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Running Fractint on a Parhelia
by Burnett, David 03 Feb '03

03 Feb '03

As the title says, I have been trying to run Fractint on a matrox parhelia. The best res I can get seems to be about 300 * 400 * 256 colours. I am not expecting to be able to use all three monitors (one would do). Does anyone have any ideas how I can coax better res out of fractint. System is Athlon 1.4 / 512m with XP pro (if that's relevant). regards David

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Re: [Fractint] Triternions
by Russ Walsmith 03 Feb '03

03 Feb '03

>From: "Gerald K. Dobiasovsky" <gerald.dob(a)aon.at> >Reply-To: fractint(a)mailman.xmission.com >To: <fractint(a)mailman.xmission.com> >Subject: Re: [Fractint] Triternions >Date: Fri, 31 Jan 2003 02:01:14 +0100 > >But why shouldn't it be possible for the fractal's cross- >section for c2=real to look identical to the cross-section >for c2=imag? Gerald, Indeed, why not? There are, never the less, some strange things going on here, but Im starting to get a handle on a few of them E.g., MrT: MrT { x=real(pixel),y=imag(pixel),v=p1 x1=y1=v1=0: a=x1^2+2*y1*v1 b= v1^2+2*x1*y1 c= y1^2+2*x1*v1 x1=a+x,y1=b+y,v1=c+v z=sqrt(a^2+b^2+c^2) z < 8 } It was y1=b y*y that turned MrT into the TMan, and its obvious that squaring all of the Y-axis values creates symmetry across the X-axis. y=sqrt(imag(pixel))^2 does this with no distortion. Then flipping the y values into the x positions gives symmetry across Y, whence TMan1. The 3-space were working in here is ruled by the X,Y,V axes, but actually defined as the intersection of three planes, as is borne out in the 3D effects seen in many of the fractal images. Yes, it seems that y=imag(pixel)*(0,1), which I once perceived as removing an imaginary component, is in fact introducing a 4th degree root which is not otherwise present, and hence expanding the space to 6D (Now I see that earlier discussions regarding this make sense) So it feels like Im gradually getting my mind wrapped around this stuff Ive also come upon a general template, a pattern by which in theory all of the proper mathematical groups are uniformly configured as fractal generators The extended version of MrT is below. MrTe { x=real(pixel), y=imag(pixel), v=p1 x1=x2=y1=y2=v1=v2=0: a1=x1^2+x2^2+2*y1*v1+2*y2*v2 a2=2*x1*x2+2*y1*v2+2*y2*v1 b1=v1^2+v2^2+2*x1*y1+2*x2*y2 b2=2*x1*y2+2*x2*y1+2*v1*v2 c1=y1^2+y2^2+2*x1*v1+2*x2*v2 c2=2*x1*v2+2*x2*v1+2*y1*y2 x1=a1+x,x2=a2-x,y1=b1+y,y2=b2-y,v1=c1+v,v2=c2-v z=sqrt((a1-a2)^2+(b1-b2)^2+(c1-c2)^2) z < 8 } This inclusive pattern is not always necessary; e.g., MrT vis-à-vis MrTe: When certain symmetries are present in the group table, just the upper left quadrant suffices. This wasnt enough though for the order 8 cyclic group in the earlier CP8 form; its expressed as CP8e below. CP8e { x=real(pixel),y=imag(pixel),v=p1,w=p2 x1=x2=y1=y2=v1=v2=w1=w2=0: a1=x1^2+x2^2+2*y1*y2+2*v1*v2+2*w1*w2 a2=y1^2+y2^2+2*x1*x2+2*v1*w1+2*v2*w2 b1=v1^2+w2^2+2*x1*y1+2*x2*y2+2*v2*w1 b2=v2^2+w2^2+2*x1*y2+2*x2*y1+2*v1*w2 c1=2*x1*v1+2*x2*w2+2*y1*v2+2*y2*w1 c2=2*x1*v2+2*x2*w1+2*y1*w2+2*y2*v1 d1=2*x1*w1+2*x2*v2+2*y1*v1+2*y2*w2 d2=2*x1*w2+2*x2*v1+2*y1*w1+2*y2*v2 x1=a1+x,x2=a2-x,y1=b1+y,y2=b2-y, v1=c1+v,v2=c2-v,w1=d1+w,w2=d2-w z=sqrt((a1-a2)^2+(b1-b2)^2+(c1-c2)^2+(d1-d2)^2) z <= 16 } If a symmetrical fractal is desired, one needs to ensure that roots of equal degree rule each half of an axis, as was done here (but while evidently always necessary, this is not sufficient). C4 is a subgroup of C8, and the CP8e frm above generates the Mset. So, progress of a sort, apparently, and Ill post some more on this later meantime, is either of these MrTs configurable into one of your rotational formulas, the better for us to continue this investigation? Ciao, Russ _________________________________________________________________ Tired of spam? Get advanced junk mail protection with MSN 8. http://join.msn.com/?page=features/junkmail

