August 2003 - Fractint - mailman.xmission.com

The "bug "is not a bug!!! Is not! Is not!
by Tim Wegner 26 Sep '03

26 Sep '03

OK folks, listen up. :-) For you Fractint is a fractal program, for me it is the World's Best Quick And Dirty Reference to Complex Transcendental Functions. Press F1, go to "Fractals and the PC", and navigate down to trig identities. I put these in the help because I went through a LOT of books finding these and wanted to collect them all in one place in order to code them up. Some books are too simple and so don't have the formulas, and other more advanced books are too abstract and so don't have them either. The formula for the complex power is as follows: z^w = e^(w*log(z)) (here"^" is ASCII text shorthand for exponentiation, in this case z^w means "raise the complex number z to the power of the complex number w") where: e^(x+iy) = (e^x) (cos(y) + i sin(y)) and log(x+iy) = (1/2)log(x^2 + y^2) + i(atan(y/x) + 2kPi) for k = 0, -1, 1, -2, 2, ... (The log function refers to log base e, or ln. The expression atan(y/x) is an angle between -pi and pi in the quadrant containing (x,y) implemented in C as the atan2() function.) The point of these formulae from a programmer's perspective is that they define the complex valued power function in terms of the real versions of e^x, log(x), sin(x), and cos(x) that are available to the C programmer. The interesting and important thing here is that the value of the complex log is not unique. There are infinitely many different values, with the imaginary members. Put another way: the complex log is not single valued. Let's look at the log. In the "bug" illustration, the complex argument is: (-5.0, 0.0) The assembler log function gives: (1.6094379124341,-3.141592653589793) Coding by hand in C gives: (1.6094379124341, 3.141592653589793) BOTH ANSWERS ARE RIGHT!!! The imaginary parts differ by 2*PI, which is acceptable. What actually causes this is that the log implementation normalizes the result to a particular 2*PI segment. Jim's fatal mistake was when he assumed that given a = 1, b = -5, g = 1, h = -1, j = 1/6, that (-a*b*g*h) is -5. Yes, it's very close to -5 in the actual computer calculation, but "close" wins only in horseshoes. The problem is that the complex logarithm of (-5,0) has an imaginary part exactly on the logarithm software implementation's boundary of 2*PI normalization (namely PI). So if the calculation (-a*b*g*h) drops a bit here or adds a bit there, even a bit of the lowest significance, it can change the imaginary component from ~ PI to ~ -PI. I leave it to the reader to plug both these logarithm values into the rest of the formula for the exponential power, and see that the result is (1.132467200411351, -0.653830243005915) in the one case and (1.132467200411351,+0.653830243005915) in the other, as Jim discovered. Here's the main point - a recurring theme in trying to create fractals with complex transcendental functions. The complex power function x^y is discontinuous because of the ambiguity in the many- valued complex log. Each decision to normalize the log to a particular 2*PI interval in an effort to define a single-valued function creates a discontinuous tear in the result in a different way. The "bug" Jim found had values right on the boundary of the tear, so that tiny perturbations and rearrangements of the code pushed the calculation over the edge of the discontinuity. Fractint is cleared of the unjust accusation! The verdict: INNOCENT! All is well with the fractal world :-) One single absolute truth of the universe is thus upheld again. It is this: There are NO software bugs, ONLY USER ERRORS!!! (Sorry Jim ... -no, not *really* sorry, I lied :-) I suggest the future Fractal of the Day explore a fractal based on a discontinuous complex transcendental function in such a way that the fractal itself has visually manifest tears. There are several examples in Fractal Creations, and I am sure many of the artists here, including the honorable Mr. Muth, know of or can discover examples. Tim Wegner

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Re: Virii, Worms, and other Seeds of Evolutionary CHAOS
by JackOTradez 01 Sep '03

