December 2002 - Fractint - mailman.xmission.com

FOTD 12-12-02 (Firefall [6])
by Jim Muth 12 Dec '02

12 Dec '02

FOTD -- December 12, 2002 (Rating 6) Fractal visionaries and enthusiasts: Fire and brimstone fill today's image, as an erupting fractal volcano on the left spews its fiery ejecta high into the atmosphere and over the cliff at the right edge of the frame. In the cloud of debris near the right edge lies a tiny object resembling a midget M-set. The resemblance is only superficial, for at the odd angle of today's image, no midgets are possible. The image is an odd slice of the open rectangle that appears in the C=-1.7435,0.0 Julia set of the Z^2.003+C fractal at Z coordi- nates 0.00019,0.07388. This unexpected and near perfect rectan- gle is one of my favorite fractal objects, one to which I often return. Its actual four-dimensional shape would appear to a 4-D person as a 2-D hole, with two extended dimensions and the remaining two dimensions very small, much as a sheet of paper in our 3-D world has two extended dimensions and one small dimension. Since two of its four dimensions are so small, this open area appears as a rectangle only when sliced in the Julia direction and at the exact C coordinates. The slightest rotation or imag(C) deviation totally destroys its straight edges and sur- rounding symmetry. At today's angle, the symmetry is totally destroyed. I named the image "Firefall" because, with a little imagination, it shows fire spewing from the ground and falling over a cliff. Others may see something else, but the falling fire is all I can see. The rating presented little problem. The slightly-above- average 6 seems just right. The image is a fast one, requiring only 2-1/4 minutes to render from the parameter file, on my worn-out machine. About the same time will be required to go online and download the completed GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Freezing rain, followed by heavy rain and melting snow brought flooding, fallen trees, power outages, and a roof leak to the Fractal Central area Wednesday. By evening, the high tempera- ture of 36F 2C had melted most of the ice that had built up on trees and sidewalks. Conditions were so bad outdoors that the intrepid cats were thankful to be permitted to remain indoors all day. The roof leak will require further investigation. Today is starting much improved, but I expect a lot of work before the day is finished, which might cut into my fractal time. To see how things work out, check back here in 24 hours, when the next FOTD will appear almost by magic. Until then, take care, and beware of fractal brimstone. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Ongoing_Eruption { ; time=0:02:17.75--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XZ-YW-new center-mag=-0.01885\ 870562667460/-0.00508405854824132/96.13768/0.1758/\ -180/-72.5978701069904417 params=98/-8/2.003/0/0.0\ 0019/0.07388/-1.7435/0 passes=1 float=y maxiter=2500 inside=atan periodicity=0 colors=000UGWTETWCSTCPSAORAMP8KO7JM6HK5HJ4HI4GH4GG\ 4GE4FD4FC4FA4EA4E74D64D54D44C34C24C24A44A64A84AA4A\ D5AF6AG7CH8CIA7JADKCEMDENEFOFFPFGRFFSFFTFEUFEWFDXE\ DYECZEC`EAaEAbEAcEAeE8fE8gE7iE7jE6lE6mE5nE5pE4qD4s\ D3tD3uD2wE2xG2yI2zK1zM1zO0zQ0zS0zU0zW0zY0zZ0z_2z`5\ yaAxbEwcIxdNyeOzfOzgOzgOzgOzgOzgOzhOziOzkOzmOztOzz\ OzmOz`OzOOzJOxKOuNOsMOpMOmMOjMOjMOlMNlMNlMNlKMlKMm\ KKmKKmKKmKJmKJnKJnJInJInJInJHpJHpJGpJGpJGpIFqIFqIF\ qIEqIEqIEsIDsIDsHCsHCsHCtHAtHAtHAtHAtHAtHAtIAtJCtK\ DsMEsMFsNGsOHsPHqRIqRJqSKqTMpUNpWOpWPpXRpWSnTTnRUn\ OWnMXnKXmJYmIZmH`mGalFblEclDelC0lA0jA0j80j80jA0jA0\ iCSiDViEZiFciGcgHcgIcgJcgKegMgfNffKffIffGefEefCeeA\ ceAceAccAccAcc8cc8bb8bb7bb7ba7ba6ba6aa6a`5a`5a`5aZ\ 4aZ4aZ4`Z3`Y3`Y3`Y2`X2`X2ZX2ZX2ZW2ZW1ZW1ZW1ZX2YX3Y\ X4YX6XX7XX8XXAXXCWYDWYFWYGWYHUYIUYKUYMTYNTZPTZRTZS\ SZUS`KZ`KYZJYYIYXIXXHXWGX } frm:multirot-XZ-YW-new {; Jim Muth ; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect e=exp(flip(real(p1*.01745329251994))), f=exp(flip(imag(p1*.01745329251994))), z=f*real(pixel)+p3, c=e*imag(pixel)+p4: z=z^(p2)+c, |z| <= 36 } END 20.0 PAR-FORMULA FILE==================================

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front end?
by Brian Rankin 11 Dec '02

11 Dec '02

Hi. Once I had a front end utility for inputing formula and parameter files to fractint.I have xince lost it and forget the name.Could any one please help. Brian.

