August 2008 - Fractint - mailman.xmission.com

FOTD 21-08-08 (A Trunk-Full [7])
by Jim Muth 21 Aug '08

21 Aug '08

FOTD -- August 21, 2008 (Rating 7) Fractal visionaries and enthusiasts: The following paragraph is science, not philosophy. Physicists are searching for the 'Higgs particle', which is the particle aspect of the 'Higgs field', the field that is theorized to interact with and give mass to other particles. This 'Higgs particle' is assumed to be so massive that the world's most powerful accelerators will be pushed to the limit in the effort to find it. My question of the day is this: what then gives mass to the Higgs particle? Is it self-massifying? When you stop laughing, try to think of a rational answer. While we're pondering that puzzling bit of science, let's discuss the fractal of today. Today's image is a scene in the East Valley area of the Mandeloid that results when Z^2 is divided by (Z^(-0.91)+500). The actual minibrot in the image is found deep in an elephant trunk of an elephant that itself lies deep in the trunk of an even larger elephant. I guess it's elephants all the way down. This endless nest of elephants inspired the name "A Trunk-Full". A certain lack of satisfaction held the rating to a not-too-bad 7. The calculation time of 2 minutes may be done away with by trekking to the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and viewing the pre-rendered image there. Wednesday was the reference standard perfect day here at Fractal Central. All measured parameters of the atmosphere were perfect for both human and cat comfort. The temperature was 77F 25C, the humidity was 50 percent, the sky was cloudless, and the north wind blew at a mild 7 knots. However, the fractal cats paid more attention to the scraggly manx cat that appears to have moved into the neighborhood. My day was slow enough to leave lots of time for leisure, yet busy enough to avoid recession concern. The next FOTD will be posted in 24 hours. Until then, take care, and I can't help it if these silly questions keep coming. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= A_Trunk-Full { ; time=0:02:04.45-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=DivideJulibrot center-mag=+188.3134121\ 224326/-0.9553325705714198/4e+008/1/165/0 params=0/0/0/0/0/0/0/0/1.09/500 float=y maxiter=2200 inside=0 logmap=244 periodicity=10 colors=000OIKNIKMIKLHKKGKKFKKEKKDOKCRKBUNDTPESSGSU\ HRWJRZKQ`MQbNPeOOgQOjRNlTNnUMqWMsXLuYLtWMtVNtTOtSP\ tRPtPQtORtMStLTtKTzbdztozsnzsmzrmzrlzqlxqkvpjtpjro\ ipoimnhkngimggmfemfbje5Wa7Va9UaBTaDSaFRbHQcJPcLOcN\ NcPNcRLiTLjVMkXMoZNp`Pl_QlZRiYRfXScWS`UTYTUVSUSRVP\ QVNRXISYES_AT`6Ta2bOGlATt6`z3gz5fz7ez9ezBdzDczFczH\ bzJazLazN`zP_zR_zTZzVYzXYuWWpWVkWTfWSaVRXVPSVONVNT\ WSYXWbX`gYdlZiqZmv_rz_vwZusZtoZtkYsgYscYr_XqWXqSXp\ OXpNVkMUgMTbLSZLQUKPQKOLJNHJMDIPEISEHVEHXEH_EGbFGd\ FFgFFjFFlFXaJmSNzIQyCOx7NwANvDNuHNz1Mx5Poihnkjmllm\ mnknjinfgnbeo_doWboS`pPZpLYpHWqEUqASq6Rq3mp6zp8zpB\ zpDzpGzzczzz`pNbpPcpSdpUepWZiVScVLYVJNLICBMDEQDHTD\ JXDM_DOcERfETjEWnEZqE`uFcxFezFhzFjzImzLozOnzRnzUmy\ Wmv_msblpelmhkjkkgnkdqjajjZmj_oc`qXasQbuJcwCdz6bzD\ `zJ_zPYzVWz`Vzfcz2kz4sz5zz6zz7zzDzzIzzOzzTzz`zzRzz\ IzzJzzKzzLzzMzzNzzOzzOzzP } frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots 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), aa=-(real(p5)-2), bb=(imag(p5)+0.00000000000000000000001), 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)/(z^(aa)+bb)+c |z|< 1000000 } END PARAMETER FILE=========================================