01 Sep '03

Folks - lose the blame game attitude and stay friends! I do not have anti-virus on any of my systems, and I have never been hit. (Cause I'm smart with these things.) However, my Parents' PC (where I have been visiting for the past week) DID get slammed by 2 of these beasties. I spent the better part of a day cleaning out the system, repairing and restoring it. This was particularly infuriating to me cause I was using their system to access my email while away. DOZENS of my emails were infected, but Mr. Norton caught them. They apparently opened an infected email from one of their friends / family and unknowingly cut the beasties lose on another system. (not Norton protected). Not everyone is a Cyber-Wiz Dude. System and Server Administrators can not be expected to be 100% foolproof. Having some experience there, I know how tough it is just to keep things running smooth when all is ok! Give 'em a break! Flies and rats do not cause garbage, guns do not cause crime, spoons did not make Rosie fat, and System Administrators do not cause spam and email virii! The fault is 100% with the authors of these toxic codes, and the email spammers. WHEN will enough be enough!? This is costing billions of dollars to an economy struggling to recover, and is especislly irritating to unemployed like me trying to find a constructive job! What needs to be done is this - hunt down and find one of these bastards, and nail him hard! It may be a teenage punk dork with pierced tounge, earrings, tatoos, clothing 10 sizes too big, who can't find a date, but so what! Give him 20 to life and let him get deliverance in the slammer! This is criminal trerrorism and should be dealt with as such. How 'bout a Federal task force, like the "Untouchables" during prohibition? Computer Gurus with an Axe in one hand and an automatic weapon in the other - Raid the spammers' and tear them up! Raid the home of the virus author, smash it all up, drag him away in chains never to see the light of day again! Confiscate all material and financial assets to the State. (Hey Echelon - are you listening? I'm willing to volunteer for this ...)

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FOTD 31-08-03 (The Tiny Crowd [5])
by Jim Muth 31 Aug '03

31 Aug '03

FOTD -- August 31, 2003 (Rating 5) Fractal visionaries and enthusiasts: Take 5/1000 part of Z^1000 and subtract Z from it before adding C, and you get an empty circle with a circumference that resembles a fine-tooth saw. A very tiny valley exists where the X-axis intersects the positive side of this saw-like border. Today's scene is located in a spiral that lies on the southern shore of this valley. I have named the image "Fractal Fern-Forest". One of the earlier attempts at coloring gave the impression of a forest floor covered with ferns. I did not save the color map because I wanted something better, but nothing better ever turned up, and I am stuck with today's inferior color palette. Only the name remains to indicate what might have been. And the rating of an average 