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Re: [Fractint] FOTD 28-01-03)
by Jim Muth 29 Jan '03

29 Jan '03

John W. wrote: <snipped> >To a certain extent each observer must observe his private >world, although in many/most(?) cases the experience is, no >doubt, similar amongst observers. Since I basically agree with the entire reply, I have little to add. My only comment might be, 'what are the philosophical and scientific consequences of our mere indirect contact with a presumed reality?' and Ricardo F. wrote: <also snipped> >Let me add some other [questions]: What is conscience >[consciousness?] . . . This question is perhaps the second greatest, right behind: Why is there something instead of nothing? The scientific method of addressing the question has been to assume that consciousness is a phenomenon somehow generated by the brain, (What else is there?), and begin searching for the mechanism. But after hundreds of years of searching, countless 'theories', and continuing claims of 'progress' and 'discoveries', we remain no closer to a definitive answer than we were when we first made the assumption. The problem is that consciousness is totally different from the material of the brain, the meat that is assumed to generate it. When we claim that consciousness originates in the brain, we are claiming that something can come from nothing, which is magic. (There is no consciousness in ordinary matter.) I am far more comfortable picturing the brain as a kind of radio receiver that detects a universal field of consciousness and focuses it into the sub-units that we call our individual self-awarenesses. Those who wish to do so may consider the universal field of consciousness to be their God. Another extremely difficult question is: What remains the same from day to day, while the contents of the mind and material of the brain constantly change? The most immediate answer would be to invoke an unchanging soul or spirit. But science does not admit the existence of such entities, and wanting to appear at least moderately scientific, I must state that I have no idea. >. . . If it [consciousness] is inmaterial, how is it that it >controls the body? . . . >From the scientific viewpoint the question is even more perplexing. It is assumed that the brain creates the mind that controls the body by controlling the material of the brain -- a circular argument that goes nowhere. There is also a conflict with causality. The physical universe is assumed to be a closed system. If what controls the brain and body is not physical, there is a physical effect without a physical cause, a clear violation of the principle of causality. >. . . What is matter? This is another puzzling question, made even more puzzling by the discoveries of quantum mechanics. Regardless of which model we use, reality at the quantum level is so alien to our intuition that it is beyond comprehension. By using numbers, we can create models of what goes on down there, and thereby make predictions, but we shall never achieve an intuitive image of things like atoms -- an image which would match the image we have of the physical world. We have discovered that we live in a material classic world composed of immaterial quantum building blocks. This is the same as discovering that we live in a house composed of the idea of a house. It is one massive puzzlement! Jim (the puzzled fractalist) Muth

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Re: searching for stupid... :)
by travelGirl 29 Jan '03

29 Jan '03

a VERY special thank you goes to russ nelson, who found 1 through 1600 in record time... still missing is 1601 through 1700, but i think the loss of those (should it come to that) will be tremendously mitigated by mr nelson's archival find and generosity... again, THANK YOU! travelGirl <carenp(a)yahoo.com> wrote: i know this is a sad and stupid tale, of a woman who made several backups, then promptly lost track of each and every one... trust me, i know this woman intimately... she's me :) would anyone happen to have backups of my PAR files from compuserve? i have at least one copy of my PARs from image #1 to #150, and one or two packets of 25 each somewhere in the 600 range, but i'm basically missing (ie, lost) everything from #s 151 through 1700... and this news, of course, makes me VERY angry... i've got the images, and could go back through them to regenerate the pars, but it would be nice to have the information (titles, creation dates and times, copyrights, notes, etc) of the original pars that were uploaded to compuserve and on those now-ancient fractint CDs... thank you in advance... :) 00 caren park --------------------------------- With Yahoo! Mail you can get a bigger mailbox -- choose a size that fits your needs

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searching for stupid... :)
by Hans-Joachim Metzger 29 Jan '03

29 Jan '03

Caren, maybe this is a sad tale, but it isn't a stupid one. Could happen to everybody. Seems I have almost all of your original par files: Ckp025 to Ckp975 (November 1992 to June 1994) and Cp1000 to Cp1600 (June 1994 to January 1997), complete with headers - must have missed 1700, though. So if you let me know your email adress I will send them to you. All the best, Hans-Joachim

