; Date: Sun, 01 Dec 2002 10:05:04 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 01-12-02 (Not a Minibrot [5]) ; Id: <1.5.4.16.20021201100712.29ffe186@pop.mindspring.com> ; --------- ; 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 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: ; ; ; ; and at: ; ; ; ; 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 PARAMETER 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 PARAMETER FILE================================== ; ;