Is it another calamity for Dr. J? He no sooner plants fresh grass in the new FractoBowl field than it has snowed! Now this is not supposed to happen here where just last weekend Dr. J got a very dark tan in just one day at the beach. Oh well, he has sent me a beautiful set of Fractoscope pictures. In fractal space, you will see, snowflakes under the right conditions, can have 3 way symmetry and have a bluish tint! This is unlike here in the real world where snow has a 6 way symmetry and is white (so I'm told, I live in San Diego).
Look closely at the samples, they sure look like strange Julia sets. Indeed they are, and with Siegel disks even!
I really doubt that humans have ever seen these before. I state this because the best Siegel disks occur when you use for the Julia set parameter, the exact edge of a Mandelbrot set component. Since these Julia sets have three way symmetry, you may observe. They must not be from the regular Mandelbrot set. z := z2 + c. Correct again, pilgrim. These are generated from yesterdays formula
1) z := z3 + c.
To plot a Julia set (named for Gaston Julia who wrote about these sets in 1918), we set the initial z according to the screen pixel and put c to some value called the Julia set parameter. Then almost 20 years ago, Benoit Mandelbrot created his sets which are a map of the Julia sets. Start these by putting c according to the screen pixel and set z to a critical value of the iteration formula.
The Mandelbrot set for Equation 1 is in fact, the FractoBowl, last night's image. The critical value for 1) is zero. Iterations with c inside the dark green ball field settle to a repeating cycle with a period of one. That is, it repeats every time. If c it at the very edge and is between attached 'buds', then the iterations will never settle into any cycle ever. They will not diverge either, just wander in a wiggly orbit, never making up their mind.
It is these wandering orbit points which give us the interesting Siegel disks. Before last night's formula
2) |2*sinh[(asinh(sqrt(-6.75)*c))/3]| <= 1
it has been very difficult to locate these. It has been suggested that the discovery of the formula (Eqn 2) has resulted in snow at the Cove. Dr. J better be careful what new formulas he investigates. He might cause Hell to freeze over! Tonight's frm file allows us to create an infinite number of Siegel snow flakes. All we need is for c to be exactly on the edge of the field and have a generating angle which is irrational. Here is how Dr. J does it.
Write Eqn 2) as a mapping of the edge of a circle to the ball field edge. Let theta (a Greek letter) be the angle in the circle.
3) 2*sinh[(asinh(sqrt(-6.75)*c))/3] = exp(i*theta)
We shall know theta, well er... we can get irrational numbers using square roots of non-square numbers.
4) theta = pi*(a + sqrt(b))
Take as an example, a=-2, b=5. Then theta=pi* 0.2360679774998... Now we solve for c, the edge point.
5) c=sinh(3*asinh(.5*exp(i*theta)))/sqrt(-6.75)
OK, let's see the frm file.
frm:Colorit-3fJS { ; (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=(real(p1)+sqrt(imag(p1)))*sqrt(-1)
c=sinh(3*asinh(.5*exp(pi*A)))/sqrt(-6.75)
z=pixel:
z=z*sqr(z) + c
|z| <=4
}
And some sample snow flakes, Figure 1.
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Figure 1. Dr. J 's has caused it to snow at FractoBall Field.
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Figure 2. Another of Dr. J 's snow flakes.
Download the frm and par files with more samples.
Stay healthy,
Jay
j3-siegel-blue01 { ; (c) Jay Hill, 1998
; -2+sqrt(5) color
reset=1960 type=formula formulafile=*.frm passes=2
formulaname=Colorit-3fJS center-mag=0/0/0.85 params=-2/5
float=y maxiter=256 inside=bof60 outside=128 periodicity=0
colors=0Wz<60>013012000<191>000
savename=blue01
}
j3-siegel-blue14 { ; (c) Jay Hill, 1998
; -3+sqrt(8) color
reset=1960 type=formula formulafile=*.frm passes=2
formulaname=Colorit-3fJS center-mag=0/0/0.85 params=-3/8
float=y maxiter=256 inside=bof60 outside=128 periodicity=0
colors=0Wz<60>013012000<191>000
savename=blue14
}