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FOTD 11-12-02 (Spirals Without End [3])
by Jim Muth 11 Dec '02

11 Dec '02

FOTD -- December 11, 2002 (Rating 3) Fractal visionaries and enthusiasts: Today's fractal might show actual new material in the hypercom- plex Mandelbrot set, or it might be an artifact of mathematical imprecision. But regardless of what it is, it's an image with spirals everywhere. There are single spirals, double spirals, even signs of triple spirals. In fact, the image is filled with spirals without end. And "Spirals Without End" is what I named it. The image is a zoom into a hidden valley that appears peeking between the converging straight arms of yesterday's "Hyperbud" image, where a new world appears to be only partly revealing itself. In contrast to yesterday's image, which suffered from resolution breakdown, today's image suffers from the near impos- sibility of finding an acceptable coloring. I finally settled on a very broad purplish palette, though the result still leaves much to be desired and drops the rating to a below-average 3. While fractaling, I sometimes wonder about the meaning of it all. Is the existence of fractals filled with a deep, as-yet- undiscovered meaning, perhaps about the nature of reality, or are fractals mere candy for the eyes? And what of the concept of meaning itself? Is it something invented by the highest known form of life on the planet Earth, or is the universe filled with it? Does meaning actually exist before a self-aware entity asks the question 'why'? Does meaning exist on a two- mile-wide chunk of ice orbiting in the cold loneliness of the Oort cloud, where observers never have and never shall venture to ask 'why'? And what of this fourth dimension with its hyper-objects? We can infer exactly how the various parts would fit together if it existed, but even knowing these things, we cannot visualize some- thing as simple as a tesseract. Does this mean that the fourth dimension does not exist? Does it mean that it does exist but we cannot visualize it? Does it mean that we have invented the whole puzzle as a mental exercise? Or does it mean anything at all? Don't expect answers any time soon. And don't expect an image soon if you try to run today's para- meter file. It takes 1-1/4 hours on my less-than-supersonic P-200 machine. The best way to see the puzzling image is to download it from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Tuesday's fractal weather was cloudy and chilly here at F.C., with a high temperature of 34F 1C. The ex-dynamic duo passed the unsatisfactory day lounging by the heat, except for when they were asking for food. This morning has brought freezing rain, which is even worse. It's a good day to remain indoors and get ahead on some work. Until tomorrow, when things will hopefully be more pleasant, take care, and try to make the best of bad situations as they arise. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ SpiralsWithoutEnd { ; time=1:15:40.15--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=HyperMandelbrot passes=1 center-mag=-0\ .04192332435179621/+0.08550882276013791/4021858 params=0/0/10/0/1e-014/0 float=y maxiter=20000 inside=255 logmap=-3100 periodicity=0 colors=0009ECAECBEDCEDDEDEEDFEDGGDHGFIGFJGFKGFLGFM\ GFNGHNGHMGHMGHMGHMGHMHHMHIKHIKHIKHIKHIKHIKHKIHKIHK\ IHKIHKIHKIHKKGMKGMKGMKGMKGNKGNMGNMGNMENMEPMEPMEPNE\ PNEPNERNERNDRNDRPDTPDTPDTPDTPDTPDVPDVRBVRBVRBVRBWR\ BWRBWTBWTBWTAYTAYTAYTAYVA_VA_VA_VA_V8_V8`V8`W8`W8`\ W8`W8bW8bW8bY6bY6dY6dY6dY6dY6d_6f_6f_4f_4f_4f_4g`4\ g`4g`4g`4g`4i`4i`4ib4ib4kb4kb4kb4kb4kd4md4md4md4md\ 4md4nf4nf4nf4nf4pf4pf4pf4pg4pg4rg4rg4rg4rg4ri4ti4t\ i4ti4vi4vi4xk4xk4yk4yk4zk4zk4zm4zm4zm4zm4zm4zm4zm4\ zn4zn4zn4zn4zn4zn4zp4zp4zp4zp4zp4zp4zr4zr4zr4zr4zr\ 4zr4zr4zt4zt4zt4zt4zt4zt4zv4zv4zv4zv4zv4zv4zx4zx4z\ x4zx4zx4zx4zy4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4\ zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx\ 4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4zx4z\ x4zx4zx6zx6zx6zx6zx6zx8zx8zx8zx8zxAzxAzxAzxAzxBzxB\ zxBzxBzxBzxDzxDzxDzxDzxEz } frm:HyperMandelbrot {; periodicity must be turned off a=(p1),b=(0,0): q=sqr(a)-sqr(b)+pixel, b=(p2+2)*a*b+p3, a=q, |a|+|b| <= 100 } END 20.0 PAR-FORMULA FILE==================================

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FOTD 09-12-02 (The Nest [6])
by Jim Muth 11 Dec '02