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searching for stupid... :)
by travelGirl 29 Jan '03

29 Jan '03

i know this is a sad and stupid tale, of a woman who made several backups, then promptly lost track of each and every one... trust me, i know this woman intimately... she's me :) would anyone happen to have backups of my PAR files from compuserve? i have at least one copy of my PARs from image #1 to #150, and one or two packets of 25 each somewhere in the 600 range, but i'm basically missing (ie, lost) everything from #s 151 through 1700... and this news, of course, makes me VERY angry... i've got the images, and could go back through them to regenerate the pars, but it would be nice to have the information (titles, creation dates and times, copyrights, notes, etc) of the original pars that were uploaded to compuserve and on those now-ancient fractint CDs... thank you in advance... :) 00 caren park --------------------------------- With Yahoo! Mail you can get a bigger mailbox -- choose a size that fits your needs

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FOTD 29-01-03 (UnNamed Fractal-III [8])
by Jim Muth 29 Jan '03

29 Jan '03

FOTD -- January 29, 2003 (Rating 8) Fractal visionaries and enthusiasts: With today's image we continue investigating the fractals that result when the numbers pi and epsilon are used as exponents of Z. In today's case 200 parts of Z^(pi) are subtracted from 200 parts of Z^(epsilon), then 1/C is added. This operation produces a large open fractal filled with debris. The eastern end of this fractal has Mandel characteristics, with a bud and a stem extending east along the positive X-axis and terminating in a flower-like feature. Smaller lightning-like side-filaments radiate from this flower. Today's image is located on one of these filaments. To be more accurate, I should say today's image is located where the filament used to be, for the image is another one that has been evaporated by an immense bailout radius and made visible by the bof61 inside fill. (In yesterday's discussion I mistakenly wrote that the image was rendered with the bof60 fill. I should have written bof61 fill.) Fans of clever titles will once again be disappointed. I have once again neglected to name the image. The phrase "UnNamed Fractal-III" that appears in the head of the parameter file should be considered more of a catalog entry than a name. But named or not, the image is well above average. The 6-minute render time requires just about the right amount of effort that should be required to see the image. Of course, those who prefer their images 'on sale' will be able as always to download the pre-rendered image from one of the FOTD web sites. Paul's site may be accessed at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Scott's site is at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Tuesday was partly cloudy with a high temperature that just reached freezing, making the dynamic fractal cats partly happy. When they rest on the porch, they prefer sun, especially when the temperature is chilly. Unfortunately, the sun will not rise above the evergreen holly trees for almost a month yet, so I suppose the intrepid ones will be disconsolate for a few more weeks. This morning is cloudy with showers of ice pellets mixed with freezing rain -- certainly not the kind of weather that will make the cats happy. I'll find my consolation when the work is finished and I can tackle the real task of the day -- finding tomorrow's FOTD. Until then, take care, and what could be worse than a day without fractals? Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ UnNamedFractal-III { ; time=0:06:05.09--SF5 on a p200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+1.25714998352473700/+0.025876131429569\ 72/9.260188e+010/1/-90/3.88578058618804789e-016 params=-200/-3.14159265358979/200/-2.7182818284590\ 5/0/1e+100 float=y maxiter=400 inside=bof61 periodicity=10 colors=000Lc0z60zA0wC0qG0kH0eL0ZN0aA00000z0Ci0aE0z\ 00a2EELW0ek0Zf0Ue0Nc0Ja0EZ08Y02W00006J1EC4J48S0CY0\ 8S04L02H00C00800200000400801002G06J08N0CE0E60G00J0\ 0L00N00P0040A00z00S00W00n00i00i00u00Z00E0000000000\ GJ0er0UZ0ZkUPS0HN0000AG06C048XUCGYHSkLcyPmzUczYUza\ JzcAzf1zk0zm000AziJyiUufcnemkeweczacfJ0cC0a60Y10W0\ 0S00P00N00Z00kA0wS0zf0zW0zJASW10e01L0H20Z00n00m00k\ 04000f0ae0qc0za0ze0zf0uk0im0Wq0Hs46wA0yE0cN4HWSGN6\ EG060G10f20S20C400400400H00W01i0Hw0Yz0nz0z000U0z20\ zE0zN0zY6zfEzqLzzUzzazCzzza0zZ0zY1zWCuWNqUZmSkiSwc\ JnYCfS4ZL0SG0JA0C404000E04U0CiEJwUSsf8qu0nz0Yz0Hs0\ 1k00c10Y60U40S20P10N02L00nz0uu0ym0ze0zW0zN0zG0z80z\ 01w08i0EY0Hc0Ji0Nn0Pu0Sz0Zs0ek0kc0sW0yN2zGAzAGeA0A\ A0z0knAccJWSUPHcH6nA0y40z00z0NZZz0zz0zz0zz0zz0zz0z\ q0ze0yU0uH0q60m00kPzazzL088zk0uY0NE00600A00C00E00G\ 00H00J00L00N00S04U0AY0GZ0 } 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 28-01-03 (UnNamed Fractal-II [8])
by Jim Muth 28 Jan '03