1 0

FOTD 20-08-08 (Blue Mandeloid [8])
by Jim Muth 20 Aug '08

20 Aug '08

FOTD -- August 20, 2008 (Rating 8) Fractal visionaries and enthusiasts: Today's image is undoubtedly the bluest of all time. But it's not really a melancholy blue, it's a dynamic artistic blue, a deep royal blue that brings out the incredible energies flowing within the number fields that make up the world of fractals. Actually, it's a slice of the Julibrot that results when Z^2 is divided by (Z^(2)+1.5). Yes, I realize that the value of real(p5) is 4, but dividing Z^2 by Z^2 creates a fractal with Z^4 attributes just as multiplying Z^2 by Z^2 creates Z^4 attri- butes. The 1.5 value of imag(p5) merely enlarges the parent fractal and moves the switchover to Z^4 stuff to a deeper level. To add to the confusion, the orientation of the image slice lies halfway between the Mandelbrot and Rectangular directions. And actually, the Mandelbrot attributes are more apparent in this halfway direction than in the straight Mandelbrot direction. (To see the straight Mandelbrot direction change the value of real(p1) from 45 to zero.) Since blue is one of my favorite colors, and there is so much of it in today's image, I rated the image at an 8. The name "Blue Mandeloid" is a bit of a misnomer however, since the image is as much a Seahorse Valley Rectangleoid as a Mandelbrot fractal. One thing not in doubt is the speed of the calculation. It takes all of 13-1/2 seconds for the included parameter file to run on a S.O.T.A. machine capable of digesting ancient DOS programs such as Fractint. The finished image, in all its deep blueness, is or soon will be posted for instant viewing enjoyment on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> There were a few more clouds than the fractal cats would have preferred here at Fractal Central on Tuesday, but there could be no complaining about the temperature of 77F 25C. The cats did not complain. With the real work finished early, my day was a bit slow. After browsing an hour or so in the philosophical section of the local library, I returned and settled down for my daily contest with the world of numbers. As today's image shows, the results of the battle were rather good. The next FOTD will be posted in 24 hours. Until then, take care, and think in higher dimensions. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Blue_Mandeloid { ; time=0:00:13.34-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=DivideJulibrot passes=t center-mag=0/-0.457742/0.9418072/0.819/90/0 params=45/0/0/90/-1.25/0/0/0/4/1.5 float=y maxiter=1500 inside=0 periodicity=10 colors=000D3AD3DD3GE3JE3ME3PF7SFBVGFYGI`HMcHQfIUiI\ XlH_oGbrFeuEhwFkxKnyPqzUtzZwzczzhxzmvzrszvpzzmzzjz\ zgzznzzpzzqzzrzzpzwozwmzwizwgzwfzwgzwfzVlzUmzTmzSm\ zRmzZgzfbznXzvSzrQznPzjOzgNzcLz_KzWJzTIzUGzVEzWCzX\ AzX8zP7zH6z95zJVzTszKczCPzDPzDOzEOzENzFNzFMzGMzGLz\ HLzHKzIKzIJzIJzHMzGPzGRzFUzEWzEZzD`zCczCfzBhzAkzAm\ zDjz9pzIBzFQzCdz6szl7zTVzVOzNZzGhzfPzcSzaUzZWzXYzU\ _zSazQczNfzLhzIjzGlzDnzBpzxJzfVzQfza5zXDzTKzPRzLYz\ HdzDkz9Yz9hz9wz9vz9uz9uz9tz9tz9sz9sz9rzSAzPFzNKzLP\ zJUzHZzFczDhzBmzx9zsDzoHzkLzfPzbTzZWzU_zQczMgzHkzD\ ozVKzTNzRPzQSzOUzNXzLZzKazIczGfzFhzDkzCmzApzSIzLVz\ Ffz49z6Lz7Wz8gz0Wz5gzn3zUTz6Yz8hzMqzKrzIrzHrzFrzDr\ zCrzArzdDzaHz_KzXNzVQzSUzQXzN_zLbzIfzGizDlzBozKFzJ\ JzINzHQzGUzFYzE`zDdzChzBkzAozLlzFoziGzdLz_QzWVzR_z\ MdzIizDnzNXzIdzDkz8uz9tz9tz9tz9sz9sz9sz9sz9rz9rz9r\ ztUzh`zEGzDBzC6zB2zC3zC3z } frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots 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), aa=-(real(p5)-2), bb=(imag(p5)+0.00000000000000000000001), 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)/(z^(aa)+bb)+c |z|< 1000000 } END PARAMETER FILE=========================================