11 Dec '02

FOTD -- December 09, 2002 (Rating 6) Fractal visionaries and enthusiasts: Well, fractal persistence paid off, and I finally found an image worthy of an above-average rating. True, it's only a slightly- above-average rating of 6, but it's still an above-average rating, and that's all that matters. I might have rated the image higher if it had been something I found rather than created with the < epsiloncross > inside fill feature of Fractint. But let's not quibble -- an above-average image is an above-average image, and I needed an above-average image. (Having repeated the phrase 'above average' a sufficient number of times, I shall mention it no more.) The expression (-Z)^(1.5)+C was iterated to draw the fractal. Iterating -Z instead of Z merely reveals a different part of the infinite complex logarithmic spiral. The 'SliceJB-new-min' formula is a slight reworking of a formula that was posted to the list several years ago by John Goering. Today's image is an oblique slice of the -Z^1.5+C Julibrot. The slice is oriented rather close to the Julia direction, which is revealed when (p1) and (p2) are set to 0.5,0.5. I named the image "The Nest" when it reminded me of some crazy bird's nest. With its effect of being illuminated from behind, the image could just as easily been named something inspirational such as "Crown of Thorns". The 'SliceJB-new-min' formula draws more rotations than any other formula in my vast and partly forgotten collection, but it does not draw every possible rotation in four dimensions. Finding a formula that does this is still one of my quests. The render time of 18 seconds is blazingly fast, but the down- load of the completed image is still available at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal weather Sunday was pleasant enough, with hazy sun, gentle breezes, and a temperature of 48F 9C. The problem was that the snow turned wet and slushy -- far too unpleasant for the cats' tender paws. They once again passed the day keeping warm and wishing they could get outside to play. This morning the snow has re-frozen and is dry, but the tempera- ture is well below freezing -- far too chilly for the intrepid ones. Having a pile of work to think about, I shall give the cold little notice -- that is unless some reason arises that I must wander outdoors. Until the next FOTD appears in 24 hours, keep warm or cool, whatever the case may be, and keep your fractals dry. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ The_Nest { ; time=0:00:18.46--SF5 on a P200 reset=2002 type=formula formulafile=julibrot.frm formulaname=SliceJB-new-min center-mag=0/0.0688935\ /0.6666667 params=0.49/0.56/0.4/0.39/1.5/0/-0.0\ 416275/0.7559499/0/0 float=y maxiter=1200 passes=1 inside=epsiloncross proximity=0.05 periodicity=10 colors=0007I00008GO7LQ6NT5QW4TZ4Wa5Zd6`g6ak7bo8cs8\ ew9hx9kyAnzBqzBtzCwzDzzDzzEzzEzzJzzOajSYiXUiaQheMh\ 000000mRj000nQk000nNm000mKnmIolGplFplDqkCrkAsk9sjE\ kiIchMXhQPgUIfYAfa3d_7bZB`XFZWIXUMVTQTRTRQXPO`NNdL\ LgJKkHIoFHrMLpSOnYSlcVjiZhoagm_dkYbjW`hUZfSWeRUcPS\ aNQ`LNZJLXHJWGHYIHZLK_NM`PO`RQaTSbVUcXWc_Zda`ecbfe\ dfgfgihhkjhmlT_aUXcVVeVTfWQhWOiXMkYJmYHnZFpZDqiu_j\ t`kt`lsamsamrbnrborcpqcqqdqpdrpesoetoftofniihdlb_o\ XVrRQuLLxONuROrUQoXRlZTiaUfdWcgX`iZYl_VoaSrbPtcMr`\ NqZOpXPoVQnSRmQSlOTkMUjJViHWhFXgDYfAZe8_d6`c4aTiUV\ jTXkTYkT_lT`lTbmTcmSenSfnShoSioSkpSlpSmlPmhMmdJzmH\ mYEnUBnR9nN6nJ3nG1hQLc_cZivHD3EQ9BaF9mLAlOAlRAlTAk\ WAkYBk`BkbBjeBjgBjjBjlAei9ag8Yd7Ub7Q_6MY5IV4ET3AQ3\ 6OK8T_9YoAbhFebKgXPjRUlLZoFbqOZiXWbeTWbUV`UVYVUWVU\ UVTRWTPWSMXSKXRIXRKYPMYNNZMPZKQ_JS_HT`GV`EWaDYaBZa\ A`YFbVJcSNeOSgLWhI_WI`JIa } frm:SliceJB-new-min {; thanx to J.R.H. Goering, July 1999 pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1), b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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)+(p4), z=r+flip(s)+(p5): z=(-z)^(p3)+c |z|<=100 } END 20.0 PAR-FORMULA FILE==================================

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Triternnions
by Tim Wegner 11 Dec '02

11 Dec '02

Does anyone know anything about these beasties? I haven't had a chance to read this in depth yet, but they appear to be 6 dimensional cousins of quaternions/hypercomplex numbers. Could probably define trig functions for these, and get some fractals out of them (there's a Mandelbrot example in the article.) The question is, would there be more to discover that is not already present with hypercomplex and quaternion slices? http://fibonacci-arrays.com/Triternions.pdf Tim

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FOTD 10-12-02 (Hypercomplex Bud [5])
by Jim Muth 10 Dec '02