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FOTD 03-02-03 (On the Inside [6])
by Jim Muth 03 Feb '03

03 Feb '03

FOTD -- February 03, 2003 (Rating 6) Fractal visionaries and enthusiasts: I'm not in much of a mood to write this morning, so I'll keep the talk to a minimum. Today's fractal is yet another of my 'evaporated' images, this one created by subtracting 4 parts of Z^(-9.3) from 7 negative parts of Z^(-1.7), then adding 1/C. I rendered the all-inside fractal with the inside set to bof61, and colored the resulting features so that the dark, gloomy foreground appears to be illuminated from behind, creating an effect rather like finding hope in a time of despair. I named the image "On the Inside" for no particular reason. The render time is 5 minutes -- not too trying. The most convenient way of seeing the image is by downloading it from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The weather was bland here at F.C. on Sunday. We found our- selves in an area known as a 'col', where nothing much happens. The fractal cats took full advantage of the partly cloudy day with no wind, a temperature of a normal 43F 6C, normal humidity, and everything else average. They spent several hours in the yard, chasing the few shafts of sunlight that filtered through the holly trees, and actually chased each other a short distance a few times. For today, I have the normal amount of work to accomplish, and then I shall likely do a lot of pondering as I search for fractals. Until tomorrow, take care, and the FOTD CD is near, that is as soon as I can pull myself out of the doldrums. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ On_the_Inside { ; time=0:05:16.37--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+0.704046611622666/+1.29535015518134/45\ 3064.1/1/-57.5/4.70417142739432137e-010 params=-7/-1.7/-4/-9.3/0/1e+100 float=y maxiter=600 inside=bof61 periodicity=10 colors=555d5Xb5Y`5Yr5Nr5Qp5TpAXoF_zpzoOdmTgmYjlbml\ gpjltjpujuuhuuhuuguuguuguuduubuu_uuYuuXupzpzSugOud\ Nu_LtVIpSGoNDlICjDAgA7e55b55`55Y55X55V55T55S55O5zp\ zVL7UK8TIASFCRDCQCDPAFO8GN7GM8FLAFKCFJDDIFDKFDNGCQ\ ICTKCWLAYLA`NAdO8eQ8jS8hT7nT7pV7rX5qY5pV5oV5nW5wb5\ vd5ue5ue5ud5ud5ub5ub5u`5u`5u_5u_5uY5uY5uX7uX7uV7uV\ 7uV7uT8uT8uS8uS8uQ8uQAuOAuOAuNAuNAuLCuLCuKCtKCtKCe\ V5Sd5Dm55u55u55u55u55u55u55r55p55m75jA5hD5eG8bKA`N\ CYODVSFTVIQYKO`LLdNIeOGhSDlTAoV8rX5uY5u`5ub5ud5ue5\ ug5uh5ue5ub5u_5uX5tT5oQ5jN7dK8_GCVFDQCGL8IG5LA5O55\ Q55T55V55Y55_55`55`55`55`55`55b55b55b55b55b55b55d5\ 5d55d55d55d55d55e55e55e55e55e55e55d55d55b55b57b5A`\ 5D`5G_5I_5L_5OY5SY5TX5XX5_X5bV5dV5gT5jT7mT7oS8rS8u\ QAuQAuQAtLDoIGhDKdAN_7QX8NVALSCKQDINFFLGDIICGKADL7\ CN58O57Q55S55T55V55X55Y55_55`55b55d55d55_55X55T55Q\ 55N5AK7FGAKDFNAIS7LX5Q`5T } frm:MandelbrotMix4 {; 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, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE==================================