1 0

FOTD 19-08-08 (Stars of Iteration [8])
by Jim Muth 18 Aug '08

18 Aug '08

FOTD -- August 19, 2008 (Rating 8) Fractal visionaries and enthusiasts: The formula for today's image adds a very small bit of Z^(-20) to Z^2 to produce a Mandelbrot set with disks of Z^(-20) stuff at the centers of all the bays and buds. The image is located on the X-axis, in the detail at the east edge of a disk almost filling the large period-2 bud of the M-set. More than anything else it resembles a surrealist's spider-web. I named the image "Stars of Iteration", which sounds a little poetic and in a way describes not only today's fractal, but all fractals. And then I rated it at an 8. I considered adding a half point for my coloring, but decided that I did not do quite enough color tweaking to merit the extra half-point. The calculation time of only 42 seconds is fast enough to cause no impatience. The finished image, which is posted on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> is convenient enough to totally take the work out of fractals. With a temperature of 88F 31C here at Fractal Central on Monday, it was just a bit too warm to be perfect, but all else was the picture of perfection. The dynamic fractal cats, who never stop reiterating, agreed. Just after lunch they spotted an unfamil- iar manx cat in the yard next door, and with a volley of hisses and growls, sent it on its way. I did not tell them that the cat was ready to move on anyway, and their empty threats had nothing to do with its departure. Otherwise the day was uneventful. The next FOTD will be posted in 24 hours. Until then, take care, and to become famous, solve the measurement problem. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Stars_of_Iteration { ; time=0:00:42.84-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=MandAutoCritInZ function=ident passes=1 center-mag=-0.9160783767764306/0/8.62e+008 params=1/2/1e-018/-20/0/0/0/0 float=y maxiter=1450 inside=0 logmap=153 symmetry=xaxis periodicity=10 colors=0000EC0FF0GI0HL0IO0JQ0LT0NW0PZ0Ra0Td0Vg1XiD\ _aPbV`cOldHwdAwgBwjCwmDwpEwsFwvGwyGvvFutFuqFtoFslF\ sjFrhFqeFqcFp`FoZFoWFnUFnSFoTEpTDqTCrTBsTAtT9uT8vT\ 7wT6xT5yT4yT4pO9hJEsFTSAcJ5cB1cC2cC2ZD2UC2PB2KA2FA\ 2AA3AA3AA3AA3AA3AC3AF3AI3AN5AS7FW9L`BReCWiEanGgsPz\ wUzvUzuUzuUztUztPysKwrHurHsqHsqHsmJmjLhgNcdPZaRUYT\ OVVJSXEPZ9M_4K`5Ja6Ha6Gb7Fb8Dc8Cc9Ad99dA8eB6eB5fC4\ fCm6Am9DmCGmFJmIMmKOmNRmQUcTXUW_OYaSdSWkJ_zAbzKYzU\ TzcOzmJztFzzGzzHzzHzzIzzIzzJzzKzzKkVLlYLm`MncMnfOl\ dQjcShbUg`We_YcZZbY``WbZVdXUfWShURjSQkRPceQXsQUrQQ\ qRNpRzoSznSzmTzlTzkTziQHhNLgLPfITeFXdD`cAdb7ha5cX5\ _T5VP5RL5PPAOSENWIMZMLbQKeVJiZIlbHpzGszIzzFrzCjz9c\ z6Wv4Pr3Mm3Kh2Ic2GZ1E_1Ce0Ak08q06v6CoCIhIOaOUVU_U_\ eZdkcamh_omYqrVsvzuzzwzzxzzyzzzzzzzzzzzzzzzzzzzzzz\ zzzzzzzzzezzNzz4zz6zz7zz8zz9zzAzzCzzDzzEzzFzzGzzIz\ zJzzezzfzzgzzhzzhzzezzbzz } frm:MandAutoCritInZ {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)), k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END PARAMETER FILE=========================================

1 0

FOTD 18-08-08 (A Quick Journey [5])
by Jim Muth 17 Aug '08

17 Aug '08

FOTD -- August 18, 2008 (Rating 5) Fractal visionaries and enthusiasts: Today's image is a minor thing worth a rating of a measly 5 points. But its calculation time of an equally measly 17 seconds means that almost no time will be wasted calculating it. The parent fractal combines just the tiniest bit of Z^(-63) with Z^2 before adding C. This parent resembles a classic Mandelbrot set obscures by scalloped disks. Today's scene is located near the X-axis in an interesting area between the two largest disks. I added a modicum of interest to the not-so-impressive scene by rendering it with the outside set to 'tdis'. Then I named the image "A Quick Journey" because I found it in less than 5 minutes. And I rated it a 5 because I found it in less than 5 minutes. The final result may be seen by running the included parameter file or by viewing the completed image on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> The sky was blue enough, but a few too many puffy white clouds kept Sunday from being absolutely perfect here at Fractal Central. The temperature of 84F 29C and low humidity were all that could be asked for, but those puffy clouds got in the way of the sun a bit too often. The fractal cats, unconcerned with minor cloud details, had a near perfect day anyway. My day was very good, which I would enjoy being able to say more often. The day's fractal was not quite so good, but no one ever claimed that this fractal world is perfect. The next FOTD, hope- fully a bit closer to perfection even if not quite there, will be posted in 24 hours. Until then, take care, and embrace your reality before it gets away. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= A_Quick_Journey { ; time=0:00:17.04-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=MandAutoCritInZ function=ident center-mag=-0.840428912973/+0.001176307923/9345305\ /1/-115/0 params=1/2/1e-018/-63/0/0/0/0 float=y maxiter=1000 inside=0 outside=tdis periodicity=10 colors=000P8W5BS`ibYg_WeXUcURaRP_ONYLLWIKXTKYcJZnJ\ ZxGYtEYpCXlAXh8Wd6W`4VX2VT9HIG48MBERIKWPQ`WWebajig\ oolmijkchiYggSefMdRBjNHkJNlGTmCZn8do5jp`GUbGVdGWfG\ XhGYjGZlG_mG`gI_bK_YLZTNZOOYJQYERYKTXPVWUWWZYVcZVa\ QM`HE_86i`xhapgbifcbedVdeOcfHcgAZjEVlIRnMNpPbzjYri\ TjhObhKVgFNgAFf68fDGbJN_QUXW`UcUXkN_sHbrJZrLVrNRrP\ NrRJrTFrVBrX7mRDhMIcGOZBTU5ZP0cUHVZXMclE`mHZnKXoNU\ pQSqTQqVLhOH`HCTA8L3GQ7OUAK_EHeHEkLBpOAnT9mX9l`8kd\ 7ii7hm6gq6fu8isAlrCnpEqoGtmIvlMwcQwVUwNQoPMhRIaTEU\ UANW6GY39Z7F`AKbDQdGVfK`hNejQklTpmZf_cXMX_OQbQJdSC\ gU6iVAiSEiPIiMRaLZULfMLTdBFv2Dw5Cw7AwA9wC7wF6wH4wK\ 3wMPBkKLgFVdBda6nZ2xWPUdVTc`TcfTclTcrTcwTcsUaoU_kU\ YhUWcPUZLTVHRQCQL8OH4NVK_gZktmwgicVeLZhNajOelPhoRl\ qSosTruUsqTtnSukRvgRwdQxaPxZPraXldcfgk`jrVly_hoddf\ i`XnXOsTEwQ5pV4iZ4bb3Wf3Pj3In2Br24v2Bt1Ir1Pp1Wn0bl\ 0ij0pi0gd7_`ESXLKTSCPYi5_ } frm:MandAutoCritInZ {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)), k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END PARAMETER FILE=========================================