28 Jan '03

FOTD -- January 28, 2003 (Rating 8) Fractal visionaries and enthusiasts: Questions, questions, questions . . . a fool can ask more ques- tions than a wise man can answer. I'm feeling a bit foolish this morning, so I shall ask a few questions that the wise men wisely brush aside. I expect no answers, because if questions such as these could be answered in an absolute manner, they would long since have ceased to be asked. Of course there is always the chance that the questions *have* been answered but those who still ask them have failed to recognize or will not accept the answers. Would fractals exist if there had been no intelligent beings in the universe to find them? Would the potential for fractals exist if there had been no intelligent beings to invent numbers? And what of numbers? Could they exist without having been inven- ted by a human type of intelligence? Things grow more complicated. Could intelligence exist without the universe? What if a few physical constants had a different value? Does the universe exist when it is not being observed? Does it exist when it *is* being observed, or is it some kind of consistent illusion. If the universe is an illusion, what is having the illusion? And if the universe is an illusion, is the same illusion common to all observers, or does each observer ob- serve his own private world. Are the observers also part of the illusion? Finally, the ultimate question: Why is there some- thing instead of nothing? No one can give absolute answers to such questions, though many hold opinions so strong that they confuse belief with knowledge and consider their opinions to be fact. And would we even want to answer such questions, when doing so would take so much of the fun and mystery out of life? Regardless of the answers or non-answers to the questions, one thing for certain is that we have another Fractal of the Day to discuss. It is another of my ghostly evaporated images, made visible by the <bof60> inside fill. In today's case, the evap- orated version proves to be better than the normal equal-itera- tion-band version. The parent fractal, which was created by subtracting quite a bit of minus pi from quite a lot of minus epsilon, is an oversized, double-lobed Mandeloid with X-axis symmetry. A meandering fila- ment extends from the northern bud, circles around to the east and joins the southern bud. Today's scene is located on a node of this filament, not far from where it joins the northern bud. I have rated the image at an 8. This rating might be a bit generous, but I'm feeling a bit contemplative this morning, and want to do something to cheer myself. I am unable to find a satisfactory name for the image for the second time in three days, so it enters the world as "UnNamed Fractal-II". Ghosts come and go quickly, and today's image lives up to its ghostly nature. It renders in less than 2 minutes, about the same time needed to download the completed GIF file from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal temperature never made it to 20F -7C here at F.C. on Monday. Combined with biting winds, the outdoor conditions were far too rigorous for the whiskers of the dynamic ones, who passed the day thankful for a warm radiator to huddle close to. Today is starting cloudy and just as cold, with snow threaten- ing. We'll see how the dynamic ones take the harsh cold. Regardless of how the dynamic ones take the cold weather, I shall take it quite well, especially since I expect to be out in it only a few minutes. Until next time, take care, and know that fractals are mystical things that are normally invisible yet can be photographed. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ UnNamed_Fractal-II { ; time=0:01:44.52--SF5 on a p200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+13.1945886431/+21.1419337772/635.8224/\ 1/150/7.2211701895863456e-011 params=-23.3/-3.1415\ 9265358979/23.3/-2.71828182845905/0/1e+100 float=y maxiter=400 inside=bof61 logmap=20 periodicity=10 colors=000000000000000L9BDK85J64NA4QE3UUSXS3_YWcX2\ febiczz6hwHZtSPrb0m0hzzEnoMogVp_bpTKiL2bD8cIEcNKcS\ QcXWcaacffck``jWZjQWiLUiGSiLSmPSpTSsXSv`Sy`Ut`Wp00\ 0`_g`ac`c_PL8WBMb1_iMYeN`bOc_PfXQiUQlTRjSSiSTgRTfQ\ UeQVcPWbPWaOSbNPcMLdLIeKEfJBfPMbUW__fWdpTizQNw2Os9\ OoFPkLPgRPcXQ_bQWhQTnTUiWUeZVa`VYcWTfWPiXLkXHbVFUU\ DLSBCR94Q8hpZco__n`WnaSmbOmbSa`WQZ_EXb2VkKQt`MmdNg\ hOalPQgxSerUcmWagY_b_YXaWScUMeTHhWLjZPmaTocXrf`tid\ wlhynkdqLbhT`_`_Rh`U``XTa_LaaEQ6tszkpnfnbashXxnTfY\ 6dSFcMObGXdKSfONgRIiVEjY9000md0XVCHMO1DZRfJPbOOZTM\ WYLSbJPgILlHIqREr_Bsi7tr4up6pn8ll9gjBchDZfEVdGQcHM\ MtyMbBQeGUhKXjP`mTdpYgrakufnwjgnlaemWYnPPoJHpD8q70\ r69t6Hu5Pv5Yw4ex4myC64K86S98_BAgCCnDEcHCTKBINA7Q96\ RC5SE4TG4TJ3UL2VN2VPATMHRJPPGWNEcLBjJ8UcUyG3mD7bAB\ S7FH5JKGHNQFQ_DTiBVsAL9VOIVQQVSYVUeVRdXOcYLcZIb_Fa\ `Caa9`b7`c4__2ZWEUOPQH_LA } 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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The Rare Event Real Growth Origin of Chaos and Maybe Fractals
by MARLENE DOCTOROW 28 Jan '03