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FOTD 02-02-03 (Spots Before the Eyes [5])
by Jim Muth 02 Feb '03

02 Feb '03

FOTD -- February 02, 2003 (Rating 5) Fractal visionaries and enthusiasts: I'm depressed today. As soon as I posted yesterday's FOTD, I turned on the radio and heard the bad news. This latest space disaster has me down. As the difficulties in the foundering U.S. space program continue to add up, it is becoming ever more apparent that neither I nor anyone else will live to see humanity reach the stars. It is almost as apparent to all but dreamers that humanity will *never* reach the stars. I am not shocked. I never expected to see men actually traveling to other solar systems in my lifetime, but back in the 60's, when I was young and enthusiastic, (i.e. a dreamer), I expected to see, before I left this world, men (and women) regularly traveling to Mars, and humanity actually colonizing the moon. Now I realize that I shall never see men on Mars, and unless a miracle happens, that I shall never see men even on the moon. In fact, I'll consider myself lucky if I live to see the earth- orbiting space station completed and in full operation. As I have realized for the past 15 years or so, earth's resources are limited. The planet simply has too few resources to support both an out-of-control population explosion and a workable space program. We didn't even have the resources to build the super- conducting-supercollider down here on earth. The problem lies in our very nature. We are totally dependent on the environment of earth's biosphere for our continued existence. To venture into space, we will always need to take a bit of that environment with us. And we have underestimated the magnitude of the task of maintaining that environment. We have put too much faith in the ability of technology to overcome our natural biological limitations. The only way humanity could ever reach deep space would be to find a way to change its nature. We would need to find a way to transfer our awareness, complete with senses, to some kind of artificially self-aware machines, (is artificial self-awareness an oxymoron?), and travel to the stars as robust androids with replaceable parts and unlimited life expectancies. But this is pure fantasy. Before it could be done, or even considered, we must determine what a self-awareness is and how it might be transferred. Since this appears to be impossible, perhaps even foolish to consider, the presence of humanity in deep space appears to be unachievable, and no amount of science and technology can change that. One good feature about exploring the fractal world is that it consumes very little of earth's limited resources. Today's image is part of the fractal that comes into being when three parts of Z^(-15) are subtracted from three parts of Z^(-1.5) and 1/C is added. It is another of my 'evaporated' images, this time brought to life by the bof60 inside fill. I named the image "Spots Before the Eyes". One glance will reveal why. If my mood had been brighter, I might have rated it a bit higher than the rating of an average 5. The render time of 16 minutes is trying, but may be overcome by downloading the completed GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The weather was unpleasant here at Fractal Central on Saturday. The sky was too cloudy, too many sprinkles of rain fell, the air was too damp, the wind was too raw, and the temperature of 36F 2C was too chilly. As a result, the fractal cats were too testy, and a spat broke out between them when both Thomas and Tippy wanted the same spot by the heat. But the quarrel soon ended, and the duo finished the day with a conciliatory treat of tuna. Today is starting the same as yesterday. Hopefully, the cats' mood will not be as sour. Until next FOTD, take care, and those stars keep getting farther away, don't they? Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ SpotsBeforeTheEyes { ; time=0:16:39.60--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=-1.28158038164121200/+0.099152116886171\ 08/17371.64/1/97.5/3.0669491252188763e-010 params=3/-1.5/-3/-15/0/1e+100 float=y maxiter=1200 inside=bof60 logmap=yes periodicity=250 colors=000htYfsYdrZbqZ`pXZoYXnZVmZTl_Rk_Pj`NibLh`K\ fZJdXIaVH_TGYREWPDTNCRLBPJAMH9KF7ID6GB5D94B7395161\ 27427637837A47D47F58H58J68M68O78L78N78P9AOBBPDDPEE\ PGFPHGPIHRJIQKJRLKOMJQMKQMKPMKQMKQZVS_WSaYScZSecNd\ _SdXWcT`cQdcNibJmbGrY4x`9wbDvdHufQuhTtjYslasnfrpjq\ rnptqpvroxvnzznwxmuvmsumqsmnqmlpmjnlhlleklcilagl_f\ lXdkVbkTakR_kOYkMXkKVjITjFSjDQjBOj9NjCMfFLcIK`LJXO\ JURIRUHNXGK_GHbFDeEAhD7mC1kD3jD4iD5hE6gE7fE8eE9dFA\ cFBbFCaFD`GE_GFZGGYGHXHIWHJVHKUILTIMSINRIOQJPPJQOJ\ RNJSMKTLKUKKVKMVJLWJKWIKXIJXHJYHIYHIZGHZGG_FG_FF`E\ F`EEaEEaHD`JD_LCZOCYQBXSBWVAVXAUZ9T`9Sc8Re8Qg7Pj7O\ l6Nn6Ms4Lp6Mm7Mk8NhANfBOcCOaEPZFPXGQUHQSJRPKRNLSKN\ SIOTFPT4UT9STDQTIPTMNTRLTVKTZITcGTgFTlDTpBTt6St8Tt\ ATtCTtETtGTtITtKTtMTtOTtQTtSTtUTtVTtXUtZUt`UtbUtdU\ tfUthUtjUtlUtnUtpUtrUtsUmtXgtZ`t`SzhUweVtbXr_YoX_l\ U`jRbgOceLdfMefMefMffMgfM } frm:MandelbrotMix4 {; 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, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE==================================