1 0

FOTD 17-08-08 (It Is Impossible [6.5])
by Jim Muth 17 Aug '08

17 Aug '08

FOTD -- August 17, 2008 (Rating 6.5) Fractal visionaries and enthusiasts: As the title of today's image implies, the image is impossible. With the value of real(p5) set to -2, the minibrots deep in the parent fractal must be of the order -2. Minibrots of a negative order are by definition impossible however. If they did exist, they would have a number of lobes 1 less that their order, but how can a minibrot have minus-3 lobes? This is what I decided to find out in the search for today's fractal, when I set the value of real(p5) at minus-2. The first problem was that, with such a large escape radius and a negative exponent of Z, the image turned out to be all blank inside stuff -- an evaporated image. This problem was quickly solved by re- rendering the blank screen with an active inside fill. In this case I first tried the 'bof61' fill. When I did this, a ghostly arrowhead-shaped thing appeared, only vaguely resembling a Mandelbrot shape with a rudimentary East Valley. A closer search of this proto-East Valley revealed the symmetries that indicate minibrots are at the center. Zooming in on a prominent symmetry, I found today's scene, with its pseudo-minibrot at the center. The 'bof61' inside fill revealed the scene, but I found that the 'fmod' fill actually works best with this image. And what does a minibrot with minus-3 lobes look like? That question can be answered by checking today's image. I'm not sure that the thing at the center is a minibrot at all, but with its symmetrical surroundings, it sure does seem to be trying to be one. After all this mathematical excitement, I could rate the image no higher than a 6.5, 1/2 point of which is due to my coloring. The calculation time of a fireball 45 seconds will burn no one. And as always, the finished image is or soon will be available on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> A perfect day here at Fractal Central on Saturday pleased humans and cats alike. The warm sun, blue sky and temperature of 77F 25C were perfect for everything but skiing. I doubt that anyone in the area had planned on skiing however. The day's FOTD might be a bit hasty, but the next FOTD is due in 24 hours. I suspect it will actually be ready on time, and it might even be a pretty good one. Until then, take care and make the impossible possible. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= It_Is_Impossible { ; time=0:00:45.13-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=DivideJulibrot passes=1 periodicity=10 center-mag=+0.21857780829516740/+0.051382415163788\ 77/284576.4/1/-87.5/0 params=0/0/0/0/0/0/0/0/-2/0.5 float=y maxiter=1000 inside=fmod proximity=0.051 colors=000gaEhcEjeElgDniDokDjdLfZSaTZYMeTGlPAsrmmX\ Cn`DldDihEglFepFbtG`xHZzHXzKVyNUwPSccctVPsXOq_MpaL\ qbNrbPsbRtbTubVvcWwcYxc_UUPzcczddzdfzdhzdjzdlzdmze\ kzfjxfiughsggqhfoielicjjbhjafk`cl_alZ_mYYmX`kWciVf\ gUuuMppPlkRhfTdbV`YXXT_TPaPKcLFeF9eHBgIDhJFjKHkLJl\ MLnNNoOPpPRrQTsLXvOWuRVtUUtXTs_SsbRreQqhPqkOpnNpqM\ oPTGdPYsLooMlkMjgMgcNe_NcXN`TNZPOXLOUHOSBMOEOQHPSK\ QUNRWQSYTT_WU`ZVbaWddXfgYhjZjm_lvQHsWYnfvp`mqWerRY\ sMQtHImRHf`G_jFUsFVpIVnLVlNVjQVhSWfVWdXWb_W`aWZdMe\ NWXfOHBXL`iNzdOz_OyWOyRPxNPwIPwEQv9Qu5QuFRnPSgYTag\ UVcBgogHp`LpUPqOTqHXrA`r4ds6as7_s9YsAWtCTtDRtEPtGN\ tHLuJIuKGuMEuNC8zihVckTbmSaoR`rP_tOZvNYzN_xMYuLXrL\ VoKUlKSiJRfJPcIO`INDSkLWeT_``cWhgQpkLxoGzrBboUAmkE\ gWIbGKcHMcIOcJPcJRcKTcLUcLWcMYcN_cO`cObcPdcQecQWRQ\ MEQC2QD3PE4OF5OG6NH7NI8MJ8MK9LLALMBKNCKOEJOGJQIISK\ IUMHVOHXQHZSG`UGcWGcXFcXF } frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots 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), aa=-(real(p5)-2), bb=(imag(p5)+0.00000000000000000000001), 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)/(z^(aa)+bb)+c |z|< 1000000 } END PARAMETER FILE=========================================