28 Jan '03

>From Osher Doctorow (Marlene is the network listname owned by my wife Marleen (notice the spellings) and I will eventually get my own user name on this network). Readers might be interested in some of the postings that I've made to http://www.mathforum.org/epigone/geometry-research and other places concerning the relationships among chaos, Rare Event Theory (RET), and pathology for functions with real ranges and real domains. To summarize in more or less ordinary English, chaos appears to derive from "pathologies" in the REAL numbers even though complex numbers are typically used for fractals and chaos. The intuitive idea of a pathology as something "ill" or "sick" or "life-threatening" is pretty close to it. Of course, that is relative to the system that one is considering - to a virus, maybe we look like pathologies. But we usually consider that life is the "non-ill" system, for better or worse, and THINGS WHICH CONTRIBUTE TO THE SYSTEM are non-pathological, while THINGS WHICH DETRACT FROM THE SYSTEM are pathological. People familiar with engineering systems may quickly get this idea. It turns out in Rare Event Theory (RET) that Rare Events (those with probability less than .05 or .01 typically, although .10 is occasionally used and there are arguments for using as much as 1/3 or even .50 in some circumstances) follow the pattern of the function p1 (x, y) = 1 + y - x with y < = x for their "non-pathological" probable influence behavior, with x and y usually taken between 0 and 1. Notice carefully that p1 is 1 if and only if (iff for short) y = x, which includes points like (0, 0), (1, 1), (x, x) for x between 0 and 1. The first number or letter in (0, 0) refers to the x value, the second to the y value, and similarly for the others. Now let's look at the typical chaotic Logistic map or equation (chaotic for certain values and intervals): 1) y = Lx(1 - x) where L is a constant (fixed real number) greater than 0 here. A closely related equation is the logistic function equation: 2) y' = kx(a - x) where a, k are constants with a > 0 and y' is the growth of y or technically rate of change of y at a particular point x which in the case of (2) is usually taken as x = t (time). Equation (2) is often used to model biological/population growth. Let's change a to 1 for simplicity (we can always make a scale change for example, although when the magnitude of a is studied we'd want to change it back). The equation y = Lx(1 - x) contains two variable factors, x and 1 - x. We can fairly easily see that x is non-pathological and 1 - x is pathological by considering that if x is mass divided by the total mass of the galaxy assumed constant or population divided by maximum supportable population (which is a fraction between 0 and 1) for example, then life increases with x but decreases with 1 - x. It also turns out that (0, 0) and (1, 1) are on y = x, but not on y = 1 - x. In fact, if x = 0, then 1 - x = 1, and if x = 1, then 1 - x = 0. So only (1, 0) is on y = 1 - x (remember that y < = x so y can't be bigger than x). That's all there is to it in essence except to note that for certain values of L and for certain ranges of x relative to this value, things get chaotic. Either equation (1) or (2) is a combination of two conflicting influences - non-pathological and pathological. That's where the chaos comes from when it does occur. I didn't say anything about complex numbers of form x + iy here, where i is the square root of -1. It turns out that many ideas in complex numbers including the complex conjugate x - iy of x + iy extend to the real numbers provided that we stick to Rare Events. How much confidence can readers have in these results in view of the fact that I'm just developing them now? Well, take a look at my postings at the above site, which extend back several years, and which refer also to some of my publications, and also my postings at http://www.superstringtheory.com/forum in the String/M Theory/Duality subforum of their Forum section, and if necessary I'll cite some of my publications directly. But pathology is already well known in super- abstract algebra - it's called the Jacobson Radical (look it up as an internet keyword) which has the star product (look that up) x o y = x + y - xy. Osher Doctorow Ph.D. One or More of California State Universities and Community Colleges

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Re: [Fractint] Triternions
by Russ Walsmith 28 Jan '03