10 Dec '02

FOTD -- December 10, 2002 (Rating 5) Fractal visionaries and enthusiasts: I see that the existence of triternions has been noticed. I had not seen them mentioned until Tim mentioned them in his recent letter, though I had assumed that they did exist and were less well-known mainly because they are less useful than quaternions. Three-part numbers would draw a three-dimensional Mandelbrot set and a six-dimensional Julibrot. But I wonder whether they would actually create new Mandelbrot material. The Quaternions and Hypercomplex numbers draw interesting Julia sets, but merely expand and distort the Mandelbrot set, introducing little if any actual new material. This might be due to the nature of the M-sets or it might be due to the way they are implemented in the program. I question the implementation because the Julia versions of the hyper-sets have 6 parameters, while the Mandel versions have only 2 working parameters. The quaternions draw a figure resembling the 3-D figure that would appear if the M-set were rotated around its X-axis. This figure, which basically consists of circles, can then be sliced through any of its parts, but the result is always a stretched and distorted version of the same Mandelbrot set that we know and love. The Hypercomplex M-set appears as two separate sets that overlap. Some interesting things happen where the two sets intersect, but once again, no new material is created. Perhaps the triternions would create a M-set with new material to explore, perhaps not. I rather suspect the latter, though it would be most interesting to actually try it and find out. Several years ago, bored with the sameness, I wrote a generaliza- tion of the Hypercomplex formula that does create new material. I have never been convinced that this new material is real rather than an artifact of mathematical imprecision, but it exists in a consistent way that would imply that it is indeed real. Today's image, which was created with this formula, and rather resembles a skewed gothic window with converging bars before it, has been named "Hypercomplex Bud". I gave it this name because it is a bud in the Hypercomplex Mandelbrot set unlike anything that appears in the familiar complex M-set. Not only is the image unique on the surface, but its inner details are surpris- ingly complex -- more intricate than anything that appears in the traditional M-set. I will show some of this incredible inner richness in tomorrow's FOTD. I had to rate today's image at an average 5. Its mathematical interest is above average, but its artistic merit is below. My alibi, as always, is lack of time. If I had more time, I could have found a better color palette. With a render time of 2/3 hour on my machine, the parameter file is a slow one. A far better way of viewing the scene is to download the finished GIF file from the internet and Paul's site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Monday was sunny but cold here at F.C. With snow on the ground and a temperature that never reached freezing, (it stopped at 28F -2C), the dynamic duo of fractal cats spent the day huddled by the heat. Late in the afternoon, a treat of turkey eased their distress, and consequently my distress also. As is usually the case, I find myself with several non-fractal tasks to accomplish before setting out on my fractal journey of the day. This is a certain sign that I must now get busy. So until next time in 24 hours, take care, and when you walk with fractals, don't expect much conversation. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Hypercomplex Bud { ; time=0:39:51.09--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=HyperMandelbrot passes=1 center-mag=-0.04037558685446038/+0.084551148225468\ 65/122.6667 params=0/0/10/0/1e-014/0 float=y maxiter=12000 inside=255 logmap=177 periodicity=0 colors=000fRKgSKhSKiTKjTKkUKlUKmVKnVKoWKpWKqXLrXLs\ YLtYMsXNrWNrWOqVOqVPpUPoUQoTQnTQnSRmRRmRSlQSkQTkPT\ jPUjOUiOUiNVhMVgMWgLWfLXfKXeKYdJYdJYcIZcHZbH_bG_aG\ ``F``Fa_Ea_EaZDbZCbYCcXBcXBdWAdWAeV9eU8fV9eV9eV9eV\ 9eV9dV9dV9dV9dV9dV9cVAcVAcVAcWAcWAbWAbWAbWAbWAbWAa\ WAaWBaWBaWBaWB`WB`WB`XB`XB_XB_XB_XB_XC_XCZXCZXCZXC\ ZXCZXCYXCYXCYYCYYDYYDXYDXYDXYDXYDXYDWYDWYDWYDWYDVY\ EVZEVZEVZEVZEUZEUZEUZEUZEUZETZETZFTZFTZFTZFS_FS_FS\ _FS_FS_FR_FR_GR_GR_GQ_GQ_GQ_GQ_GQ_GP`GP`GP`GP`HP`H\ O`HO`HO`HO`HO`HN`HN`HN`HN`HNaIMaIMaJMaJMaKMaKLaKLb\ LLbLLbMLbMLbMKbNKbNKcOKcOKcPJcPJcPJcQJcQJdRJdRIdRI\ dSIdSIdTIdTIeUHeUHeUHeVHeVHeWGeWGeWGfXGfXGfYGfYFfZ\ FfZFfZFg_Fg_Eg`Eg`Eg`EgaEgaEhbDhbDhcDhcDhcDhdDhdCi\ eCieCieCifCifBigBigBjhBjhBjhBjiAjiAjjAjjAjjAkk9kk9\ kl9kl9km9km9km8ln8ln8lo8lo8lo8lp7lp7mq7mq7mr7mr6mr\ 6ms6ms6nt6nt6nt5nu5nu5TVn } frm:HyperMandelbrot {; periodicity must be turned off a=(p1),b=(0,0): q=sqr(a)-sqr(b)+pixel, b=(p2+2)*a*b+p3, a=q, |a|+|b| <= 100 } END 20.0 PAR-FORMULA FILE==================================

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FOTD 08-12-02 (Arcing [4])
by Jim Muth 08 Dec '02