1 0

FOTD 16-08-08 (Few Discontinuities [7])
by Jim Muth 16 Aug '08

16 Aug '08

FOTD -- August 16, 2008 (Rating 7) Fractal visionaries and enthusiasts: It is an inescapable mathematical fact that fractals with frac- tional powers of Z must have discontinuities. These discontinui- ties sometimes make interesting patterns in themselves, but just as often they spoil the perfection of the fractal in which they are found. The discontinuities cannot be avoided, but as the FOTD image for today unexpectedly shows, they can be minimized. The formula for today's image contains Z^(3.5), which demands that the image have discontinuities. The breaks are there indeed, but they are so few and so unobtrusive that they could easily be missed when the image is glanced at casually. With its one minor break on the left side, today's image is a scene in the East Valley area of its parent fractal, which is an oversized Mandelbrot set. I named it "Few Discontinuities" and rated it at a 7. The calculation time of just under 5 minutes is fast enough to hold boredom at bay, yet slow enough to allow ample time for anticipation. The image is also available on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Pleasant temperatures and plenty sun made Friday an acceptable day here at Fractal Central. A sudden thunder-shower at 5pm was too brief to spoil things. The fractal cats agreed as they stretched in the returning sun. For me the day was acceptable. I could think of a few things that might have gone better, but why quibble with minutiae. The next in the apparently endless series of FOTD's will be posted in 24 hours. Until then, take care, and if life is all a dream, where will we be when we wake up? Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= FewDiscontinuities { ; time=0:04:53.03-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=DivideJulibrot center-mag=+3.798906332849346/-0.0029529560051977/\ 5.019158e+007/1/20/0 params=0/0/0/0/0/0/0/0/3.5/15 float=y maxiter=15000 inside=0 periodicity=10 colors=000jbjgciddhaegZefWfeTgdQhcNibKjaHk`Fk`Lh_Q\ fZVdY_bXd`WiZVnXUsVTxTTsWRnZQj`OecNafLXhKTkIOnHKpF\ FsEBuDApFAlHAhIAdKA`M9XN9TP9PR9LS9HU9DVBFXCGZEH_FJ\ aHKbILdJMeLOgMPiOQjPSlRTmSUoTVpUWoVXnWXnXYmYZlZZl_\ _k`_ja`jbaicahdbhebgfcfgdfhdeiediedhbeh_ehYfhVfhTg\ hQghOhhLhgIhgGigDigBjg8jg6kg3kg1ke2ic3ib4i`5hZ6hY7\ gW8gU9fTAfRBePCeODeMEeKFeJGeHHeFIeEJeBGi8Dl5Ao28r5\ Ar7CsADtCFuFGvHIwKJxMLyPMzROxTQvWRtYTr`UpbWpeXogZo\ j_olaonbom_nlYmlWmkTljRkjPkiNjhKihIigGhfEgfBge9fd7\ eb3gd5ee6cf8ag9`hBZiCXjEWkFUlHSmIRnKPoLNpNLqOKrQIs\ RGtTFuUDvVBvW1xZAwbJtfRrj_mmhlolknngmkdke`h`YfVUcQ\ Q`KNYFJV9HR3GT4FV5EX6EZ7D`8Cb9BdABfBAhC9jD8lE2Nn8m\ EAe9CY5AN0DQ1GS2JU3MW4PY5R_6Ua7Xc7_f8bh9djAglBjnCm\ pDprEtuFrtEptEotEmsDlsDjsDisDgrCerCdrCbqBaqB_qB`s9\ ZqBXpCVnDUmESkFQjGPhINgJLeKKdLIcMGaNE`ODZQBYR9WS8V\ T6TU4SV1PV3RW5TW7VW8WWmak } frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots 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), aa=-(real(p5)-2), bb=(imag(p5)+0.00000000000000000000001), 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)/(z^(aa)+bb)+c |z|< 1000000 } END PARAMETER FILE=========================================

