; Date: Sun, 14 Mar 2004 10:52:58 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 14-03-04 (The Great Mystery [not rated]) ; Id: <1.5.4.16.20040314105553.0d771bb2@pop.mindspring.com> ; --------- ; ; 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: ; ; ; ; 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 PARAMETER 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 PARAMETER FILE================================== ;