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A Note on Real vs Complex Fields
by MARLENE DOCTOROW 01 Feb '03

01 Feb '03

In reply to the member who seemed upset about my comments about the complex multiplication not reducing to real operations, I have to say a few things. 1. There's no room for indicating that a member is old-fashioned in my opinion. In any case, I won't waste time replying to such a person in the future after two occurrences of that behavior. 2. A positive or neutral orientation is the most easily detected part of reading somebody's internet or email postings, and I don't reply to people with negative orientations after two occurrences of that behavior. 3. The real and complex fields are NOT ISOMORPHIC, and the two- dimensional Euclidean plane regarded as a vector space R2 over the real field is not algebraically isomorphic to the complex plane over the complex field because R2 has no complex multiplication. This is not old-fashioned - it is upper division algebra. 4. The positive part of what I said in my previous posting is the part that is obscured by readers' anger: Rare Event Theory is the only way that I know of to get partially around the difficulty of item 3. Osher Doctorow

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FOTD 01-02-03 (The Cave Entrance [8])
by Jim Muth 01 Feb '03

01 Feb '03

FOTD -- February 01, 2003 (Rating 8) Fractal visionaries and enthusiasts: I tried not to see the entrance of a cave in today's image, it appeared so trite, but there it is, clear as can be, a view from inside a cave, looking out toward an unearthly scene and the alien blue sky beyond. There are even puddles of liquid nitrogen on the cave floor, reflecting bits of the scene beyond. With all this evidence, I had no choice but to name the image "The Cave Entrance". I rated the image at an 8 because of its unusually strong resemblance to an actual 3-D scene. The colors are a bit surreal, but this doesn't detract from the impression of a real scene. The puddles of nitrogen on the cave floor are actually small, foreshortened spiral areas where the iteration count of the surface features becomes greater than the count of the features beneath, and the normally hidden features show through. These areas resemble reflections because the features showing through are extensions of the features directly above. To create today's image, I once again used the C4G almost- triternion formula. While searching the East Valley area of a square Mandeloid, I noticed today's curious feature in the pattern around a sub-midget. A zoom-in, a bit of stretching, a bit of rotation and a bit of skewing produced today's image. Perhaps the most unusual aspect of the image is the criss- crossing elements that appear to go on indefinitely, almost invisibly, in the distant blue of the sky. When enlarged and