1 0

FOTD 15-08-08 (Little Critters [8])
by Jim Muth 15 Aug '08

15 Aug '08

FOTD -- August 15, 2008 (Rating 8) Fractal visionaries and enthusiasts: In today's fractal I divided Z^2 by (Z^(0.000001)+5), or some- thing like that, and then added C. At least, that's what I think happened when I entered the parameters. But regardless of what actually happened, the resulting parent fractal resembles an oversized Mandelbrot set, though deep inside it is different. I found today's scene on the edge of the infinitely divided main spike, in the area just east of the large minibrot. This is quite a rich area, filled with surprises, an area which until now I have largely ignored. I named the image "Little Critters" when I noticed the dark insect-like things sitting on the webs surrounding the central minibrot. Then I rated the image at an 8, which includes a small reward for my coloring. The brief calculation time of 1-1/3 minutes will make everyone but the most impatient fractalist happy. And even those impatient few will be only slightly discombobulated. To avoid all possibility of discombobulation however, the finished image may be accessed on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Too many clouds and a threat of rain kept Thursday from being perfect here at Fractal Central. But the temperature of 82F 28C was quite pleasant and the rain held off until nightfall, so the fractal cats had nothing to complain about. I had nothing to complain about either. So I'm not complaining. The next Fractal of the Day will be posted in 24 hours. Until then, take care, float like a duck and sting like a flea. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Little_Critters { ; time=0:01:20.40-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=DivideJulibrot center-mag=-10.49111003\ 76656/+0.00000459624450347/5.459246e+011/1/177.5/0 params=0/0/0/0/0/0/0/0/2.000001/5 float=y inside=0 maxiter=1800 logmap=200 periodicity=10 mathtolerance=0.05/1 colors=000K8_K9bK9dKAgKAiKEkKHmKKoKNqKQsKUsKXvJ`vG\ cxDfvAjw7nxAqyKuzUvuZwqcwlhuhmscrq_voVzmRzkUziXzfZ\ zbazZczXfzVhFSdEP`CNXBKTAHP8FL7CH69D479345222`_IZZ\ HYXGXWGWVFVUFTTESSERQDQPCPOCNNBMMBLLAKJ9JI9HH8GG8F\ F7EE7DC6BB5AA5994884773552442331221110AM38H26D1481\ 240oYOjULeRJ`OHXLFSIDNFAIC8E969644324Yz3Xz3Wz3Vz3V\ z3Uz3Tz3Tz3Sz3Rz3Rz3Qv3Ps2Po2Ok2Ng2Nd2Mc2Lb2L`2K_2\ JZ2JY2IW2HV2HU1GT1FS1EQ1EP1DO1CN1CL1BK1AJ1AI19H18F\ 08E07D06C06A059048047036024023012001W0NU0LS0KR0Jq4\ 5n45k45h45e35b35_35X35U35R25O25L25I25F25J79MCCQGFT\ LIWPL_UObYRfbUifXlk_pobstevxhuwitvisuirtiqsipriori\ nqimpiloiknijmiilihligjfgicfh`fgYffVedSecPdbMd`JdZ\ G_ULVSPRRUMQYHPbDQfEReESeETeEUeFUeFWeFYdF_dKadPcdU\ edZgdbidfkajm_noYqqWusUuuSuwQvzOwzMxzKyzIzzGzzFzzE\ zzDzzCzzBzzAzz9zz8zz7zz6zz54z49z7Ez9JzCOzETzHYzJbz\ MgzOlzRqzTuzVezRQzNlzVkzY } frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots 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), aa=-(real(p5)-2), bb=(imag(p5)+0.00000000000000000000001), 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)/(z^(aa)+bb)+c |z|< 1000000 } END PARAMETER FILE=========================================