5 reminds me to save color palettes that appear all right but easy to improve upon. All the detail in the image is natural high-iteration stuff. The image is a totally traditional one, with a blank black inside and an outside of equal-iteration bands. When it comes to fractals, I am a traditionalist. When it comes to the philosophy of existence, I am a non-tradi- tionalist, but we will not go into that aspect at this time. It is time to post today's FOTD, which renders in 16 minutes. The image can be viewed in far less time by downloading it from the web sites of Paul or Scott at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Other than being rather warm, with a temperature of 93F 34C, Saturday was a near perfect day here at Fractal Central. The weather experts were warning of rain all day, but their warnings were another false alarm. No rain fell to spoil the cats' sunny afternoon in the yard. When evening came, a cool north breeze arose to comfortably end one of the best days the fractal duo has had in several weeks. And I saved a can of tuna. Today once again brings a forecast of rain. It is starting partly cloudy, but after all the empty warnings of yesterday, I feel that a dry and comfortably cool day can be expected. As is apparent, I have little confidence in the ability of the weather forecasters to accurately forecast weather. The chance of the next FOTD appearing in 24 hours is 99.99 percent. The chance it will rate higher than a 5 is 55 percent. The chance I will have a good day is 80 percent. The chance that this is the end of today's FOTD is 100 percent. Until next time, take care, and if I could turn on precognition at will, I would be the world's best forecaster. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Fractal-FernForest { ; time=0:16:29.48--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=ident passes=1 center-mag=+1.99617324961325700/-0.000000334263081\ 95/8.949466e+009/1/27.4998800176821874/3.186600636\ 06257538e-005 params=0.005/1000/-1/1/0/0 float=y maxiter=2000 inside=0 logmap=227 periodicity=10 colors=000MIcMJdMKeNMfNNgNPhNQiORjOTjOUkPWlPXmPYnQ\ _oQ_pQ`pR`qR`rRasSatSbuSbvSbvRatR`rQ_pQZnPYlPXjOWh\ OVfOUeNTcNSaMR_MQYLPWLOUKNSKMQKLPJKNJJLIIJIHHHGFHF\ DOEBRD9WD8cM9gU9ja9mj9rr9wz9zwCzuEwrHrpJmmMhkOchQ_\ fTUcVTaYRZ_PXbNUdLRgMSfMSfMSfMSfMTfMTeMTeMTeMUeMUe\ NUeNUdNUdNVdNVdNVdNVdNWcNWcNWcNWcOXcOXbOXbOXbOXbOY\ bOYbOYaOYaOZaPZaPZaPZaP_`P_`P_`P_`P_`P`_P`_P`_Q`_Q\ a_Qa_QaZQaZQbZQbZQbZQbZQbYRcYRcYRcYRcYRdXRdXRdXRdX\ ReXReXReWSeWSeWUfWZfWciWhnVmvVruVrqVmkVhkWciVZhVUf\ VSdVScVSbVS`VS_VS_VS_VR`VRaVRaVRbVRcVRcURbURbURbUR\ aUQaUQaUQ`UQ`UQ`UQ_UQ_UQ_UQZUQZUPZTPZTPYTPYTPYTPXT\ PXTPXTPWTPWTOWTOVTOVTOVTOUTOUTOUUOUVOTWOTXNTYNSZNS\ _NS`NRaNRbNRcNQdNQeNQfMRgNQhOQiPPjQPkROlSOmTNnTNoU\ MpVMqWNrXOsYPtZQuZRv_Sw`TxaUybVzcWzdXzdYzcZzb_za`z\ aaz`bz_czZczZczYczXczWczWczVczUczTczTczSczRczQczQc\ zPczOczNczNczMczLczLczLcz } 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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Re: [Fractint] Spelling
by p.hentges＠gmx.net 30 Aug '03

