**Fractal of the Dayby Jim Muth**

FOTD -- December 14, 2007 (No Rating)

Fractal visionaries and enthusiasts:

Well, here it is in all its glory -- the parent fractal for the month of December -- the universe we have been exploring all this month. And what a world it is. To begin, its size is deceptive. The magnitude of an unusually small 0.0068 makes its area 10000 times the area covered by the Mandelbrot set. The center point of the image, -128,-30, is no mistake. We actually are this far from the origin.

Studying the image, we notice that it is filled with rings, as have been most of the images so far this month. Every filament terminates in a decorative ring. Then we notice the main bay, if that's what it is. This nearly circular feature bears little resemblance to the cardioid shape of the classic Mandelbrot main bay. Where is East Valley, and where is Seahorse Valley? Actually, these features are in the fractal, though well hidden.

We next notice the largest bud on this circular bay. Intuition says that it is the main bud, and would therefore have a periodicity of 2. But actually, its periodicity is 13, as can easily be checked by counting the number of arms in its star. If there is a main bay in this parent fractal, it is very distorted. And look at the shape of the chaotic area leading from the large bud to the largest ring. It does rather resemble an East Valley elephant.

The elephant resemblance is even more pronounced in the chaotic area of the slightly smaller bud north of the large one, which strangely enough has a periodicity of 12. It appears that nearly the entire shore line of the circular bay is the southern shore of the main bay of this fractal, and the farther we travel counter-clockwise, the deeper we go into an East Valley environment.

It is now simple to find the actual period-2 bud and Seahorse Valley. Simply count clockwise around the buds from the period-13 bud until the count reaches 2. There is our main bud, complete with Seahorse Valley and the main spike, all of them so small that a blow up is needed to make them visible at all. As we continue beyond the period-2 bud, the periodicity rapidly increases until we find ourselves deep in the north side of East Valley.

This area of the fractal is actually the most interesting. The smaller circular bay just to the east is something entirely new. Nothing like it appears anywhere in the M-set. It is highly fragmented, and at first appears to be non-critical. But its shore is dotted with tiny irregular buds in which some very curious minibrots are well hidden.

The fractal is also peculiar in that the orbits of the points diverge so fast as to be virtually invisible. Occasionally a single dot appears, but this is a rarity. Such rapidly diverging orbits are likely true of all fractals created with negative exponents of Z.

I named the image

Thursday was most unpleasant here at Fractal Central, as wet sloppy snow and cold rain fell most of the day, while the temperature hung just above freezing. The fractal cats spent the day by the heat, ignoring the weather.

My day was uneventful. The next FOTD will be posted here in 24 hours. Until then, take care, and even though the concept of an ultimate reality might be a human invention, I really do believe in the virtual reality of fractals.

Jim Muth

jamth@mindspring.com

jimmuth@aol.com

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