re-colored, this distant sky area changes and becomes another cave entrance. We literally have a cave whose entrance goes on forever. Luckily, the parameter file runs in far less time than forever. In fact, even on my worn-out machine with the balky fan it runs in 8 minutes, and draws an image well worth the short wait. And even that short wait can be avoided if the completed GIF file of the image is downloaded from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Friday remained cold, damp and dreary all day, with the tempera- ture hovering just above freezing and occasional drizzle. The cats remained gloomy all day, and needed a treat of tuna in the evening to make them fit to live around. Today looks like another repeat. I have no idea how the duo will take three indoor days in a row. With most of the work caught up and not much else to do, I'm going to take today as easy as possible. I'll probably spend a good part of the day somewhere in Fractal Land, searching for tomorrow's FOTD. Until then, take care, and prepare for the imminent release of the long-awaited FOTD CD, which will contain every FOTD image from April 1997 through December 2002, plus every FOTD discussion and a lot of spurious blather. It will also contain a number of my earlier fractal images, some of them created (blasphemy) by programs other than Fractint. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ The_Cave_Entrance { ; time=0:08:07.30--SF5 on a P200 reset=2002 type=formula formulafile=c4gsave.frm formulaname=C4G passes=1 center-mag=-1.74974156125\ 1203/+0.00000002376090472/1.527103e+008/1/128.9/1\ 7.7763268937502694 params=-0.72/-0.53 float=y maxiter=1600 inside=0 logmap=246 periodicity=10 colors=000UJXTJZSJ`QKaPKcNKbMKaKL`JL_ILZGMYFMXDMWC\ NVANU9NTANSBLRCKQDJPEIOFHOGHOHIOIJOJJOKKOLLOMLONNQ\ NPSMRTMTVLVXLXYLY_K_aKabJcdJeeIggIhiIjjHllHnnGpoGr\ qGsrJpsLmtNkuQhuSevUcwW`xZYx`WybTzdRzkYurcpcUkukhc\ Uemobiq_esXauUYwSZrQ_mO_iM`dKa_JaWHbRFLMDNICLJBKJA\ KJAMJAOJAQKARKASLATMAUMAVNAWOAXOAYPAZPA_RA`TAaVCaW\ DbYFc_Gc`IdbJedLeeMfgNgiPgjQhlSinTioVjqWksYktZlv`m\ xamybhwXduR`tLXrFTq9UmDVjGVgJWcMX`PXYSYUVYRYZOa_Kd\ _Hg`EjaAma7pb4sb1va4t`6s_9rZBqYDpYGoXInWLmVNlUPkUS\ jTUiSWhRZgQ`fQcePedOgcNjbMlaMn`PjYSgVVcSX`P_XMbUKe\ QHgNEjJBmG8oD6lH9iLCfOFdSHaWKZZNWbQUeS_oxZnwZnvZnu\ ZmuZmtZmsZlsZlrYlqYkqYkpYkoYjoYjnYjmYjmXfjXcgW`dWY\ aVj_ViXVOUULRUIOTEMTBJS8GS5DS2BTQRTOLTMGVOIXPKZRM`\ SNbTPdVRfWThXUjZWl_Yna_pb`rcbtedvffwggtcaq_XnWRlTM\ iPHfLBcH6aE1bG2bH3bI3cJ4cL5cM5dN6dO7dQ7eR8eS9eT9fV\ AfWBfXBgYCg_Dg`DhaEhbFhcF } frm:C4G {; p1 is (plus or minus) 1 or i x=real(pixel), y=imag(pixel)*p1 a=b=0: a1 = a^2-p1^2*b^2 b1 = 2*a*b a=a1+x, b=b1+y z = sqrt(a^2 + b^2) z < 4 } END 20.0 PAR-FORMULA FILE==================================

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Re: Mandelbrot and Julia Sets Explored by RareEvent Theory
by MARLENE DOCTOROW 01 Feb '03