1 0

FOTD 14-08-08 (Cosmic Un-Conquest [Rating Unbounded])
by Jim Muth 13 Aug '08

13 Aug '08

FOTD -- August 14, 2008 (Rating Unbounded) Fractal visionaries and enthusiasts: Since space is unbounded though not necessarily infinite, and today's image reminds me of a globular cluster in space, I decided to give today's image an unbounded rating. Regardless of what this means when applied to a fractal rating, today's image is different enough to be well worth a look. I named the image "Cosmic Un-Conquest" in sympathy with mankind, who, because he must carry a clumsy physical body with him wherever he goes, will never conquer even the solar system, much less the infinite multiverse. A temporary mind of limitless fancy trapped in a body of wet kleenex can do little more than sit here on earth, look up at the night sky and wish it were otherwise. But such is not the case with fractals, where we can venture wherever our computers and fanciful minds can take us. In today's case we journey to the land of Z^2/(Z^(-2)+11)+C, which at first appears to be a blank screen consisting totally of inside stuff. But, as the 'bof61' fill reveals, this inside stuff is active. Today's image is unusual in that it is dependent on both the screen resolution and the choice of passes. Whenever one of these is changed, a different image will result, since different dots will be chosen or passed over. Perhaps the biggest drawback in the image is its rather slow calculation time of 10 minutes. But this drawback may be turned into an advantage by visiting the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> where the image is posted for effortless viewing. Fine midsummer weather prevailed here at Fractal Central on Wednesday, with a cat-pleasing temperature of 82F 28C and sunny skies. In my department things went pretty well also. As for the next FOTD, it will be posted in 24 hours, perhaps not as cosmic as today's but interesting nevertheless. Until then, let your mind soar where your body cannot. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Cosmic_Un-Conquest { ; time=0:10:07.25-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=DivideJulibrot passes=1 float=y center-mag=-1.1400168468679/+1.308367613934373/\ 1.048206e+007 params=0/0/0/0/0/0/0/0/-2/11 maxiter=1500 inside=bof61 logmap=yes periodicity=10 colors=000Fm4Em4Dn4Cn4Bn4Bo5Ao59p58p57p56q55q54q53\ r52r52r5Mp7eq8do9dnAcmBclCbkDbjEaiFagG`fH`eI_dJ_cK\ ZbLZaMZ`NYZOYYPXXQXWRWVSWUTVTUVRVUQWUPXTOYTNZSM_SL\ `TJ_SK`RL`RMaQNaQObPPbPQbORcOScNTdMUdMUeLVeLWeKXfK\ YfJZgJ_gI`hHahHbhGciGdiFdjFejEfkEgkDhkCilCjlBkmBlm\ AmnAnn9on9oo8poAqpzzzAsqAtqAuqAvrAwrAxsAysAyu3yt3y\ s3yrUUU3yqzpc3yo3yo4yn4ym4ym4ylkhz4yj4yj4yi5yh5yg5\ yg5yz5ye5ye5yd5yczmb6yb6ya6y`6y`6y_6yZzyY7yz7yz7yz\ 7yz7yz7yz7yz7yz8yz8yz8yz8yz8yz8yz8yz8yz00000000000\ 00000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000\ 0000000000000000000000000 } frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots 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), aa=-(real(p5)-2), bb=imag(p5+0.00000000000000000001), 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)/(z^(aa)+bb)+c |z|< 1000000 } END PARAMETER FILE=========================================

1 0

FOTD 13-08-08 (Fractal in Flight [7.5])
by Jim Muth 13 Aug '08

13 Aug '08

FOTD -- August 13, 2008 (Rating 9) Fractal visionaries and enthusiasts: Today's fractal takes us into fractional powers of Z. In the image for today the exponent is 2.5. The value of 100 that I gave to imag(p5) assures that the parent fractal will be huge, and on the surface will resemble the Mandelbrot set. But deep inside the 2.5 characteristics will appear. Today's image is located on the edge of the infinitely divided main spike of the monumental parent fractal, in the area just east of the large minibrot. I named it "Fractal in Flight". I must have had a reason for choosing such a name, but the reason now totally eludes me. The rating of a 9 includes a full point for my coloring efforts. The unusual colors are the main feature of the image. There is no risk of impatience appearing when the included para- meter file is run. The calculation time of one brief minute can hardly be bettered. The image may also be seen on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> A very pleasant summer day prevailed here at Fractal Central on Tuesday, with blue skies decorated with puffy clouds, and a temperature of 81F 27C. The cats, being fractal by nature, took the conditions in fragmentary bits and pieces. I took the day as it came, finishing the work on schedule with time for a pretty good fractal. The next fractal, a cosmic one, (not a comic one), will be posted in 24 hours. Until then, take care, and when you find your reality, cling to it. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Fractal_in_Flight { ; time=0:01:03.45-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=DivideJulibrot passes=1 logmap=92 center-mag=-170.9755598748956/+0.1185530913812762/\ 1.643467e+009/1/172.5/0 params=0/0/0/0/0/0/0/0/2.5\ /100 float=y maxiter=1725 inside=0 periodicity=10 colors=000A6IA7KA8MA9OAAQABSACUADWAFYAH_HKaPMcROeW\ MhVKcUIYTGTSEORCIPADN98L9BN9EPAHREPTITWMXYQ_gS`pZa\ zaczfez`XvSPrIGm78c00W23V45U67T79S9BRBDQCFPEIOGKNI\ MMEKLFKKFKJFKMFKOFKQHKTHMVMKXLJ_LHaLFcKDfKBhJAjJ8m\ J6oI4qI2sI1wklsjjojhljfhiddibai`YiZUhXRhVNhTJhRGgP\ CgNhgLhgJhhKhhLhhchhchhchhchhchhchhcThQVhRXhSZhSLP\ a85kBBjDGiFLhHQgJVgM_fOdeQidSncUscSreQrgOqhMqjKqlI\ pzGpzEpzCozAoz8oz6nz4nz2nz6ozApzEqqIrnMskQthTtfXuc\ `v`dwYhxVlySpzPszNotQlnTihWfbZbYa_SdXMgUGjRBlUFkXI\ kZLkaOkcRkbPgaNcaL_`JW`HS_FOZDKZBGY9CY79aX4zv0zw1z\ w2zw2zw3zw4zw4zw5zw5zw6zw7zw7zw8zw9zw9zxAzyAzzIzzQ\ zzYzzezzmzztzzdzzQzzBzzHzzMzzRzzWXzaYzfZzk_zp`zubz\ tdzsfzshzrjzrkzqmzpozpqzoszouznvznpzdjzVdzMZzCUz3S\ z6Rz8zzAzzDzzFzzHzzJzzMzzOzzQFzSEzVzzXzzZzz`zzTzzL\ zzDzzCfzBdzAaz9Zz8Xz7Uz6Sz6Qz7Oz8Nz9LzAKzBIzCHzDFz\ EEzEQz``zvZzsYzpXzmVzjUzg } frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots 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), aa=-(real(p5)-2), bb=imag(p5+0.00000000000000000001), 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)/(z^(aa)+bb)+c |z|< 1000000 } END PARAMETER FILE=========================================