30 Aug '03

Virii is not correct English - according to the Oxford Advanced Learner's Dictionary the plural of virus is simply "viruses". pierre --On 29 August 2003 17:58 -0600 Tim Wegner <twegner(a)swbell.net> wrote: > Paddy wrote: > >> Virii >> Sorry Tim I'm English. > > English??? If you were Roman I would understand! > > Hmm - shouldn't the Latin plural of virus be viri? > > The list administrator is going to get me for being off topic ... > > Tim > -- COMPUTERBILD 15/03: Premium-e-mail-Dienste im Test -------------------------------------------------- 1. GMX TopMail - Platz 1 und Testsieger! 2. GMX ProMail - Platz 2 und Preis-QualitÃ¤tssieger! 3. Arcor - 4. web.de - 5. T-Online - 6. freenet.de - 7. daybyday - 8. e-Post

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Re: Fractint Digest, bounce messages
by Jim Muth 30 Aug '03

30 Aug '03

At 11:43 PM 8/24/03 -0700, Kevin Sexton wrote: >Before you went off on it it might have been better to politely >let the admin know what happened, Yes, this would have been a better response. >maybe it was just a one time thing, email and listservers >aren't infallible. Yes, it could have been a one-time thing. >Sometimes a mail to you bounces through no fault of your own >but an intermediate server problem. It's possible that over the >past 6 years enough emails bounced that they tripped the 'too >many bounces' counter. Not possible. I only lurked on the list. During the 6 years I lurked, I posted maybe 2 or 3 messages to the list >I would have just tried resubscribing and maybe send a mail to >the admin letting them know there was a problem. On most days, I would have done the same. >Mailing lists are set up to automatically bounce bad addresses, >it doesn't matter if it's a large or small list. Yes, it works best this way. >Most unsubscribes are really due to an address no longer >existing. These addresses should be purged from the list. >There will be on any list that has been around a while bounces >that do not result in an automatic unsubscribe. These take the >administrators time to sort through the headers, and figure out >who bounced (it really isn't always easy, the original address >may not be anywhere in the bounce.) Being a list administrator does take some effort. >I manage a small list, around 10 people, I got about 40 >messages in the last year and several bounces per message, and >when I do get a bounce message, I may have to send test >messages to figure out who has a bad address. Then you know the work involved in being the administrator of a mailing list. >It's even possible that someone else subscribed with a bad >mindspring address and the server matched the headers to your >address. Yes, this is a possibility. >Another possibility is someone subscribed with a forwarding >address like you can get at bigfoot.com, and forwarded to >mindspring. If they cancel mindspring, the bounces come from >mindspring, with no clue to it being another address. Yes, this is another possibility. I'm sure there is a simple, easily explained reason why I was unsubscribed. I didn't blow my top because there was no reason; I blew my top because it happened. On most days I would have shrugged it off and resubscribed. It was the line in the notice which said that, before resubscribing, I should solve *my* problem, that set me off. I found the notice just as I finished a 6-hour argument with a customer who insisted that the errors in his job were caused by my incompetence rather than by his inability to accurately communicate instructions by words or fax. I'll settle down in a day or two. Jim M.

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FOTD 30-08-03 (The Tiny Crowd [5])
by Jim Muth 30 Aug '03