28 Jan '03

>On Sat, 25 Jan 2003 00:22:36 +1300 Morgan L. Owens wrote: > >"Does that make me a bridesmaid?" > Yo Morgan, since all of us are god and were making the world out of thoughts, you are that which you choose to be. I perceive this Earth, and in fact the entire cosmos, is a venue where our ideas, beliefs, desires, attitudes and expectations precipitate into physical reality as objects and events. So we are, for practical purposes, figments of our own imagination here, and thus we are free to define ourselves in any way we choose. Of course the dogmas of limitation that are so strongly embraced in the conventional wisdom of common social consciousness, will vehemently contradict this suggestion. But as you have apparently looked long and deep into many mind-blowing fractals, you surely have evolved beyond the herd mentality to some extent. You see, I conceive of fractals as Cosmic Mandalas, objects of such hypnotic intricacy and beauty that they shall ultimately prove to be powerful tools in the service of expanding our consciousness and awakening us to our grander identity So lets get on with it and if were going to move so quickly into the new freedom of generating fractals of any dimensionality whatsoever, it seems important to have at least a small inventory of fundamental forms that can be utilized as references. An attempt to create a few of these follows. In the interest of uniformity, certain simplifications are employed that should help make the transition upwards from 2D to 3D to 4D and so forth an intuitive process. The first thing to do is to forget that imaginary numbers ever existed, or remember that they never existed; i.e., unimagine them if you will Thus the complex plane reverts to the Cartesian plane, and we have two axes, X and Y, both measured by real numbers. Let a point in the Cartesian plane be identified by x,y: Then the Mandelbrot set (M) can be generated by the following formula, where brackets indicate a subscript. a[1] = b[1] = 0 s[n] = a[n]^2 + b[n]^2, t[n] = 2*a[n] *b[n] a[n+1] = s[n] + x, b[n+1] = t[n] + y if sqrt(s[n]) < 2, then the point x,y is in M. Back to the familiar .frm form in a moment, but in order to see more clearly whats going on here, well take a quick refresher course in CP. CP in this context stands for cyclical permutation, and as an example take the sequential set of numbers 1 2 3 4. Call this configuration of these four numbers, c1. Now permute c1 cyclically to c2 = 2 3 4 1, then to c3 = 3 4 1 2 and again to c4 = 4 1 2 3. So it may be said of c2 that it is the operation permute the four numbers to the leftward one slot, and so c2^1 = c2, c2^2 = c3, c2^3 = c4 and c2^4 = c1. Hence, c2 is a 4th root of identity. As c3^2 = c1, then c3 is a 2nd root of identity. These four permutations form the order 4 cyclic group (C4). It is isomorphic (i.e., abstractly identical) to the complex number system: Thus c1 = 1, c2 = (0,1), c3 = -1 and c4 = (0,-1). There is much more exposition on these ideas at http://fibonacci-arrays.com/Triternions.pdf