01 Feb '03

>From Osher Doctorow Morgan and Russell, Thanks for some interesting ideas. We may be coming from at least partly opposite directions, but that's fine. I'm 64 years old and missed the computer boat so to speak, although I can read many programs (somewhat like saying somebody can read some French but it isn't their native language or even a specialty or hobby). Curiously enough, I went into the super-abstract mathematics direction outside computers and only fairly recently have been coming back to the more concrete reality and some of it involves computers. One of the difficulties in reducing computer programs to their real and imaginary components also occurs in non-computer mathematics and quantitative sciences, where one comes up against the fact that MULTIPLICATION of complex numbers doesn't decompose (doesn't correspond to any operation in the reals for Re(z) and Im(z)) according to any known theory accepted by the mainstream. So there are no difficulties as long as one just adds and subtracts complex numbers - then there's a pretty clear correspondence to the real numbers and one could use either a complex or two real programs or a combination of two real programs. When multiplying two complex numbers, if you did manage to somehow incorporate complex multiplication into real programs you'd really be extending the real programs to complex programs because it's a known fact that complex multiplication is NOT APPLICABLE TO THE REALS EXCEPT WHEN THE IMAGINARY PARTS OF BOTH NUMBERS MULTIPLIED ARE 0. So what is surprising about Rare Event Theory (RET) insofar as it relates to complex numbers is that it seems to be able to arrive at similar results to complex numbers INCLUDING complex multiplication - but not identical results in general. In other words, RET has potential for exploring things and arriving at results that may be too complicated at present with complex multiplication - but the results may for example have an inequality in the wrong direction or something like that. RET also is useful to establish CAUSES and EXPLANATIONS which may not be so clear using complex multiplication or even complex numbers, because RET is a literally causal language (one of its forms or representations is in fuzzy multivalued logics in fact, where "a implies b" literally corresponds to 1 + b - a, which you will recognize as 1 + y - x or p1(x, y) with a, b being fuzzy multivalued real number values which are attached to the propositions. The probability- statistics meaning or translation or representation of 1 + y - x is literally: probability that x causes y. Here y < = x (is less than or equal to x). The number 1 + y - x is always between 0 and 1 when x and y are betwen 0 and 1. Thanks also to Guy for starting the comments, and let me know on fractint if you see any further ways to get some of this into computer program form or to even generalize or modify or specialize anything. Osher Doctorow

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One Typo
by MARLENE DOCTOROW 31 Jan '03

31 Jan '03

>From Osher Doctorow In equation (1) of the last posting, the second = should be --> where the arrow indicates converting to p1 representation. Osher Doctorow

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Exponential Chaotic Explosion by Rare Event Theory
by MARLENE DOCTOROW 31 Jan '03

31 Jan '03

>From Osher Doctorow Robert L. Devaney in "A survey of exponential dynamics," 2001 or 2002, http://math.bu.edu/people/bob/paper/augsburg.pdf as well as in various publications including a book has examined the exponential function analog of polynomial Julia/Mandelbrot sets which has an explosion when (c1 + ic2)exp(z) has c2 = 0 and c1 > 1/e. Here z = x + iy. According to Devaney, the appearance of so-called indecomposable continua in the chaotic Julia set for c1 > 1/e (in my notation) accounts for the explosion. In the explosion, the Julia set jumps from being nowhere dense to becoming the whole plane. The Rare Event Theory analysis is quite simple and proceeds as follows. 1) c exp(z) = (c1 + ic2)(x + iy) = c1 (cos u - sin u) = p1(c1 sin u - 1, c1 cos u) = 1 where c2 = 0 and x = cos(u), y = sin(u) because exp(z) = cos(theta) + isin(theta) for angle theta in the usual complex representation and we set u = theta. The equating to 1 optimizes p1. Setting the middle expression of (1) equal to 1 results in the following. 2) c1 cos(u) - c1 sin(u) = 1 and therefore: 3) c1(cos(u) - sin(u) = 1 But dividing both sides of (3) by c1 not equal to 0 results in: 4) cos(u) - sin(u) = 1/c1 This is impossible for u between 0 and pi/2 radians and for c1 = 1/e since both cosine and sine functions are between 0 and 1 there. If 0 < c1 < 1/e, then 1/c1 > 1/e and again (4) is impossible. The only place where (4) is not contradictory is where c1 > = 1 which is most of the interval between 1/e and infinity except for (1/e, 1). The fact that the chaotic and non-chaotic regions have reversed in the Rare Event Theory representation for u between 0 and pi/2 is probably not that dangerous because an entire quarter of the complex plane, where u is between pi/2 and pi, is excluded since cosine is negative there and sine is positive there and thus (4) would be contradictory. The pathology is in the function itself c exp(z) more than in its domain or range. Osher Doctorow

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FOTD 31-01-03 (UnNamed Fractal-V [7])
by Jim Muth 31 Jan '03