30 Aug '03

FOTD -- August 30, 2003 (Rating 5) Fractal visionaries and enthusiasts: When I opened my e-mail box this morning, I noticed a message with some real fractal substance in it from Osher Doctorow on the Fractint list. The message speculates that perhaps the proper application of fractal geometry could clarify some of the seemingly insoluable puzzles of modern physics. I can't quite grasp how fractals could clarify things such as quantum theory, but I have long watched and wondered how such a revolutionary discovery as fractals could find so little practical application and have so little impact on our under- standing of things. It will be interesting to see where, if anywhere, Osher's ideas go. I am still browsing that old physics text book of mine, which was published in the 1880's. At the time, there were many puzzles which have since been explained. To begin, we were seen living on a world hundreds of millions of years old, illuminated by a sun that could have been burning for no more than ten- thousand years. Then there was the problem of the Ether, which everyone knew existed because waves need a medium in which to propagate, but which was impossible to detect and needed to possess some impossible qualities. Finally, there was a very puzzling situation with thermal radiation, which according to classical physics should have been in the form of extreme ultra- violet radiation. These problems were not solved until we went beyond classical physics. The discovery of nuclear energy solved the puzzle of the generation of the sun's heat. The theory of relativity made the Luminiferous Ether unnecessary, and quantum theory explained why hot coals glow orange-red instead of emitting gamma rays. There are puzzles equally as perplexing in the world of physics today. Could fractals point the direction to the solution of these puzzles? Will fractals solve the puzzles, or better yet, make them irrelevant? And 120 years from now, will the idea of quarks and superstrings be considered as quaint as the idea of Luminiferous Ether seems today? Today's fractal is another scene in the Z^3+C Mandeloid. It is located deep in the West Valley, which is a mirror of the East Valley. For a change, today's image was rendered with the out- side set to the 'fmod' option. I named the image "The Tiny Crowd" because it is filled with many cubic midgets. The rating is an average 5. The 19-minute render time is slow, but speed may be found by downloading the finished GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Friday here at Fractal Central was very warm and muggy, with a temperature of 90F 32C and a dew point of 77F 25C. A brief shower in the afternoon cut short the cats' day in the yard almost as soon as it started, leaving them sulky and making a consolation treat of tuna and turkey necessary. They ended the day in resignation. Today is starting warm and muggy again, but there is more sun than yesterday at this time. Whether this bodes good or evil will not be determined until the day has ended. In any case, I shall return in 24 hours with another fractal, this one not in the cubic M-set. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ The_Tiny_Crowd { ; time=0:19:32.77--SF5 on a P200 reset=2002 type=formula formulafile=slices.frm formulaname=MandelbrotCube passes=1 center-mag=-0.\ 3875828936411013/-0.00020565676691499/4.793506e+009 params=0/0/0/0 float=y maxiter=7200 inside=0 proximity=0.015 outside=fmod periodicity=10 colors=0I0zzzzzzzzyzzyzzyzzwzzwzzwyzwyzuwzuwzuuzsu\ zsszsszsqzqqzqoyqoyomwomuoksoksmiqmiomfokflkdjkdhk\ bhibei`di`difefkhdojbsl`woZzqXzsVzuSzwQzzOzzQzyQzw\ SzuSzsVyqVuqXqoXmlZijZbhZZe`Vd`QdbMabE_dAXd4Vf0Tf0\ Tf0Vd0Xd2Xb4_b4_b7a`Ad`CdZCeZEeZHhXKhXKjXMlVOlVQoS\ SoSSqSVqQXsQZuOZuO`wObwMdyMbwHdyMdzOdzQfzVfzXfzZiz\ `izdizfkzikzkkzomzqmzsmzuozyozzozzqzzqzzqzzszzszzs\ zzuzzuzzuzzwzzwzzwzzwzzyzzzzzzzzzzzzzzzzzzzzzzzzzz\ zzzzzzzzwzzuzzqzzozzkzzizzdzzbzzZzzXzzSzzQzzMzzKzz\ EzzCzy7zw4zu0zs0zq0zo0zl0zo0zl0zl0zl0zj0zj0zj0zh0z\ h2zh4ze7yeAueCqdEmdHkeEidKfaMd_Od_QbXS`VV`VXZTZXQ`\ VNdVNfSLiQIkQImOIoMIqKIsKIuHIyEIzEIzCIzAIz7Iz7Iz4I\ z2Iz2Iz7IzAIzEIzHIzKIzOIzQIzVIzXIzZIzbLzdNziQzkTzm\ VzqXzs_zwazydzzezzhyzjyzlwzowztzztzzrwzqsznqzjmzei\ zcfzabzXZzXXyTSwROuPKuIHsICqI7oI4mI0kI0kI0mI0oI0qI\ 0sL0uQ0wT0yX0za0ze0zj0zo0 } frm:MandelbrotCube {; Jim Muth real(c),imag(c) z=p1, c=pixel+p2: z=z*(sqr(z))+c, |z| <= 16 } END 20.0 PAR-FORMULA FILE==================================

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Spelling
by Paddy Duncan 30 Aug '03

30 Aug '03

Virii Sorry Tim I'm English. Paddy Duncan

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Brief Discussion of Fractals on geometry-research
by mdoctorow＠comcast.net 29 Aug '03