At any rate, the point being made here is that the complex number concept is not at all requisite to the process of generating fractals. In the following frm files, the default imaginary values for Y are changed by imag(pixel)*(0,-1) to real. CP4 (XAXIS){ x=real(pixel), y=imag(pixel)*(0,-1), a=b=0: a1=a^2+b^2, b1=2*a*b a=a1+x, b=b1+y z=sqrt(a^2 + b^2) z < 4 } There are other versions of C4, so this formula is called CP4 to identify it as being based on the group generated by the CP process. This file makes an interesting version of M. E.g., note that the inside orbits no longer exhibit any curvature. Moreover, certain color mappings make visible field lines that suggest extensive connections between M and its satellites and even (by single threads) to several points at infinity. Now take c1 = 1 2 3 4 5 6, c2 = 2 3 4 5 6 1 and etc. These form the order 6 cyclic group (C6), which comprises the roots c2^6 = c6^6 = c3^3 = c5^3 = c4^2 = c1. The formula that follows is based on this group. CP6 (XAXIS){ x=real(pixel),y=imag(pixel)*(0,-1),w=p1, a=b=c=0: a1=a^2-2*b*c b1=-c^2+2*a*b c1=b^2+2*c*a a=a1+x,b=b1+y,c=c1+w z=sqrt(a^2+b^2+c^2) z < 4 } Somehow, Fractint seems to have reserved z for comparison with the bailout value; any other symbol Ive tried distorts the set? The CP6 formula is a close cousin of TMan and TGirl, which are based on a variant of C6, and the familial resemblance is there. There are 58 other variations of C6 that should also generate images, but suffice it to say that, in constructing formulas for many of these, complications mat ensue... On to C8, where c2^8=c4^8=c6^8=c8^8=c3^4=c7^4=c5^2=c1 CP8 (XAXIS){ x=real(pixel),y=imag(pixel)*(0,1),v=p1,w=p2 a=b=c=d=0: a1=a^2-c^2-2*b*d b1=2*a*b-2*c*d c1=2*a*c+b^2-d^2 d1=2*a*d+2*b*c a=a1+x,b=b1+y,c=c1+v,d=d1+w z=sqrt(a^2+b^2+c^2+d^2) z < 4 } There are many, many possible basic variations here, perhaps even thousands. Some interesting results can come of just changing a few signs In closing, well look at C10: c2^10=c4^10=c8^10=c10^10=c3^5=c5^5= c7^5=c9^5=c6^2=c1. Cp10 (XAXIS){ x=real(pixel), y=imag(pixel)*(0,-1), u=p1, v=p2, w=p3 a=b=c=d=e=0: a1=a^2-2*b*e-2*c*d b1=-d^2+2*a*b-2*c*e c1=b^2+2*a*c-2*d*e d1=-e^2+2*a*d+2*b*c e1=c^2+2*a*e+2*b*d a=a1+x,b=b1+y,c=c1+u,d=d1+v,e=e1+w z=sqrt(a^2+b^2+c^2+d^2+e^2) z < 4 } It should be clear from these patterns that imaginary/complex numbers are not inherent to fractals per se. To reintroduce them into any of these formulas, particularly into systems such as CP6 and CP10, where 4th order roots are not already presently, creates a bit of havoc but hey, havoc and chaos may complement one another nicely, e.g., as in Lee Skinners renditions of Jim Muths TMan1. BTW, some of these, such as Fig Tree, are to my eye reminiscent of Japanese watercolors, an impression that fractals dont often convey. Peace, Russell _________________________________________________________________ STOP MORE SPAM with the new MSN 8 and get 2 months FREE* http://join.msn.com/?page=features/junkmail