31 Jan '03

FOTD -- January 31, 2003 (Rating 7) Fractal visionaries and enthusiasts: It looks like it's going to be a good day here at F.C. My luck appears to have finally turned for the better. A few minutes ago I received a phone call. Expecting a customer, I answered the phone on the second ring. But instead of a customer, I heard a near-hysterical voice informing me that I had just won a free three-day trip for two to Orlando in Florida. Unfortunately, I am too busy to take a trip at this time, so I told the voice to simply send the cash equivalent of a three-day trip to Orlando. I am now eagerly waiting for the money to arrive, but it certainly would ease my mind if I could remember entering a contest where the prize was a free trip to Orlando. My fractal luck also appears to be on the upswing. The latest formulas in the undefined triternion series were posted to the Fractint list on Thursday. Instead of complex numbers, these formulas use real numbers, but the result is the same. While noodling around with one of the formulas, I stumbled upon today's image. I could have given the image many names. I see feathered wings quite clearly. But to keep a good thing going, I decided to end the month with another nameless fractal. (Who said fractals must be named?) Since it is the fifth such image, I attached the identifier "UnNamed Fractal-V". But, as a glance at the image will show, the lack of a name does not imply a lack of quality. The parent fractal resembles a square Mandelbrot set. And since parents give birth to similar offspring, the midgets tend to be square also. The squared-off midget at the center of today's image lies in the East Valley of the large midget on the main stem of its parent. Because of the unusually vividly defined square area surrounding the mis-shapen midget at the center, today's fractal effort rates a 7. The render time of only 2 minutes is another good feature of the image. As always, the image may also be seen by downloading it from one of the FOTD web sites: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal weather Thursday was cloudy with occasional flurries of snow and a temperature that hung around freezing. The fractal cats spent the day by the heat, occasionally demanding food. Their mood was barely acceptable. Today's weather promises to be a repeat, so the cats' attitude will likely be a repeat also. My mood will improve markedly when I finish the modicum of work now waiting to be done, and start searching for the next FOTD. Until tomorrow, take care, and is infinity an odd or even number? Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Un-Named_Fractal-V { ; time=0:02:30.99--SF5 on a P200 reset=2002 type=formula formulafile=c4gsave.frm formulaname=C4G passes=1 center-mag=-1.74967514907\ 1832/0/1.409911e+007 params=0.5/1 float=y maxiter=1500 logmap=151 inside=0 symmetry=xaxis periodicity=10 colors=000IHMKBQM6SO5UQ4WS3YU3ZW2`Z1b`0db0ee1gg2hi\ 3jk3lm4mo5oqAqsAsuAuwAwzAz6cU5bR4bO8BV6`j5yzwMIuPG\ tSErVCqYBo`9nc7me6y_ZjjRWtJSnMOhPKcSGYVCTY8N`5IcEF\ bNDaVA`c8_k6ZlDblJemPhmWlnaongr`f5Tb8MZBFWE8SH1PJ6\ UOBXTGaYKdaPhfUlkZppbrrYomUkgQg`MaVHYODUI9QB5N54I4\ 3D3292141KA0KAGD88tzPpuNlqLhmKdiIaeGY`FUXDQTBPS2SR\ 8VRDXRJ_QOaQTdQZfQciPhkPnnPspPxnOulNskNpiMnhLkfLie\ KfcJdaJa`I_ZHXYHVWGSVFQTFNRELQDIODGNCDLBBKB9MACNAE\ OAGQAIRAKSAMUAPVARWATYAVZAX_AZa9ab9cc9ee9gf9ig9ki9\ nj9pk9rm9tn9vo9xjHjePX`XJYUHVRGSPEPMDMKBJHAHF8EC71\ 12BA5874552221dlfaic_gaYd_WbYU_WSYUQVROTPMQNKOLIMJ\ GJHEHFCECACA898676444222_72Y61W61V61T51S51Q51P41N4\ 1L41K31I31H30F30E20C20A20910710610400300100YH6WG5U\ F5WE5YD4_C4bB4eA3h93k83nC2qF2tI2wK1zM1zO1zQ0zS0zUT\ zWQzYOz_MzaKzcIzeGzgEziCzkAzm8zm6zm4zm2zm0zm0zm0zm\ 0zm0zm0zm0zm0zm0zm0zm0zm0 } frm:C4G {; p1 is (plus or minus) 1 or i x=real(pixel), y=imag(pixel)*p1 a=b=0: a1 = a^2-p1^2*b^2 b1 = 2*a*b a=a1+x, b=b1+y z = sqrt(a^2 + b^2) z < 4 } END 20.0 PAR-FORMULA FILE==================================

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