; Date: Sun, 02 Oct 2011 17:52:10 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 02-10-11 (Rectangular Reason [No Rating]) ; Id: <1.5.4.16.20111002175213.11f7b612@pop.earthlink.net> ; --------- ; ; FOTD -- October 02, 2011 (No Rating) ; ; Fractal visionaries and enthusiasts: ; ; The math expression that was iterated to create today's image is ; Z^2+C, the same expression that gives us the well-known classic ; Mandelbrot set with all its perturbed sets, as well as all the ; Julia sets. Yet the image is clearly nothing at all like a ; Mandelbrot set, pure or perturbed, nor does it resemble any ; Julia set. This is because it is a slice through the Z^2+C ; Julibrot in what I call the Rectangular direction, which is ; defined by imag-C,imag-Z. The rectangular shape of the slice ; shows why I call this particular orientation the Rectangular ; direction, and also explains the name I gave to the image -- ; "Rectangular Reason". ; ; The large open area that fills a good part of the frame is a ; different view of the main bay of the M-set. The narrow bluish ; stripes running horizontally at the top and bottom of the open ; area are what I call 'bridges'. In this case they are sideways ; views of the west branches of the valleys leading to the large ; north and south period-3 buds of the M-set. The horizontal open ; areas above and below these 'bridges' are sideways views of the ; buds themselves. ; ; I gave no rating to the image, since almost all its interest is ; in its mathematics. But don't downplay the math interest. To ; make things much clearer than I can explain in words, gradually ; decrease the value of the real(p1) parameter from 90 to zero in ; increments. (Keep the imag(p2) parameter unchanged.) The ; images are very fast and the process will take little time. By ; the time the value is reduced to 50 the destination will be ; apparent. Doing a 360-degree rotation would make an interesting ; animation, but I doubt that I'll get the chance. (Decreasing ; the imag(p2) parameter of today's image to zero while leaving ; the real(p1) parameter unchanged brings us to the Elliptic ; orientation, but that's another story.) ; ; To create today's image, I oriented the 4-D Julibrot figure so ; that the imaginary Z-axis is perpendicular to the Mandelbrot set ; on the screen and ignored the parts of the Julibrot that did not ; fit into our 3-space, leaving an imaginary invisible 3-D slice ; of the Julibrot intersecting the screen. Then I rotated the ; screen image around the line perpendicular to the X-axis at the ; point -0.2. The resulting image is what would be seen by a ; viewer located in the plane of the screen just to the left or ; right of the screen when the invisible 3-D slice was sliced at ; the point of -0.2 on the real-C axis. ; ; All this talk does not mean the image is a slow one. In fact, ; it finishes in a lightning fast 5 seconds. ; ; The finished image is available on the FOTD web site at: ; ; ; ; The high-definition version is available at. ; ; ; ; The classic original web site may be accessed at: ; ; ; ; Heavy clouds and a temperature of 50F 10C made today a day to be ; forgotten here at Fractal Central. The occasional showers ; helped none at all. And the experts assure us that more rain is ; on the way tomorrow. The fractal cats always live in the ; present, so they usually forget a day's events at the first ; stroke of midnight. Many humans are not so forgetful. They ; hold long term grudges and resentments that fester until war ; breaks out. FL and I fall between the cats and the long-term ; resenters. ; ; The next FOTD, a continuing adventure in 4-D hyperspace, will be ; posted in 24 hours. Until then, take care, and I just read that ; there is some evidence that the sub-atomic particles known as ; neutrinos might travel faster than the speed of light. The ; claim is not yet confirmed, (In fact, I doubt it), but if this ; proves to be true, then neutrinos can move backward in time. ; This would surely play havoc with our scientifically accepted ; ideas of an effect following its cause, but after a century of ; relativity and quantum mechanics, we should not be so surprised ; when we find one more thing that violates our common-sense view ; of reality. ; ; ; Jim Muth ; jimmuth@earthlink.net ; ; ; START PARAMETER FILE======================================= Rectangular_Reason { ; time=0:00:05.66-SF5 on P4-2000 reset=2004 type=formula formulafile=basicer.frm formulaname=SliceJulibrot4 center-mag=0/0/0.73/1/\ 90/0 params=90/0/0/90/-0.2/0/0/0/2/0 float=y maxiter=5000 inside=0 periodicity=6 colors=000D08F09H0AK0BQ0CW0E_0Gb0Ih0Kn0Mp0Ot0Qm5Pc\ 7OU9NEJMJSMN`MSiLXrL`zPZuUYpZWkcVghTbmSYrQTvPPzOOz\ OOzNNzNNzG_zHVzIRzJMz`ezW_zSUzOOzQLzPKzOKzNJzNJzMJ\ zmIzmIzmTzmRzmPzmNzmLzmJzmdzmTzmgzmZzmQzm8zmAzmCzr\ EzrGzrxzrHzrIzrIzrIzrIzrGzrHzrHzrIzrXzrUzrSzrQzrOz\ rMzrKzrRzrPzrOzrNzrLzrKzrJzrzzrszrkzrdzrXzrPzrJzru\ zrazrGzrHzrHzrHzrHzrIzrIzrIzrWzrUzrSzrQzrOzrMzrKzr\ dzr`zrYzrVzrRzrOzrLzrizrdzrXzrPzrTzrQzrNzppgppgppg\ ppgppgppgpogpogpogpohpohpohpohpohpnhpnhqnhqnhqniqn\ iqniqniqmiqmiqmiqmiqmiqmjqmjqmjqljqljqljqljqljrljr\ lkrlkrkkrkkrkkrkkrkkrkkrkkrklrjlrjlrjlrjlrjlrjlrjl\ rjlrjlsiksijsiisihsigsifsifsheshdshcshbshashash`sg\ _sgZsgYsgXsgWsgWsgVsgUsfTsfSsfRsfRsfQsfPsfOseNseMs\ eLseLseKseJseIseHsdGsdGsdFsdEsdDsdCsdBscAscAsc9sc8\ sc7sc6sc5tb6sc5sc5sc5sc5sc5sc5sc5sc5sc5sc5sc5sc5sc\ 5sc5sc5sd5sd5sd5sd5sd5sd5 } frm:SliceJulibrot4 {; draws all slices of Julibrot pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1*0.0055555555555556), b=pi*imag(p1*0.0055555555555556), g=pi*real(p2*0.0055555555555556), d=pi*imag(p2*0.0055555555555556), ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g), sg=sin(g), cd=cos(d), sd=sin(d), p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd), q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd), r=u*sg+v*ca*sb*cg, s=v*sin(a), esc=imag(p5)+9 c=p+flip(q)+p3, z=r+flip(s)+p4: z=z^(real(p5))+c |z|< esc } ; END PARAMETER FILE========================================= ; ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint ;