FOTD -- August 29, 2009 (No Rating) Fractal visionaries and enthusiasts: Check the philofractal list for a longer discussion. Today's image is more a fractal study than any kind of art work. But as a combination of Z^2 and Z^(-2) it is still worth a look. The image is named "Various Distortions". I gave it this name when I saw in it the three types of distortion that can afflict Mandeloids. These distortions surely have sound mathematical causes, but other than the non-critical type of distortion, I have no idea what these causes might be. The first type of distortion appears in the dark Mandeloid fragments vaguely showing in the background in the northeast and southeast parts of the fractal. This distortion appears when the starting point of Z is set to a non-critical value. This particular distortion can be readily seen by calculating the type=mandel Mandelbrot set and setting the parameters to something between zero and one. The distortion simply cuts off part of the set. The second type of distortion is clearly visible in the two prominent Mandeloids on the X-axis. In the Mandeloid on the left, the main period-2 bud is too large and the higher-order buds are bunched toward East Valley, while in the Mandeloid on the right, the main bud is too small and the higher-order buds are spread out toward the main bud. I have no idea of the mathematical cause. The last type of distortion is visible in the two smaller asymmetrical Mandeloids at the top and bottom of the image. In this case the entire system of buds is skewed to one side or the other. In the buds in the image, this skewing is not too severe, but it sometimes becomes so severe that the main period-2 bud is impossible to find without checking the actual orbits of the points. Once again I have no idea of the cause. Since the image is more a mathematical study than a work of art, I could give it no rating. But with a calculation time of under 15 seconds, at least on my recently rehabilitated middle-aged 2000mhz machine, there is no reason for anyone to miss seeing the distortion. Of course, the trip to the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> is still more fun than calculation. Heavy clouds and periods of rain spoiled Friday here at Fractal Central. The temperature of 70F 21C was irrelevant. The fractal cats checked the outside several times, but when they saw too much rain and no sign of the end of the world. they went back to sleep. My day was rather busy but not frantically so. The next FOTD is scheduled to be posted in 24 hours. If an unexpected antiquing expedition comes up, the FOTD might be delayed however. Until next time, take care, and prepare. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER FILE======================================= VariousDistortions { ; time=0:00:14.88-SF5 on P4-2000 reset=2004 type=formula formulafile=basic.frm formulaname=MandAutoCritInZ function=ident passes=1 center-mag=-0.861346/0/0.814578 params=0.1/-2/0.55\ /2/0/0/0/0 float=y maxiter=350 inside=0 logmap=yes symmetry=xaxis periodicity=10 colors=000wkCri0mg3he6cc9ZbCU_BPXAKU9FR9ANH5JO1FV1\ Ba07h03z9DzINzQXzZfzfpzamzXjzSgzNdzIazDZz8Wz4UzCQz\ KNzSJz_GzgDz_AzS8zL6zD4z62zWSzMBzNAzNAzNAzNAzNAzNA\ zNAzNAzNAzNAzNAzNAzNAzXSzehncimajl_jkYkjWliUlhSmgR\ mfPneNodLocJpbHqaFq`Dr_CrZFs_Ht_Ju_Lu_Nv_Qw_Sw_Ux_\ Wy_Yy_iwOuvDtrFsoHrlJqiLpfNobPn_RnXSmUUlRWkNYjK_iH\ ahEchBdiEejGekIelLemNemPenSfoUfpWfqZfr`frbfodble_j\ fXghUeiRbjO`lLYmIWnFTpCRq9Or6Ms3Pp4Rn5Tk6Vi7Xf8_d9\ aaAc_BeXCgVDjSElQFnNGpLHrIItGJvPXxYjzexzguzhrziozj\ lzkjzlgzmdznazoZzpXzqUzrRzsOztLzuJzvLDwzFxzGyzIzzJ\ zzLzzMzzOzzPzzRzzSzzUzzVzzWzzTzzQzzOzzLzzJzzGzzDzz\ Bzz8zz6zz9zzBzzDzzFzzHzzJzzLzzNzzPzzRzzTzzVzzTzzRz\ zQzzOzzNzzLzzKzzIzzHzzFzzEzzCzzBzkEzeGz`IzWKzRMzMH\ zKCzI7zG2zE7zFBzGFzHJzIOzJSzKWzL_zMczIfzEizAlz6oz2\ gz3_z4Sz5Kz6Cz79z67z65z53z51z5vzNqzOlzPgzQbzRYzSTz\ TOzUJzVEzW9zX5zYCzZWzUSzP } frm:MandAutoCritInZ {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)), k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END PARAMETER FILE=========================================