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Author Topic: Exploring the Buffalo, Celtic, and Burning Ship  (Read 2451 times)
Description: Where you zoom does make a difference
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stardust4ever
Fractal Bachius
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Posts: 513



« on: March 03, 2012, 02:25:28 AM »

I consider myself a seasoned veteran fractal explorer of the Mandelbrot Set. One of the principals of deep-zooming the Mandelbrot is that whenever you select a new inflection point to zoom into, eventually all the patterns you have previously seen will duplicate themselves, and you eventually will find the original pattern you zoomed into wrapped around the centroid twice. The zoom depth becomes denser as well as the surrounding patterns become compressed in such a way that the original pattern is doubled. I call this "periodic doubling" and by exploiting the periodic doubling effect, I have been able to achieve breathtaking patterns such as this:
http://www.fractalforums.com/images-showcase-%28rate-my-fractal%29/magnum-opus-x/

Around the same time period that I felt I had ultimately conquered the Mandelbrot set, Panzerboy released his absmandvar plugin, which introduced three new fractals to the Fractal Extreme Library, including Burning Ship, Celtic, and Buffalo variants. These three fractals use an abs() command on either the imaginary, real or both axis of the Z^2 in the original formula, Z=Z^2+C The Burning Ship applies the abs() command to the imaginary side, the Celtic applies the abs() to the real, and the Buffalo applies abs() to both axes.

If we track the location of Z1 as it bounces about the set from iteration to iteration, for the original Mandelbrot set, Z1 can be located within any of the four quadrants within the complex plane,

IVI
IIIII

But the abs() function limits Z1 to only half, or even one of the four quadrants. Burning Ship allows only quadrants I and IV to be used, Celtic allows only quadrants I and II to be used, and the for Buffalo function, Z1 resides only within quadrant I. Whenever a fractal explorer deep zooms within the absmandvar sets, the respective Z values from the original four quadrants get reflected across lines of symmetry, creating the beautiful kaleidoscope-like fractal patterns which are seen in the abs() variations. However, unlike the Mandelbrot set, whereas zooming into a like region on the sides of the fractal creates similar patterns and mostly only affects the rotation angle, zooming into seemngly congruent areas within the fragmented abs() Mandelbrot variations can create vastly different results. I have attached a Jpeg image containing a specific pattern of 8 seemingly congruent circles I found within the Buffalo set, at a depth of 84 zooms, and the patterns that resulted at a depth of 120 zooms, created by zooming into the centroid of the circular formations.

With Panzerboy's Fractal Extreme plugin, the images which yielded the most detail were formed by zooming into the NorthEast side of the formation (quadrant I). Because I have rotated the fractal 100 degrees counterclockwise, the paths with the most detail appear on the upperleft side of the image, but had the fractal been left unrotated, the upper right side would have produced the most detail, and the lower left side would have produced the least.

In summary it is possible to drastically change the level of detail in the resultant patterns, by zooming into different directions from the centroid. The same effect applies as well to the Celtic and Burning Ship variants. afro


* fractal buffalo exploration.jpg (245.87 KB, 1280x960 - viewed 436 times.)
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