'm very fascinated by the flakes he is finding in his fractal universe.
Last night Dr J caught a close up of a very rare 6 cornered snow flake while watching the FractoLympics. These usually form in fractal space under conditions occurring at the 5000 iteration mark. It is certainly not likely at La Julia, nor at Dr. J's nearby home. It is much more likely to rain, sometimes a very heavy rain. With more rain and flooding nearly every night, Dr. J has decided to stay late at his laboratory. Tonight he has tried to create some snow flakes of his own. And unlike the flakes he gathered up to examine earlier, these he hoped would have many sides, even more than 6. He has succeeded beyond his wildest expectations, achieving 16 sides. Since he has these in his laboratory, he can get a very close look, Figure 1.
His flakes (Julia sets with Siegel disks) require the Julia set parameter be right on the edge of a cycloid component.
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Figure 1. Part of a sixteen sided snow flake, up very close.
Stay warm,
Jay
j16-siegel-close-2 { ; (c) Jay Hill, 1998
; uses irrational angle = (-10+sqrt(116))*pi
reset=1960 type=formula formulafile=fotn.frm
formulaname=Colorit-16JS passes=2 center-mag=0.446616/0.547247/19.28
params=-10/116 float=y maxiter=5600 inside=bof60 outside=128
periodicity=0 colors=0Wz<60>013012000<191>000
savename=Blue16c2
}
frm:Colorit-16JS { ; (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(16)/15), c=w-sqr(sqr(sqr(sqr(w)))), z=pixel:
z=sqr(sqr(sqr(sqr(z)))) + c
|z| <=4
}
