October 15, 2003 This FOTD image and discussion originally appeared on April 20, 1999. I present the image today in an improved version with fine-tuned colors. The discussion has been edited to remove some irrelevant stuff, and a few new comments about the fourth dimension have been added. ======================================================== CLASSIC FOTD April 20, 1999 (Hyper-Spherical) Fractal enthusiasts: What does a 4-D hypersphere look like? We shall never know. Speculation is futile. A hypersphere is not a visual object; it is an abstraction. Trying to visualize a hypersphere is as hopeless as trying to visualize any other abstraction, such as honor. But it is still fun to try to see what a hypersphere would 'look like' if we were four-dimensional beings. Of all the higher-dimensional figures, the hypersphere is the simplest. Any 3-dimensional slice of a hypersphere of any number of dimensions is a 3-D sphere. For example, any 3-D slice of a 4-D hypersphere is a sphere. Any 2-D slice is a circle, and any 2-D slice of a hypersphere of any number of dimensions is a circle. Today's fractal is a revelry of circles, which I assume, though I have not checked, are slices of 4-D hyperspheres. The three-dimensional effect is due entirely to the coloring, to which I devoted an unusual, (for me), amount of time and effort. The image resembles nothing as much as a scene in orbit around a giant planet in some far away galaxy. The entire picture is composed of circles colored to appear as spheres. These circles are arranged into interesting sheets and tendrils, which need investigating. Anyone out there who would like to investigate further, go ahead. I've got too many more fractals waiting, to have time to spend more than one day on a given image. I named the picture "Hyper-Spherical" more out of hope than fact. I hope that these circles are slices of near-perfect 4-D hyperspheres, but I have a strong suspicion that they are not. The GIF file has been posted in all its planetary glory to: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> The weather at fractal central has not changed. Partly cloudy skies with occasional sprinkles of rain and a temperature of 57F 14C made it a chilly spring day, yet a perfect day for finding fractals. I shall now close down the fractal shoppe for another evening, and call it a day. But I'll return tomorrow with more fractal blather. Until then, take care, and keep the fractal faith. Jim Muth jamth@mindspring.com jimmuth@aol.com START 19.6 PARAMETER-FORMULA FILE========================= Hyper-Spherical { ; time=0:12:57.80--SF5 on a P200 reset=1960 type=formula formulafile=mandnewt.frm formulaname=MandNewt12 passes=1 center-mag=+0.7897\ 05601237999/-0.02843030167377993/477729.2/1/30 params=-4.3808/-0.5139/-1.2595/1.0881/-0.81/2.519 float=y maxiter=250 inside=bof60 logmap=yes periodicity=0 colors=000zzYyxYxuXwqWvmVukUtiTsgSrf\ RqeQpdPocOnbNmaMl`Lk_KjZJiYIhXHgWGfVFeUEdTDcSDbRCa\ QC`PB_OBZNAYMAXL9WK9VJ8UI8TH8SG7RF7QF6PD6OA5N75M98\ LBAK75J66I66H008WT8WT8WT8WT8ZV8`Y8c_8fa8ic7kf7nh7q\ j7tl7vo7yqAwqDurHrrKpsNnsQltTjtWhu_eubcveavh_wkYwn\ WxrTxuRyxPyuMqsIipF`nBTk8Lj9KhAJgBIfCHeCGcDFbEEaFD\ `GCZHCYIBXJAWK9UL8TL7SM6RN5PO4OP3PP4QP6RQ7SQ8TQAUQ\ BVQCWREXRFYRHZRI_RJ`RLaSMbSNcSPdSQeSRfTTgTUhTViTXj\ TYkUZlU`mUanUboUdpUeqVgrVhsVitVkuVlvWmwWoxWpwYmuZk\ t`hrafqccpd`nfZmgWliTjjRilOgmMfoJhpKipLkqMmrNosOps\ PrtQstRrqSqoSplSojSnhSmfSmcSlaSk_SjYSiVShTSgRSfPSe\ MSdKScISbGSbDSaBS`9S_7SZ4SY2SY0SZ1R_1R_1R`2Qa2Qa2Q\ b2Qc2Pd2Pd3Pe3Pf3Of3Og3Oh3Oh4Ni4Nj4Nj4Nk4Ml4Ml5Mm5\ Mn5Lo5Lo5Lp5Lq6Kq6Kr6Ko7Jl7Ji8Ig8Id9Ha9GZAGWAFTBER\ BEOCDLCDIDCJEDKGFKHGLIIMKJNLLOMMOOOPPPQRRRSSSTTSVV\ TWWUXYVZZW_`W`aXbcYcdZdc_dcwTgwSgwRgwQg } frm:MandNewt12 {; Jim Muth z=g=pixel, a=real(p1), b=imag(p1), c=real(p2), d=imag(p2), e=real(p3), f=imag(p3): h=z^a+(g-1)*(z-b) j=c*z^d+g z=z-e*h/(f*j), .0000000000000000000000000000005 <= |h| } END 19.6 PARAMETER-FORMULA FILE===========================