08 Dec '02

FOTD -- December 08, 2002 (Rating 4) Fractal visionaries and enthusiasts: I named today's image "Arcing". My excuse is that the bluish streaks surrounding the central area of activity could be interpreted as a series of electric arcs. The actual reason is that I rather enjoy using the word, which regardless of how it is spelled, never seems to be spelled correctly. The presumed correct spelling is 'arcing', but spelled this way, the word looks as if it should be pronounced 'arse-ing', which more sensitive fractalists could take to mean something un-nice. The spelling 'arking' looks as though it would be pronounced correctly, but seems unrecognizable when seen on paper. The participial form of the word 'picnic' is 'picnicking', and this word gives no trouble. Why then can not the participial form of the word 'arc' be spelled 'arcking'? . . . I guess it looks like a mis-spelling of 'racking'. This little side trip into the world of correct spelling is in no way meant to divert attention from today's image, which is rated well above yesterday's lowly 2, but still comes in at a slightly-below-average 4. The slit-like hole at the center is actually a very odd slice of a midget that appears in the east valley area of a larger midget that lies on a filament extending from the northwest shore of the main bud of the M-set. At this angle, the 2,4,8... series of elements has basically disintegrated into a disorganized chaos, though the overall pattern is still rather well defined. A glance at the < tab > screen will show that the image has been greatly stretched and skewed to undo the inherent stretching and skewing of the scene. Such stretching and skewing is an important part of examining the odd slices of the Julibrot, where features in their natural form are often stretched and twisted beyond recognition. This natural stretching is an inevitable part of the odd slices. It exists for mathematical reasons, one of which is that objects appear far larger in the Julia aspect than in the Mandelbrot aspect, and the odd slices include parts of both aspects. I will have more to say about this stretching in a future FOTD. Right now, it's time to reveal that, with a render time of 4 minutes, the image is barely worth the effort of rendering it from the parameter file. Those who choose not to render may find the pre-rendered image ready for download on Paul's site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and on Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal weather Saturday was sunny but chilly, with a temperature barely above freezing. The snow still on the ground kept the dynamic ones safely indoors by their radiators. But I could not avoid a trip outdoors to do some things that needed to be done. This morning is sunny and chilly again. I expect a similar day. Until the next FOTD appears in 24 hours, take care, and see you then. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Arcing { ; time=0:04:06.95--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XZ-YW-new passes=1 center-mag=-0.00000000004526324/-0.000000000006343\ 38/1.471077e+011/0.0132/172.255691302663337/-56.45\ 35740250172111 params=-26.03289895321512/95.611438\ 3373516/2/0/-1.256850515434251/0.3795596138199557/\ -1.256850515434251/0.3795596138199557 float=y maxiter=2500 inside=0 periodicity=10 colors=000cdOXZPPTRHNSAHTELYHObLRfOUkSXoV_tYbxWapU\ `iT_aR_VQZNOYGNY9LZDJ_GH`JFaNEbQCcTAdX8e_7eb8c`8b_\ 9`Y9_XAYVAXUBVSBUREVQGWQJXPLYPNYPQZOS_OV`NXaNZaN_d\ K`fI`iFakDanAbp8bs5cu3cw1_m3Wc5SU7OK8PWPPgePsvMrnK\ rgIq`GqUDqMBpF9p87p18l39i5Ae6Bb8C_ADWBETDEPFFMGGJI\ HFKICLJ8NK5PK2QN6OQAMSELVHJXLH_PGbTEdWCg_Bic9lg7nj\ 6oa9pTBqKErCGoDHmEHjFHhGHfGHcHHaIH_JIXKIVKITLIQMIO\ NIMNIOSPQWVS`aUegWjn`tyZqvYntXlrVipUfmTdkRaiQZgPXd\ NUbMR`UVHPSQLPZGNgCKp8Iy7Hv7Ht7Gr7Gp7Fn7Fk7Ei7Eg6E\ e6Dc6D`6CZ6CX6BV6BT17Q49R6BR8DSAFSCHTEJTGLUINULPVN\ RVPTWRVWTXXVZXX`Y_`_ZaYZaWYaUYaTXbRXbPWbOWbMWcKVcJ\ VcHUcFUdETdCTdATd9R`9QY9PV9OS9NP9MM9LI9KF9JC9I99H6\ 9G39J6CL8FOAIQCLTEOVGRYJU_LXbN_dPbgReiThmewj_lkVjk\ PilKglEfm9dp1bm4ck7chAdfDdcGdaJeZMe_HfYLeXPeWTeVWe\ U_dTcdSgdRjdQhcQgbPfbPdaPc`Ob`O`_N__NZZNXYMWYMVXFK\ `JPZMUXQYVTbTXgR_kPcpNftL } frm:multirot-XZ-YW-new {; Jim Muth ; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect e=exp(flip(real(p1*.01745329251994))), f=exp(flip(imag(p1*.01745329251994))), z=f*real(pixel)+p3, c=e*imag(pixel)+p4: z=z^(p2)+c, |z| <= 36 } END 20.0 PAR-FORMULA FILE==================================

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FOTD 07-12-02 (FMOD Modules [2])
by Jim Muth 07 Dec '02

