FOTD -- March 28, 2008 (Rating 8.5) Fractal visionaries and enthusiasts: One of the reasons I am fascinated with fractals is because they are four-dimensional things, and I waste no end of time futilely trying to visualize a four-dimensional anything. Many phenomena in 4-D hyperspace have similar things in our familiar 3-D space, which help us to understand hyperspace. But there is at least one thing in 4-D space (I am sure many others exist.) that is totally unlike anything in our 3-D space. That thing is the motion called double rotation. In 3-D space an object rotates about a line as its axis, and due to the gyroscopic effect, that line will retain its orientation. But in 4-D space a 4-D object rotates about a plane as its axis, which makes the situation more complex. In a simple 3-D type of rotation, the points of a 4-D object describe a circle, while an entire plane of the object remains fixed in place as its points turn on themselves. And just as in 3-D space, the gyroscopic effect will keep the object rotating in the same direction. But what of that axis plane, where the points remain stationary, while turning in place? This is the weird thing. Since the axis-points are not moving, they are not subject to the gyrosco- pic effect, and the entire axis-plane can turn on itself around its center point, carrying the entire 4-D object with it, while the entire object continues to rotate in the first direction. The object is now in a state of double rotation, a motion totally alien to our familiar 3-D objects. While fantasizing a walk on a 4-D hyperplanet recently, and trying to imagine the apparent motion of the heavenly bodies, I realized that I was perilously close to visualizing double rotation, (as projected into 3-D space of course.) What I imagined is extremely difficult to put into words, but I'll give it a try in a near-future FOTD discussion. As for today's image, it lies in the parent fractal that results when 0.01 part of Z^(-2) is subtracted from the classic Mandelbrot set. This fractal is a distorted M-set, filled with debris. Today's image is located in the East Valley area of the remains of the large minibrot on the negative X-axis of the M-set. I named the image "In a Soiled M-brot", which the parent fractal most certainly is. I rated it at an 8-1/2 -- 8 points for the underlying fractal, 1/2 point for my coloring work. The calculation time of only 49 seconds means that no one will be disappointed with the result. Some may prefer to avoid the calculation however, so for convenience, the finished image is or soon will be posted on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Good weather in early spring never lasts long in Central Pennsylvania, so we were not too disappointed here at Fractal Central when Thursday turned out cloudy, chilly and drizzly. The temperature of 45F 7C kept people well wrapped, while the drizzly mist kept the rain gear handy. The fractal cats kept cozy near the heat. My day was on the busy side, though not too busy to affect the fractal. The next FOTD will appear in 24 hours. Until then, take care, and there is more to empty space than meets the eye. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER FILE======================================= In_a_Soiled_M-brot { ; time=0:00:49.43-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=MandAutoCritInZ function=ident float=y center-mag=-1.230634995848692/-0.488602834129332/\ 1.201185e+011/1/140/0 params=1/2/-0.01/-2/0/0/0/0 maxiter=1500 inside=0 symmetry=none periodicity=10 colors=000WJLVIJUHITFGSEFRDDQtCPrAMp9Mn7MlkMjhMheM\ fbMd_MaYMZVMWSMSPMPMMMKMJHMGEMDBMA8aEKZDIXCGVBESAC\ Q9AO88dhkjGbgF_eEXcDU`CRZBOXBLVAIS9FQ8CO799aiAZeoo\ rmiskcsjYthStfMueHucPsbWracq`jp_roZynHWXJYVL_UNaTP\ cRQdQSfPUhNWjMXkLZmJ`oIbqHcrGanF`kEM87_hEZeDYbCWZC\ VWBUTATQASN9QJ8PG8OD7NA6nsA`dh_aeZ_cYY`XWZWUWVSUUQ\ RTOPSLNRJKQHIPFFODDNBAM98qrcoo`mlZkiXifVgcTe`RcYPa\ VN_SKYPIWMGUJESGCQDAOA8NKTMIQMHOMGMMEJMDHMCFMACM9A\ M88f3mb4e_5Z4c`4c`4aZ4_X6YV7WT8URASPBQNCOLDMJFJHGH\ FHFDJDBKB9L97zzzwwwsssooohfhc`aZXYXUUVSSTQQQOOOMMM\ KKKDCKA9z6gw7cs7`o7Yk7Vg7Td7Rb7P`7NZ7KX7IV7GT7ER7C\ M7AJ78zUKrOAgI9XC7sTHnQFjNEfKCbIBYFAUC8Q97Lr_MnYMk\ WMhUMeSMbQMZOMWMMTKMQIMNGMJEMGCMDAMA8bmz`hzZdzX`zW\ WzUSzSOzRJzPFHNBBDxNErLFlJGfHH`FIVDJPBKJ9LD7zyKwwK\ rrVmhYhcUbYQ_TMWNITIEPCAzqHmaD_M9TVESTDRRCRPCQNBQL\ APJAPI9OG9OE8NC7NA7M86YLN } 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=========================================