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Minibrot Chaos
by Raymond Filiatreault 27 Jan '03

27 Jan '03

Occasionally I like to search for interesting minibrots, although I rarely find nice ones such as some of Jim's FOTDs. Recently, I found one which is most unusual. Certainly not in the sense of being nice from an artistic point of view, but because the entire inside of the minibrot was escaping (with no inside fill) as if it was a window opening on another part of the parent fractal. Inside that "window", I even found another part with all the framing of a minibrot but again seemingly opening up on the parent fractal. I was thus wondering if anybody had such experience before. I was using the HyperMandel formula and the parameters were random ones generated by Fractint's evolver. The following .par file is for the higher zoom "minibrot". By zooming out, you can then find the inside of the parent minibrot (turn the logmap OFF if necessary). Numerous other similar minibrots can be found on the filaments of that parent minibrot. Raymond MiniChaos1 { ; Raymond Jan/03 ; Version 2002 Patchlevel 4 reset=2002 type=formula formulafile=fractint.frm formulaname=hypermandel center-mag=+0.31055536680379110/+0.16071949514401270/1.28043\ e+009 params=-0.4605548265022736/-0.05972472304452651 float=y maxiter=2000 inside=0 logmap=585 sound=off colors=000IfmIciH`eGXa<3>CHKBDFABAKEB<3>bTFfWGjZGn`HqcIteIvg\ JxhJyiJziJzjJziJyiJxhJvgIteHqcG<2>fWDbTCYPAUM9PI7EE6FF5<3>GU\ BHYCH`EIcFIfG<2>JmJJoJJpKJqKJrLJrL<2>JpKJoKImKIkK<3>E`IDYHCU\ HBQGAMG8FF7FF8GJ<3>FJaHKeILi<3>NNuONwPOxPOyPOzPOz<2>POxONwON\ u<3>LKiKJeJIa<3>EEJCCFHIJ<3>XZ`_bdcfhfik<2>mrtotvquxrwyswzsx\ zsxzswyrwxquwotu<2>ilmfijcff_bc<3>LNNHIIFED<3>YIRaJUeKXiKZmL\ apMcsMeuNgwNhyNizOizOizOjzOi<2>uNfsMdpLbmK`iJYeJVaHSYGPTFLOE\ IKDDFEE<3>WUL_YMcaNgdPjhQnkR<2>urTwtUyuVzvVzvVzwVzvVzvVyuUwt\ U<2>qmRnkQjhP<3>WUKSQJOMHJIFFEE<3>YGGaGHeGHiHImHIpHI<2>wIJyI\ JzIJ<2>zIJyIJwIJuHJsHIpHI<3>aFGYFFTEFOEEKDDDDDDHKELOFPTGTYHX\ a<2>IfmJipJksKnuKowKqyKqzKrzLrzLqzKqyKowKnuJksJip } frm:HyperMandel {; Chris Green. ; A four dimensional version of the mandelbrot set. ; Use P1 to select which two-dimensional plane of the ; four dimensional set you wish to examine. ; Use floating point. a=(0,0),b=(0,0): z=z+1 anew=sqr(a)-sqr(b)+pixel b=2.0*a*b+p1 a=anew |a|+|b| <= 4 } _________________________________________________________________ The new MSN 8: advanced junk mail protection and 2 months FREE* http://join.msn.com/?page=features/junkmail

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Get me going...
by Charles Druesedow 27 Jan '03

27 Jan '03

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