07 Dec '02

FOTD -- December 07, 2002 (Rating 2) Fractal visionaries and enthusiasts: On some days you just can't find a good fractal. That's what I told myself as I reluctantly designated today's abysmal image as FOTD for Dec. 7. My alibi is that, though I had expected a slow day on Friday, a pile of unexpected work arrived, and after snow- removal duty, I was left with basically zero time to devote to fractals. But even a much-below-average image is better than no image at all. Today's image is a zoom into the 'New View' image of Dec. 4. To add a bit of life to an otherwise lifeless scene, I colored the image with the inside areas set to < fmod >. I named the resulting mess "FMOD Modules". The result vaguely resembles a surreal landscape, with stretched and tortured shapes hanging in the sky above the straight horizon. In the foreground, an eroded, brilliantly yellow peak juts upward from nowhere, while in the distance, mysterious features lurk. The preceding paragraph is an exercise in saying something about nothing. Speaking of the fourth dimension can also be considered saying something about nothing. But at least the fourth dimension is a real abstraction, in the sense that we know exactly how it would appear if it did exist and we had the eyes to see it. And who knows? These days, when physicists speak of 10, 11, or 26 dimensions as though they actually exist, who knows what else might exist. Perhaps, hidden in some undiscovered and undiscov- erable parallel world, a hypercubic crystal of four-dimensional salt actually exists. At least it's worth thinking about for a few seconds. With its abysmal rating of 2 and its render time of 14 minutes, the image is unlikely to be worth the effort of running the parameter file. A download of the completed image is strongly recommended. The download can be found at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Friday was a partly cloudy, breezy and cold day here at F.C. The snow cover and temperature that never got above freezing kept the intrepid duo snug indoors all day. I was kept quite busy removing snow and by an unexpected job that came in around noon and had to be finished by today. But all went well, and things are back on track. Hopefully, the next FOTD, which will appear in 24 hours, will have a bit more class. Until then, take care, and better things are to come. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ FMOD_Modules { ; time=0:14:38.04--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XZ-YW-new center-mag=-0.41302\ 339764054050/+0.00629609963586154/116.7456/0.9579 params=4.5/22/2/0/0/0/-1.25/0.03 float=y maxiter=5000 inside=fmod logmap=16 periodicity=10 colors=000000ZWFQVHRVJSVLTUNUUPVURWTTXTVYTXZSZ_S`_\ SbaUacW`eY`f__ha_jcZleZmfYmgYmhXmiXmjXmkSnlNomIpnD\ qq6tt8wwAzzBwzDtzFqzGnzIkzKizLgzNezOcz4ay6_x7Yx8Xw\ 9YwBZvC_vD`uEauF_tIZsKXrMWrOVqQTpSSoURoWPnZOm`MlbL\ ldKkfIjhHijGilLknPloTmpYnqapreqsjrtnsuuuxrtvpstmrs\ kqqhppfondnmamk_ljXkhVjgSieQhdOgbLfaJe_GdZEcXBbW9a\ U7`TLbZYcdjejwfoshloiikjfgkccl`_mYWnVSoSNqROpQOoQP\ nQPmPPlPQlPQkPQjORiORhORhOSgNSfNTeNTdMTcMUcMUbMUaL\ V`LV_LWYKV_LV`LUaMUbMTcMTdNSeNSfNRgORhOQiPQjPPkPPl\ QOmQOnQNoRNpRNqRLoPKmOJkMIiLGgKFeIEcHDaFC`EAZD9XB8\ VA7T85R74P63N42L31K23M75OC6QH8SM9TRBVWDX`EZeG`jHao\ HanHblHcjHdhHefHeeGdgFdhEdiEcjDckCcmBcnBboAbp9bq9b\ r8Yg7UX7UZ8V`8Wb8Wc8XegVTiVSlWRoZSnWRnUQmSPmPPmNOl\ LNlINkGMkELkBLj9Kj7Jj5Ji6Ki6Li6Li6Mi6Mi6Ni6Oi6Oi6P\ i6Pi6Qi6Ri6Ri6Si6Si6Ti6Ti6Ui6Vi6Vi7Wi7Wi7Xi7Yi7Yi7\ Zi7Zi7_i7_i7`i7ai7ai7bi7b } frm:multirot-XZ-YW-new {; Jim Muth ; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect e=exp(flip(real(p1*.01745329251994))), f=exp(flip(imag(p1*.01745329251994))), z=f*real(pixel)+p3, c=e*imag(pixel)+p4: z=z^(p2)+c, |z| <= 36 } END 20.0 PAR-FORMULA FILE==================================

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FOTD 06-12-02 (Passion Flower [5])
by Jim Muth 06 Dec '02