29 Aug '03

>From Osher Doctorow Ph.D. I have been developing an expanded theory called Geometric Physics which may lead to a TOE (Theory of Everything). I discussed fractals in connection with this on geometry-research(a)forum.swarthmore.edu (which can also be reached by accessing http://www.mathforum.org) on 28 Aug. 03 at 19:35:07. The good news is that Geometric Physics has a good possibility of moving Fractals to center stage in mathematics, physics, engineering, and Life Sciences (biology, botany, etc.). The bad news is that I don't have the influence to do it. I am a college mathematics teacher/researcher at one or more of California State Universities and Community Colleges, which are way underfunded compared to Research Universities like UCLA or Berkeley, which in turn are underfunded. Hopefully, some members of FractInt will be interested in Geometric Physics and have more influence than me. Geometric Physics is motivated by the paradoxes, anomalies, infinities, and in fact the general lack of interdisciplinary work and open-mindedness to new ideas in physics, engineering, and to a lesser but still real degree mathematics. Basically, I decided to start physics from almost scratch. The one thing I did was concentrate on ideas related to Cosmology and Astrophysics, where the latter is probably the most tolerant of new ideas of any science. Geometric Physics defines the Universe as containing 11 Fundamental Elements: Expansion (E), Contraction (C), Rare or Rarity as in Rare Events/Processes (R), Non-Rarity (N), Knowledge as Semantics or Meaning (K), Information as Syntactics or Form of Messages (I), transmission (T), inside (i), outside (o), potential (p), actual (a). The 11 Fundamental Elements form a language via concatenation, and they can be taken as elements of Fuzzy Multivalued Lukaciewicz or Rational Pavelka Logic with logical implication (x-->y) = 1 + y - x where x, y are elements of the language in quantified form, that is to say with truth values not just 0 or 1 for true or false respectively but any real numbers between 0 and 1 inclusive, and where 0 < = y < = x < = 1 for the non-trivial cases. Michael Barnsley's contraction maps can be used to generate fractals by iterated function systems. By allowing both Expansion and Contraction, including to different degrees or distances and with different speeds in different directions at different times, one can also generate arbitrary curvature of the Universe and hence geometry and General Relativity. Quantum Theory turns out to be contractive and much less restricted than in the Mainstream. Statistical Thermodynamics including in black holes is recovered by laws of both increasing Information and increasing Knowledge with time. Temperature is derived from speeds and velocities averaged in specific circumstances. Not everybody likes the results. I have posted them recently to Researchmathematics(a)yahoogroups.com, http://www.superstringtheory.com/forum, and a few other forums. Loop Theory of Lee Smolin and Abhay Ashtekar and Carlo Rovelli is not in the "cards" with Geometric Physics, but superstringtheory is including M Theory and brane theory and duality theory. General relativity is in the cards, possibly with modifications, but both Quantum Mechanics (QM) and Quantum Field Theory (QFT) undergo many theoretical modifications. This doesn't make many QM and QFT people very happy when they learn about it, especially ones who are committed to Mainstream theories. Osher Doctorow Ph.D.

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Re: [Fractint] FOTD 28-08-03 (Dog-Bone [4])
by Tony (Anthony) Hanmer 29 Aug '03

29 Aug '03

Hi Jim, It's time for me to apalogise for being mostly a lurker on the Fractint list, with little to contribute. I really enjoy your posts - look forward to the daily fractal fix and am rarely at all disappointed, mostly fascinated. Thank you very much and please indeed keep it up! To our intrepid Multifract developer - wow, those long formulae are really masterpieces of evolving creativity, along with the pars you send to provide glimpses into how they work. The annotations to the .frm files also really help. Thank YOU too very much. Maybe the list is in rather a slump. I have not stopped working on L-systems and IFSs - just not posted much for a long time, hoping to refine some work to the point of a mathematical milestone worthy of major attention, though maybe I'm dreaming. LONG LIVE FRACTINT AND ITS PRODUCERS AND EXPLORERS! Tony Hanmer Tbilisi, Republic of Georgia (usually) _________________________________________________________________ Help STOP SPAM with the new MSN 8 and get 2 months FREE* http://join.msn.com/?page=features/junkmail

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FOTD 29-08-03 (Mission Cubical [5])
by Jim Muth 29 Aug '03