1 0

FOTD 12-08-08 (No More Perfection [No Rating])
by Jim Muth 12 Aug '08

12 Aug '08

FOTD -- August 12, 2008 (No Rating) Fractal visionaries and enthusiasts: NEW FORMULA ALERT! The DivideJulibrot formula included in today's parameter file is a new version that has been slightly changed from the earlier version. The earlier version will still work, but the version in today's parameter file is some- what faster. As for today's image, take it or leave it. For some reason I dislike perfection, probably because it leaves no room for improvement. And this afternoon when I took another look at yesterday's image, it struck me as being just a bit too perfect. A tiny rotation took care of the perfection however. A rotation of a mere 0.0005 degree from the Julia orientation wreaked total havoc on yesterday's image. (A slightly larger rotation makes the rectangle go away completely.) But the wreckage struck me as being good enough to serve as FOTD for August 11, especially since I had little time for a further fractal search. The color palette is the same as yesterday's, with only a slight change. Since yesterday's image was the real thing, I gave no rating to today's image, which is a parody of yesterday's. The name "No More Perfection" explains itself. The parameter file runs in 1-1/4 minutes, about the same time it takes to view the finished image on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Monday here at Fractal Central was actually a bit too chilly for perfection, and the heavy thunder-shower in mid-afternoon made it even cooler. An afternoon temperature of 63F 17C is hardly ideal for mid-August in the northern hemisphere, but the day was salvaged in the evening by a lovely double rainbow spanning the eastern sky. The fractal cats enjoyed the few intervals with brilliant sun, and endured the periods of rain. My day was rather busy, yet pleasant enough. I expect a repeat tomorrow. The next FOTD will be posted in 24 hours. Until then, take care, and the farmers must be praying for rain. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= No_More_Perfection { ; time=0:01:14.44-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=DivideJulibrot passes=1 center-mag=-0.0000104151/+0.002511651686/798281.4 params=89.9995/0/89.9999/0/-1.7435/0/0.00019/0.073\ 88/2.003/0.0003385 float=y maxiter=3500 inside=255 logmap=136 periodicity=10 colors=000EJMILNNNOSQQXTUaWYmdftiongjj`cdUW_NOUHGE\ 2DJ7BOBATG9hF6mK9mPBmUDmZFmcHmkImhJmfKmdLmbMmcNmhO\ mcPcVQcTRcRScPTcNUcLVcJWcIacHXcHScHOmHJgHEsG5mHAgI\ FaJKWKPQLUONTKMYHMbEMgBMl6Ds7Ir8Mq9Qp9UoAZnBbmCflC\ jlzauzcpzelzghziczk_zmWzoUzqSzsRzuPzwOzxNzyLzzKzzI\ zzHzzGzzLzzPzzUzzYzzbzzfzzkzzozznzznzzmzzmzzlzzlzz\ kzzkzzjzzjzzizzizzhzzhzzgzzgzzfzzfzzYzzezzezzdzzdz\ zczzczzbzzbzzazzazz`zz`zz_zz_zzZzzZzzYzzYdzXezX9zz\ BzzCzzDzzEzzFzzGzzHzzIzzJzzKzzLzzMzzNzzOzNPzNQzORz\ PSzPTzQUzQVzRWzSXzSYzTZzT_zU`zVazVbzWzzWzzzzzFzzzz\ zGzzzzzHzzzzzHzzzzzIzzzzzIzzzzzJzzzzzJzzzzzKzzzzzL\ zzzzzLzzzzzMzzzzzMzzzzzNzzzzzNzzzgzOgzzfzPfzzfzPfz\ zfzQfzzfzQfzzfzRfzzfzRfzzfzSfzzzzTzzzzzTzzzzzUzzzz\ zUzzzzzVzzzzzVzzzzzWzzzzzXzzzezXTzzUzXUzzUzXUzzVzX\ VzzVzXVzzWzXWzzWzXWzzWzXXzzXzXXzzXzXYzzYzXYzzYzXYz\ zZzXZzzZzXZzz_zX_zz_zX_zz } frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots 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), aa=-(real(p5)-2), bb=imag(p5+0.00000000000000000001), 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)/(z^(aa)+bb)+c |z|< 1000000 } END PARAMETER FILE=========================================

1 0