FOTD -- March 14, 2004 (Rating none) Fractal visionaries and enthusiasts: Anyone looking at today's image would recognize it at once. They would know that it is a picture of the Mandelbrot set. And they would be wrong. One might examine the image as carefully as possible, and they would find no evidence that it is not the Mandelbrot set. But it is still not the set. The Mandelbrot set is the slice that cuts a two-dimensional hole through the center of the Z^2+C Julibrot in the plane of the two C-axes. There is only one such slice, therefore there can be only one Mandelbrot set. But what then is today's image, in which the formula Z^2+C is iterated? It is a slice of the Julibrot cut in an oblique direction halfway between the Mandelbrot and Julia orientations. Since a double rotation is needed to turn from the Mandelbrot to the Julia orientation, there are many slices oriented in the half- way position. Today's slice is only one of them. It can be easily demonstrated that today's image is not the familiar M-set by changing the starting point of C. Instead of merely shifting the image around on the screen, changing initial C in today's image actually changes the shape of the fractal as the image shifts. Changing initial Z also shifts the image and changes the fractal's shape. Discovering this second Mandelbrot set, which I call the shadow set, was perhaps the greatest surprise of my entire fractal career. It has linear dimensions 1.4142 times that of the familiar M-set and an area double that of the familiar set. An entirely new set of coordinates is needed to find the familiar objects. In the shadow set, East Valley terminates at 0.35355 on the X-axis, while the largest midget is centered at -2.47485. This shadow set has the curious property that the C and Z coordinates of every point are of equal value. The set also is curious because of the fact that some points as far from the origin as 2.8284 are trapped. This means that the trapped points of the Julibrot are not contained within a hypersphere of radius 2. I think a radius of 2.8284 might do the job, but I am not sure. And yes, it is possible to get a marginally greater resolution of a Mandelbrot object by examining the shadow set, though the screen coordinates of the object being sought must be multiplied by a factor of 1.4142 before the search begins. I have never stopped to figure in how many ways a plane may be oriented halfway between the Julia and Mandelbrot directions. I could probably make quite a project of finding all of them. And I also have doubts that the present SliceJulibrot2 formula can draw them all. But it's something to think about. I did not rate today's image, since its interest is almost totally mathematical. The render time of under three seconds makes up for the lack of artistic merit. The name "The Great Mystery" fairly describes the very simple image of today's FOTD. As always, the completed GIF file is available for download on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Saturday was sunny but breezy and chilly here at Fractal Central. With the sun now rising well above the holly trees during the middle of the day, the fractal cats did manage over 1/2 hour in the yard, but finally, the temperature of 43F 6C chilled their ears beyond endurance, and they came inside to enjoy their tuna. Today is starting with less wind and higher temperatures, but the clouds are moving in. The cats have yet to express their opinion of the conditions. As for me, I'm going to take it easy for the rest of the day. I might even get dragged to one of the countless local antique emporiums to look at the stuff that others have thrown away. One thing certain, (unless the world ends), is that the next FOTD will appear in 24 hours. Until then, take care, and believe it or not, here in Maryland they are talking of taxing our potato chips. They call it a 'snack tax'. Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ The_Great_Mystery { ; time=0:00:02.97--SF5 on a P200 reset=2003 type=formula formulafile=allinone.frm formulaname=SliceJulibrot2 center-mag=-0.7071/0/\ 0.612855 params=45/0/45/0/0/0/0/0 float=y maxiter=255 passes=b inside=255 logmap=yes symmetry=xaxis periodicity=10 colors=000JC0ME0PG0SI1VK2YM3`O4cQ5fS6iU7lWEgYL__SS\ aZKceCekJgnQiqXktbmwempgmijmblmWnmQnmQmmQmmRmmRmmR\ lmRlmRlmSkmSkmSkmTjmUjmVjmWimXimYimZim_hm`hmahmbhm\ cgmdgmegmffmgfmhfmiemjemkemldmmdmodmqcmscmucmwcmyb\ mzbmzbmzamzamzamz`mz`mz`mz_mz_mz_mzZmzZmzZmzZmzYmz\ YnzYozXpzXqzXrzWszWtzWuzVvzVwzVxz3yz6zz9zzBzzEzzGz\ zJzzLzzOzzQzzTzz_zzZzzZzzZzzZzzZzzZzzZzzZzzZzzZzzY\ zzYzzYzzYzzYzzYzzYzzYzzYzzYzzYzzXzzXzzXzzXzzXzzXzz\ XzzXzzXzzXzzXzzWzzWzzWzzWzzWzzWzzWzzWzzWzzWzzWzzVz\ zVzzVzzVzzVzzVzzVzzVzzVzzVzzEzzJzzNzzRzzbzzazzazza\ zzazzazzazzazzazz`zz`zz`zz`zz`zz`zz`zz`zz_zz_zz_zz\ _zz_zz_zz_zz_zzZzzZzzZzzZzzZzzZzzZzzZzzYzzYzzYzzYz\ zYzzYzzYzzYzzXzzXzzXzzXzzXzzXzzXzzXzzWzzWzzWzzWzzW\ zzWzzWzzWzzVzzVzzVzzVzzVzzVzzVzzVzzozznzznzznzzmzz\ mzzmzzlzzlzzlzzlzzkzzkzzkzzjzzjzzjzzjzzizzizzizzXz\ zYzzZzz_zz`zzazzbzzczz000 } frm:SliceJulibrot2 {; draws most 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), c=p+flip(q)+p3, z=r+flip(s)+p4: z=sqr(z)+c |z|<=9 } END 20.0 PAR-FORMULA FILE==================================