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: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Scott's site is at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> 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 20.0 PAR-FORMULA 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 20.0 PAR-FORMULA FILE==================================