FOTD -- December 01, 2002 (Rating 5) Fractal visionaries and enthusiasts: Today's FOTD features the first of the odd slices of various Julibrots, a theme that will continue throughout December. The Julibrot being sliced in today's image is the best known of them all -- the figure produced by the formula Z^2+C. This Julibrot figure is the assemblage of all the Julia sets into a single four-dimensional unit. It is also the assemblage of all the perturbed Mandelbrot sets, though in this case, the slices are cut through the Julibrot in the completely perpendicular direction. The minibrots that I am so fascinated with are actually slices of holes which are like gas bubbles in a four-dimensional swiss cheese. Their appearance depends on the point at which and the direction in which the holes are sliced. When sliced in the Julia direction, these holes will appear as various julia sets, their exact appearance depending on the point at which they are sliced. When sliced in the Mandelbrot direction, they will appear as perturbed Mandelbrot sets. They will appear as perfect midgets only when the slice is taken at a critical point of the generating formula, which in the case of the formula Z^2+C is zero. The question immediately arises, if these holes can be sliced in at least two different directions, why can they not be sliced in other directions, and in fact an infinity of directions? And if other slices are possible, what might these slices look like? Today's image answers that question. The object at the center of the image, which rather resembles an artillery shell, is part of the Z^2+C Julibrot. But it is obviously not a Mandelbrot midget, nor is it a Julia set. It is a slice through a 4-D hole, another slice of which appears in the M-set as a midget in the Seahorse Valley area. The slice in today's image is taken in a direction rotated 30 degrees from the Julia orientation. In this direction, the hole appears as nothing in particular. It can be said to be only a hole in a fractal, but the pattern that appears around both the Mandelbrot midget aspect and the corresponding Julia sets is still there, though it is distorted into a blunt triangular shape. The hole itself has two sides that are perfectly straight and parallel, a thing that is never seen in J-sets or M-sets. And a glance at the <tab> screen will show that the image has been both stretched and skewed. This is a hint of very important features of the Julibrot figure, which I will discuss in future messages. After much indecision, I have named today's image "Not a Minibrot". Since it is clearly *not* a Minibrot, the name is justified. The rating of 5 for an artistically routine but mathematically interesting image is also justified. Those curious few who would like to see the undistorted Mandelbrot midget can find it in the M-set at today's real and imaginary p(3) or p(4) coordinates. With a render time of only 2 minutes, the parameter file may be the best gateway to the world of 4-D. But for convenience, Paul and Scott have posted the image to their web sites at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal weather Saturday here at Fractal Central featured sun, clouds, showers and wind, with a temperature of 48F 9C. In the evening the temperature started dropping like a rock, accompanied by a few flurries of snow. The cats took their outdoor time in the late morning, when it was sunny and mild. This morning is starting sunny again, but it is very cold and windy. The cats will have a less satisfying day. Since it takes little but fractals to satisfy me, my day will also be satisfying. And I shall return in 24 hours with another 4-D fractal and more talk. Until then, take care, and walk with a fractal in your pocket. Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ Not_a_Minibrot { ; time=0:02:05.84--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XY-ZW-new function=flip/ident passes=1 center-mag=-0.00000000002510085/-0.000000\ 00183999569/1.549816e+008/0.2727/-99.5810541174990\ 504/-33.4839088781078544 params=90/60/2/0/-0.74860\ 53932854075/0.09514208701232971/-0.748605393285407\ 5/0.09514208701232971 float=y maxiter=2400 inside=0 logmap=159 symmetry=none periodicity=10 colors=000SFJSHNSKSSNWSP_SScSVhSXlS_pSbtSdxUhuWlxY\ rxZwx`rubmpdhkecfgZaiT`kSYlRWnQUpPRrOPsNNlLQeKT_IW\ THZMFaGEd9Cg3Bj8AeCAaH9XL9TP9PU8KY8Gb7Bf77g73eDVbI\ c_NjUSoQXmMXvQ_tUZrYXpaWndVmhTklSipQgtPewOdvVevaev\ hevofvvfvzfgrhTiiE`j0Sk4Tg8TcCT`GTXKUUOUQSUMWUJ_VF\ cVCgV8kV5jZ4jb3jf2ii2im1iq0ht0hx0hz0hz0cy0Zs2Um3Pg\ 5La6cZ0e`2fa5gb8hcBidEjeHkfKlhNmiQnjTokWplZqmarnds\ ogkkfdgeXcdQ_cIWbBSa4P`BQcHRfNSiTTlZUodVrjVukUnlUg\ mT`mTUnTNoSGpS9pS2qNCqJLnMMlPMiSMgUNdXNb_N`bOYdOWg\ OTjPRmPPoPOlQNjRMhRLfSLcSKaTJ_TIYUIWUGVREVOCULAUM9\ UP7TS5TV3SY1S`0Sc2Zf4di5jl4go4dr3bu9_xA`xBaxCbxDcx\ EdxFexGfxHgxIhxJixKjxLkxMlxNmxOnxPoxQpxRqxSrxTrxUr\ xVrxWrxXrxYrxZrx_rx`rxarxbrxcrxdrxerxfrxgrxhrxirxj\ rxkrxlrxmrxnrxorxprxqrxrrxsrxtrxurxvrxwrxxrxxrxxrx\ xrxxrxxrxxrxxrxxrxxrxxrxxrxxrxxrxxrxxrxxrxxrxxrxxr\ xxrxxrxxrxxrxxrxxrxxrxxrx } frm:multirot-XY-ZW-new {; draws 6 planes and rotations ;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J ;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994, z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p3, c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p4: z=z^(p2)+c, |z| <= 36 } END 20.0 PAR-FORMULA FILE==================================