06 Dec '02

FOTD -- December 06, 2002 (Rating 5) Fractal visionaries and enthusiasts: Today's image catches one of those Mandelbrot midgets at an angle where it is most distorted. The actual midget can be seen by checking the M-set at the coordinates of (p4) of today's image. At the angle of today's slice, little is left of the hole, which once was the midget and now lies invisibly at the center of the image, or even of the basin itself, for that matter. At this angle, the basin appears to have split into two branches, though actually it is the angle of the slice that creates the split, cutting the single curved basin in two places. The surrounding areas of chaos, which somewhat resemble comets, are far better defined in the M-set. I named the picture "Passion Flower". At one time I thought I saw a flower blossom somewhere in the scene, though the bloom is now far less apparent. I rated the image at an average 5, which is all I could do for it. (The corresponding midget, which appears in the M-set, would rate a 6 or 7.) With a 3-1/2 minute render time, the parameter file is one way of viewing the scene. A more efficient way is to download the completed GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Today's image is a slice of the four-dimensional Julibrot. The Julibrot cannot be accurately considered an object, because objects are things that exist objectively in the objective world, and to our specialized senses at least, the objective world is three dimensional. I have spent many years trying to visualize a four-dimensional hyper-object, knowing all the while that my effort was futile. The greatest problem is that the surface boundary of a 4-D figure must be three-dimensional, and my imagination invariably produces flat two-dimensional bounding surfaces for all my mental figures. A 4-D hyper-object must have on its bounding outer surface as many dimensions as there are altogether in our space. On the 3-D surface of a 4-D planet, there would be six cardinal directions. Three parameters would be needed to fix a location, and two parameters would be necessary to define a direction. The horizon would not be a circle, but a sphere that totally surrounds and encloses a visitor from the third dimension, as the heavens surround the planet Earth. On such a world, the sunrise would first appear as a brightening concentrated in a particular area. Then a brilliant star-like dot would suddenly pop into existence at the brightest point, grow to the familiar blinding disk, then shrink again to a dot and vanish. At sunset the same thing would occur at another point of the field of view. At night, assuming perfect clarity of the atmosphere, the stars would appear as twinkling fireflies as they rose above and sank beneath the horizon, passing through the 3-D visitor's field of view. The view on a 4-D world would be interesting indeed -- provided, that is, that the visitor could find a way to shield his retinas from light from the side, and also a way to avoid crumpling into a wrinkled heap as a 2-D sheet of plastic wrap does in our 3-D world. It snowed at a moderate rate all day Thursday here at F.C., piling up 8in 20cm by the time it ended at 11pm. Combined with a temperature of 27F -3C, the conditions proved far too harsh for the intrepid duo, who passed the day alternating between the window and the radiator. All things considered, their moods were surprisingly cheery. Unable to get out yesterday, I passed the day getting ahead in the work. As a result, after clearing the sidewalk, I will have extra time this afternoon to devote to fractal hunting. Hope- fully, I'll find something worthwhile. Check back tomorrow to see whether I actually find it. Until then, take care, and fractals can and do make life interesting. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Passion_Flower { ; time=0:03:35.04--SF5 on a P200 reset=2002 type=formula formulafile=julibrot.frm formulaname=SliceJB-new passes=1 center-mag=0/0/2.\ 632523e+010/0.006156/-8.21933390591836854/89.48120\ 55080059565 params=0.22/0.51/0.505/0.326/2/0/-0.15\ 40762385267798/1.030994820300086/-0.15407623852677\ 98/1.030994820300086 float=y maxiter=2500 inside=0 colors=000WIFXJGYKHZLHZMI_NJ`OJ`PKaQKbRLbSMcTMdUNd\ VOeWOfXPd`XeZTfXPgVLhUHiSDjQ9mR3jP6hN9eLBcKE`IGZGJ\ XEMUDOSBRP9TN7WJ2_L6YNAWPDURHSTKQVOOXRMZVL`ZJbaHde\ FfhDhlBjo9ls7ms6kqDimKgiRkkVmmWfeXbYW_YXYYYWYYUYZR\ Y_PY_NY`Laa9ceEcgJYaNRWSLQXEKd4Gb6F`8FZAFXCEVEETGE\ SIDQKDOMDMOCKQCISCHUBFWBDYBB_A9aA7cA3dD6gA9j7Cl4Fm\ 2Mk8SiEZfKecMeaOd`PcZQbYRaXS`VT_UUZTVYRWXQXWPYWN_V\ M`ULaTJbSIcRHdQFePEfODgNBhMAiM9jPIlSQmVToYWp_Zrbas\ ecuhevjguiiuhkshlrglpfmofmmenlenjdnicogcoebpdbpbaq\ a`q_`rZ_rX_rWaoXbmYckYdiZfgZgd_hb_i``kZ`lXamUbnSbp\ QcqOcrMdtJdvHewFixDly8lxBiwEfvHcuJ_sJWqJQoJPmJOkJN\ iJMgJLeJKcJJaJJ_JIYJHWJGUJFSJEQJDOJCMJBMBDLFCKJBKN\ AJR9IV8IZ7Ha6He5Gi4Fm3Fq2Eu1Cz0Ex1Fw2Gu3Ht4Ir5Jq6K\ o7Ln8Ml9NkAOiBPhCQgDReESdFTbGUaHV_IWZJXXKYWLZUM_TN\ _RL`SNaTPbUQcVSdWTeXVfYWgYYhZZi_`j`akaclbemcfndhod\ ipekqflrgnshotiqujrvktzmw } frm:SliceJB-new {; by John R. H. Goering, July 1999 pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1), b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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)+(p4), z=r+flip(s)+(p5): z=z^(p3)+c |z|<=9 } END 20.0 PAR-FORMULA FILE==================================

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FOTD 06-12-02 (The Laugh [5])
by Jim Muth 06 Dec '02

