; Date: Sun, 14 Sep 2003 11:52:00 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 14-09-03 (The Whole Seahorse [5]) ; Id: <1.5.4.16.20030914115236.0d579e20@pop.mindspring.com> ; --------- ; ; FOTD -- September 14, 2003 (Rating 5) ; ; Fractal visionaries and enthusiasts: ; ; I am constantly fascinated by the concept of the fourth spatial ; dimension. Even though four-dimensional space and objects are ; not visual things, I want to know what a 4-D object looks like ; when seen all together at a single moment, and not as a 3-D ; object changing shape over a period of time. ; ; I want to see an object with a three-dimensional surface. (The ; very idea seems absurd.) I want to see an entire plane of a ; hypersphere remain fixed in position, with every point rotating ; on itself as the hypersphere rotates. I want to see a hyper- ; sphere in a state of double rotation, where only the single ; point at the center remains fixed, while all other points trace ; out the circular spirals known as surfaces of double revolution. ; ; I want to look at the night sky and see not a flat surface with ; stars scattered over it, but a solid three-dimensional surface ; with stars embedded all through it. I want to be on the ocean ; and see the sphere of the horizon surrounding me in all direc- ; tions including the directions we would call up and down. And ; finally, I want to see the Z^2+C Julibrot in its full 4-D glory, ; with its arms shooting off to infinity. ; ; There are these things and so much more that I would like to ; see, but alas, I never will see them, for my visual perception ; is limited to three dimensions. I know that 4-D objects are ; possible because I can create them at will in my computer and ; examine any section I choose. As an example, I have created for ; today's FOTD the 4-D object known as Seahorse Valley. ; ; As I said, Seahorse Valley is a four-dimensional object, and as ; such can be examined from many impossible-to-visualize angles, ; one of which is today's image. The outer border of today's ; fractal is in the shape of the familiar Julia set with a C-value ; of -0.75,0. This is not unexpected, since the view is turned ; only one degree from the Julia orientation toward the Oblate ; orientation. But the interior of the fractal is filled with ; strange features unseen in either Mandelbrot or Julia sets. ; ; This stuff filling the interior of the fractal is an entirely ; new view of Seahorse Valley. I named the image "The Whole ; Seahorse" because it shows more of the Valley than can ever be ; seen in the M-set alone. ; ; The narrow straight line cutting down the center of the outer ; fractal is actually a side view of the two branches of Seahorse ; Valley coming together. The extreme maxiter of 1,500,000 is ; needed to close the gap between the two branches because, from ; this angle, the branches of the valley have been magnified 57 ; times. ; ; The orbits of the points in this area are so close to periodic ; that the periodicity checking must be turned off to avoid false ; calculation terminations. Combined with the extreme maxiter of ; many of the parts of the image, this results in a very slow ; image, which takes just under an hour to render on my machine. ; ; Considering the slowness of the image and the fact that I have ; posted several similar images in the past, I could rate today's ; image no higher than an average 5. But it is still an interes- ; ting image with coloring that is not bad, and the completed GIF ; image will be ready for download from the FOTD web sites within ; the hour. ; ; The URL for Paul's site is: ; ; ; ; Scott's site is at: ; ; ; ; The temperature reached 75F 24C on Saturday here at F.C., but ; the rain and showers continued all day The cats made a brave ; effort to remain in a good mood, but late in the afternoon their ; frustration got the better of them and a squabble broke out over ; something minor. They did a lot of yelling and tail thrashing ; under the dining room table. A few blows were thrown, but no ; damage was done, and the dynamic duo ended the day eating tuna ; and sulking at each other. Had it not been for the excitement ; among the duo, the day would have been uneventful. ; ; Today is starting sunnier, but lots of clouds are around and ; there is still a chance of rain. Only time will tell how it ; will go. I plan on another uneventful day, which might or might ; not happen. Until next time, take care, and be of stout spirit ; and slim body. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE================================ The_Whole_Seahorse { ; time=0:59:42.90--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XY-ZW-new function=ident/flip passes=1 center-mag=-7.77156e-016/4.996e-016/0.986\ 1933/1/-35/3.88578058618804789e-016 params=89/90/2\ /0/0/0/-0.75/0 float=y maxiter=1500000 inside=0 logmap=yes periodicity=0 colors=00092F85H87J7AL7CN6FO6HQ6KS5MU5PV4RX4UZ4W`3\ Zb3`c2ce2eg2hi1jj1ml0nn0op0pq5qoArnCslDtjEuhFvfGwd\ HxbIy`JzZKzXLzVMzTNzROzPPzNQyLRwJSuHTsFVqFUoEUmEUk\ EUiEUgEUeDUcDTaDT_DTYDTWDTUCTSCTQCTOCSMCSKCSIBSGBS\ EBSCBSABS0B76YB7`E7cI8eL8hP8kS9mW9pZ9sbAueAxhAzgBx\ gBvfBtfBsfBqeBoeCndCldCjdCicCgcCebCcbDbbD`aDZaDY`D\ W`DU`DTaEVbFWbGXcHYdIZdJ_eK`eLafMbgNcgOdhPeiQfiRgj\ ShjTikUjlVklWlmWmlXnmXomXpmXqnXrnXsnXtnXuoXvoXwoXx\ pXypXzpXzrWzpXzoXznXylXxkXwjXuhXsgXqfXodXmcXkbWi`V\ g_UeZTcXSaWR_VQYTPWSOURNSSNQTOQUPSVQTWRVXSWYTYZTZ_\ U``VaaVcbVdcVfdVgeUifUjgUlhUmiUojUpkUrlUsYVeKWT0XA\ 6WGCVMIUSOTYTTcSVeSWgSYiSZkR`mRaoRcqRdsSdxReuQerPf\ oOflNgiNggMhdLhaKiZJiWIjTIjRHkOGkLFlIElFElDDdMCXVB\ PcBIlCMjCPhDSfDVdEYbE``EdZFgXFjVGmTGpRGsPIsQKsQMsS\ OsUQsWSsYTt_UuaVvcWweWxgXyiYzk_zmazoczqezsgzuizvkz\ wmzxozyqzzszzuzzwzzwzzwzz } 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================================== ;