29 Aug '03

FOTD -- August 29, 2003 (Rating 5) Fractal visionaries and enthusiasts: I have on my book shelf a 115 year-old text book on the princi- ples of physics, which I found in my father's house a few weeks after he died. I'm not sure how he came upon the book. Most likely, he got it from his father. But it is one of my favorite books to open and browse when I find a few minutes with nothing else to do. The book is fun to read for several reasons. To start, it is interesting to see the accepted beliefs of the 1880's as they were actually taught at the time, and not as they are described by present day writers. Five forms of matter are described -- solid, liquid, gaseous, ultragaseous, (now called plasma), and Etheric. The Luminiferous Ether was still assumed to be the transmission medium of light, but its existence was not naively accepted without question. On almost every page of the chapter on 'Ether waves' a new puzzle is mentioned. Experiments are pointed out that contradict both other experiments and observa- tion. The problem of the nature of the Ether was considered, as the book states, 'the most recondite in physics'. But perhaps what draws me most strongly to the book is the sense of wonder it imparts to simple experiments that today are taken for granted even in high-school physics labs. The 19th century was an age of enthusiasm and discovery. This sense of wonder reminds me of the sense of wonder that surrounded the discovery of fractals, and then all too soon vanished as people grew bored with pretty pictures and turned to more practical things. Today's not-too-practical image is another scene in the Z^3+C Mandeloid. It pictures a cubical midget very close to the north- west shore of the north bay of its parent, in an area where the tendrils have become quite jagged. This is an area where Siegel Disks are found in the corresponding Julia sets, though I doubt that I would know a Siegel Disk if I saw one. After a few minutes thought, I named the image "Mission Cubical". Since there is nothing really out of the ordinary about the image, I could rate it at only a perfectly average 5. The render time of 6 minutes is not too slow, but a more convenient way of viewing the image is to download it from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Thursday was as pleasant a day as might be expected in this part of the world at this time of year. The mostly sunny skies, temperature of 88F 31C and moderate humidity allowed the cats to have a satisfying romp in the yard. And while they were romping, I finished the graphic work without incident. If only all days could be as uneventful . . . But today is starting cloudy with rain forecast, meaning it is unlikely that the day will be as pleasant as yesterday. We won't know for sure until the day has ended however. To see exactly how things work out, check back here in 24 hours. Until then, take care, and be fractally correct in everything you do. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Mission_Cubical { ; time=0:06:33.31--SF5 on a p200 reset=2002 type=formula formulafile=slices.frm formulaname=MandelbrotCube passes=1 center-mag=-0.29534585848656850/+0.690623707339427\ 90/4.225576e+007 params=0/0/0/0 float=y maxiter=5000 inside=0 logmap=400 periodicity=10 colors=0000qw5ow9nwDmwHkwLjwPiwUgwYfwaewecwibwmawk\ `uj`ti_sh_rg_qfZpeZodYncYmbYlaXj`Xi_WhZWgYWfXVeWVd\ VVcUUbTUaST_RTZQTYPSXOSWNRVMRULRTKQSJQRKSRIQQGOQFM\ QDKPBIPAGP8EP6CO5AO38O27O38M39L39J4AI4BG4BF4CD5DC5\ DA5E95F76F66G46H33E16H29J3CM3FO4HQ5KT5NV6QX7T_7Va8\ Yc9`f9chAfjBhmBkoCnqDqtDtvEwxFvxEuxEuxEtxEtxDsxDrx\ DrxDqxDqxCpxCoxCoxCnxBnxBmxBlxBlxBkxAkxAjxAixAix9h\ x9hx9gx9fx9fx8ex8ex8dx8ez5dx8cwAbvCauE`sG`rI_qLZpN\ YnPXmRWlTVmTWkVXiWXgXYfYYdZZb__a`__a`Yb`XcaVdaTebS\ fcQgcOhdMidLjeJkeHlfGmgEngCohBph9qi7rk4ti6sh8sfAre\ BrcDrbFq`Gq_IqYKpXLpWNpUPoTRoRSoQUnOWnNXnLZmK`mI`l\ JamJbmJcmJdmJenJfnJgnJgnJhnJioJjoJkoJloJmoJmpJnpJo\ pJppJqpJrqJsqJsqJtqJuqJvrJwrJxrJyrJyrLxpMxoNxnOxmQ\ xlRxkSxjTxiVxhWxgXxfYxe_xd`xcaxbbxadx`ex_fxZgxYixX\ jxWkxVlxUlzRmxTmwUmvVmuWmtYmsZmr_mq`npbnocnndnmenl\ gnkhnjinijohlogmofnofnofn } frm:MandelbrotCube {; Jim Muth real(c),imag(c) z=p1, c=pixel+p2: z=z*(sqr(z))+c, |z| <= 16 } END 20.0 PAR-FORMULA FILE==================================

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