06 Dec '02

FOTD -- December 06, 2002 (Rating 5) Fractal visionaries and enthusiasts: Today's image catches one of those Mandelbrot midgets at an angle where it is most distorted. The actual midget can be seen by checking the M-set at the coordinates of (p4) of today's image. At the angle of today's slice, little is left of the hole, which once was the midget and now lies invisibly at the center of the image, or even of the basin itself, for that matter. At this angle, the basin appears to have split into two branches, though actually it is the angle of the slice that creates the split, cutting the single curved basin in two places. The surrounding areas of chaos, which somewhat resemble comets, are far better defined in the M-set. I named the picture "Passion Flower". At one time I thought I saw a flower blossom somewhere in the scene, though the bloom is now far less apparent. I rated the image at an average 5, which is all I could do for it. (The corresponding midget, which appears in the M-set, would rate a 6 or 7.) With a 3-1/2 minute render time, the parameter file is one way of viewing the scene. A more efficient way is to download the completed GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Today's image is a slice of the four-dimensional Julibrot. The Julibrot cannot be accurately considered an object, because objects are things that exist objectively in the objective world, and to our specialized senses at least, the objective world is three dimensional. I have spent many years trying to visualize a four-dimensional hyper-object, knowing all the while that my effort was futile. The greatest problem is that the surface boundary of a 4-D figure must be three-dimensional, and my imagination invariably produces flat two-dimensional bounding surfaces for all my mental figures. A 4-D hyper-object must have on its bounding outer surface as many dimensions as there are altogether in our space. On the 3-D surface of a 4-D planet, there would be six cardinal directions. Three parameters would be needed to fix a location, and two parameters would be necessary to define a direction. The horizon would not be a circle, but a sphere that totally surrounds and encloses a visitor from the third dimension, as the heavens surround the planet Earth. On such a world, the sunrise would first appear as a brightening concentrated in a particular area. Then a brilliant star-like dot would suddenly pop into existence at the brightest point, grow to the familiar blinding disk, then shrink again to a dot and vanish. At sunset the same thing would occur at another point of the field of view. At night, assuming perfect clarity of the atmosphere, the stars would appear as twinkling fireflies as they rose above and sank beneath the horizon, passing through the 3-D visitor's field of view. The view on a 4-D world would be interesting indeed -- provided, that is, that the visitor could find a way to shield his retinas from light from the side, and also a way to avoid crumpling into a wrinkled heap as a 2-D sheet of plastic wrap does in our 3-D world. It snowed at a moderate rate all day Thursday here at F.C., piling up 8in 20cm by the time it ended at 11pm. Combined with a temperature of 27F -3C, the conditions proved far too harsh for the intrepid duo, who passed the day alternating between the window and the radiator. All things considered, their moods were surprisingly cheery. Unable to get out yesterday, I passed the day getting ahead in the work. As a result, after clearing the sidewalk, I will have extra time this afternoon to devote to fractal hunting. Hope- fully, I'll find something worthwhile. Check back tomorrow to see whether I actually find it. Until then, take care, and fractals can and do make life interesting. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Passion_Flower { ; time=0:03:35.04--SF5 on a P200 reset=2002 type=formula formulafile=julibrot.frm formulaname=SliceJB-new passes=1 center-mag=0/0/2.\ 632523e+010/0.006156/-8.21933390591836854/89.48120\ 55080059565 params=0.22/0.51/0.505/0.326/2/0/-0.15\ 40762385267798/1.030994820300086/-0.15407623852677\ 98/1.030994820300086 float=y maxiter=2500 inside=0 colors=000WIFXJGYKHZLHZMI_NJ`OJ`PKaQKbRLbSMcTMdUNd\ VOeWOfXPd`XeZTfXPgVLhUHiSDjQ9mR3jP6hN9eLBcKE`IGZGJ\ XEMUDOSBRP9TN7WJ2_L6YNAWPDURHSTKQVOOXRMZVL`ZJbaHde\ FfhDhlBjo9ls7ms6kqDimKgiRkkVmmWfeXbYW_YXYYYWYYUYZR\ Y_PY_NY`Laa9ceEcgJYaNRWSLQXEKd4Gb6F`8FZAFXCEVEETGE\ SIDQKDOMDMOCKQCISCHUBFWBDYBB_A9aA7cA3dD6gA9j7Cl4Fm\ 2Mk8SiEZfKecMeaOd`PcZQbYRaXS`VT_UUZTVYRWXQXWPYWN_V\ M`ULaTJbSIcRHdQFePEfODgNBhMAiM9jPIlSQmVToYWp_Zrbas\ ecuhevjguiiuhkshlrglpfmofmmenlenjdnicogcoebpdbpbaq\ a`q_`rZ_rX_rWaoXbmYckYdiZfgZgd_hb_i``kZ`lXamUbnSbp\ QcqOcrMdtJdvHewFixDly8lxBiwEfvHcuJ_sJWqJQoJPmJOkJN\ iJMgJLeJKcJJaJJ_JIYJHWJGUJFSJEQJDOJCMJBMBDLFCKJBKN\ AJR9IV8IZ7Ha6He5Gi4Fm3Fq2Eu1Cz0Ex1Fw2Gu3Ht4Ir5Jq6K\ o7Ln8Ml9NkAOiBPhCQgDReESdFTbGUaHV_IWZJXXKYWLZUM_TN\ _RL`SNaTPbUQcVSdWTeXVfYWgYYhZZi_`j`akaclbemcfndhod\ ipekqflrgnshotiqujrvktzmw } frm:SliceJB-new {; by John R. H. Goering, July 1999 pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1), b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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)+(p4), z=r+flip(s)+(p5): z=z^(p3)+c |z|<=9 } END 20.0 PAR-FORMULA FILE==================================

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