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Probability distribution functions and the Triangle Map
by Morgan L. Owens 31 Jan '03

31 Jan '03

Just spent a couple of hours with a statistics text. Looked at Kerry Mitchell's tent map formulae. Went hnnnnnnnnnn...... pdfs. My initial idea was to iterate the usual nonlinear transformation with one hump that lead to Feigenbaum's discovery of his number. Traditionally, a parabola is used, with its height governed by a parameter lambda that runs from 0 to 4. But a piecewise linear curve - a triangle, in fact, could be used instead and (among other things) leads to exactly the same Feigenbaum constant. I thought, hnnnn, what about a Gaussian curve? Okay, it's not exactly confined to the interval (0,1), but why should that be any hindrance to me? I mean, I'm not even going to be confining myself to the reals, let alone a particular finite interval of them. I mean, some of these distributions are even supposed to be _discrete_! Kerry Mitchell had chucked in a user-definable "c" parameter to adjust the size of the triangle, and to provide a parameter for selecting elements of its Julia space. I left it in for the latter reason, but also tossed in the various parameters that are used to describe the various distributions (I could have used one of these instead of leaving in "c", but which one?). I also put in an offset parameter "o", since many of these distributions are centred on the origin, and they're probably more useful at, say o=1. I haven't done any massaging of these parameters - they still have their statistical significance - so some hunting around may be needed to find something interesting (I found a few things during my initial explorations, including the ubiquitous almond bread man). The bailout is also user-definable, but if none is supplied it defaults to 100. Generally speaking, a full zoom-out may be a good start. Starting with values like 1 or 2 may be beneficial (note, of course, that for some values the resulting distribution may be undefined - a look at the formula should be enough in most cases to pick the bad choices). There are Julia and Mandelbrot sets for each distribution. The Mandelbrot sets for some proved tricky - for those, like the exponential distribution, with a critical point at infinity, I've arbitrarily initialised z to 100. Some I'm not yet convinced have anything to offer. The Laplace distribution may even be buggy - I used a hacked version of Sylvie Gallett's Graph formula to check that the formulae I'd written had the right shape, but despite this laplace_distro_m just doesn't seem to want to deliver. Morgan L. Owens "Now on to more important matters. Whoopdy-doo." gaussian_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif sigma = real(p1) o = imag(p1) c1 = 2*sqr(sigma) z = pixel, c = p3/sqrt(pi*c1): z = c*exp(-sqr(z-o)/c1) |z|<bailout } gaussian_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif sigma = real(p1) o = imag(p1) c1 = 2*sqr(sigma) z = o, c = pixel/sqrt(pi*c1) : z = c*exp(-sqr(z-o)/c1) |z|<bailout } exponential_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif mu = real(p1) o = imag(p1) c1 = mu z = pixel, c = p3/c1: z = c*exp((o-z)/c1) |z|<bailout } exponential_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif mu = real(p1) o = imag(p1) c1 = mu z = 100, c = pixel/c1: z = c*exp((o-z)/c1) |z|<bailout } laplace_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) o = imag(p1) z = pixel, c = p3*a/2: z = c*exp(-abs((z-o)/a)) |z|<bailout } laplace_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) o = imag(p1) z = o, c = pixel*a/2: z = c*exp(-abs((z-o)/a)) |z|<bailout } cauchy_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) o = imag(p1) z = pixel, c = p3/(a*pi): z = c/(1+sqr((z-o)/a)) |z|<bailout } cauchy_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) o = imag(p1) z = o, c = pixel/(a*pi): z = c/(1+sqr((z-o)/a)) |z|<bailout } rayleigh_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif sigma = real(p1) o = imag(p1) c1 = sqr(sigma) c2 = -2*c1 z = pixel, c = p3/c1: z = z-o z = c*z*exp(sqr(z)/c2) |z|<bailout } rayleigh_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif sigma = real(p1) o = imag(p1) c1 = sqr(sigma) c2 = -2*c1 z = o, c = pixel/c1: z = z-o z = c*z*exp(sqr(z)/c2) |z|<bailout } rayleightail_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif sigma = real(p1) a = imag(p1) o = real(p2) c1 = sqr(sigma) c2 = 2*c1 c3 = sqr(a) z = pixel, c = p3/c1: z = z - o z = c*z*exp((c3-sqr(z))/c2) |z|<bailout } rayleightail_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif sigma = real(p1) a = imag(p1) o = real(p2) c1 = sqr(sigma) c2 = 2*c1 c3 = sqr(a) z = o, c = pixel/c1: z = z-o z = c*z*exp((c3-sqr(z))/c2) |z|<bailout } lognormal_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif sigma = real(p1) zeta = imag(p1) o = real(p2) c1 = 2*sqr(sigma) z = pixel, c = p3/sqrt(pi*c1): z = z-o z = c*exp(-sqr(log(z) - zeta)/c1)/z |z|<bailout } lognormal_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif sigma = real(p1) zeta = imag(p1) o = real(p2) c1 = sqr(sigma) c2 = 2*c1 z = exp(zeta-c1)+o, c = pixel/sqrt(pi*c2): z = z-o z = c*exp(-sqr(log(z) - zeta)/c2)/z |z|<bailout } logistic_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) o = imag(p1) z=pixel, c=p3: z = z-o z = c*exp(-z/a)/(z*sqr(1+exp(-z/a))) |z|<bailout } logistic_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) o = imag(p1) z=100, c=pixel: z = z-o z = c*exp(-z/a)/(z*sqr(1+exp(-z/a))) |z|<bailout } pareto_jul { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) b = imag(p1) o = real(p2) c1 = a+1 z = pixel, c = p3*a/b: z = z-o z = c/((z/b)^c1) |z|<bailout } pareto_man (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) b = imag(p1) o = real(p2) c1 = a+1 z = 100, c = pixel*a/b: z = z-o z = c/((z/b)^c1) |z|<bailout } weibull_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) b = imag(p1) o = real(p2) c1 = b-1 z = pixel, c = p3*b/a^b: z = z-o z = c * z^c1 * exp(-(z/a)^b) |z|<bailout } weibull_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) b = imag(p1) o = real(p2) c1 = b-1 z = o, c = pixel*b/a^b: z = z-o z = c * z^c1 * exp(-(z/a)^b) |z|<bailout } gumbel_1_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) b = imag(p1) o = real(p2) z = pixel, c = p3*a*b: z = a*(z-o) z = c/exp(b/exp(z)+z) |z|<bailout } gumbel_1_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) b = imag(p1) o = real(p2) z = 100, c = pixel*a*b: z = a*(z-o) z = c/exp(b/exp(z)+z) |z|<bailout } gumbel_2_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) b = imag(p1) o = real(p2) c1 = a+1 z = pixel, c = p3*a*b: z = z-o z = c/(z^c1*exp(b/z^a)) |z|<bailout } gumbel_2_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif a = real(p1) b = imag(p1) o = real(p2) c1 = a+1 z = 100, c = pixel*a*b: z = z-o z = c/(z^c1*exp(b/z^a)) |z|<bailout } geometric_distj { if(real(p4)==0) bailout=100 else bailout=real(p4) endif p = real(p1) o = imag(p1) c1 = 1-p z = pixel, c = p3*p: z = c*c1^(z-o-1) |z|<bailout } geometric_distm (XAxis) { if(real(p4)==0) bailout=100 else bailout=real(p4) endif p = real(p1) o = imag(p1) c1 = 1-p z = o+1, c = pixel*p: z = c*c1^(z-o-1) |z|<bailout }

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Re: [Fractint] Mandelbrot and Julia Sets Explored by RareEvent Theory
by Russell Walsmith 31 Jan '03

31 Jan '03

I believe that I can clarify the first part of this statement (and Osher Doctorow can correct me if it's wrong): What is being said here is that the Mset can be generated as easily from real numbers as from complex. E.g., in the form of an frm: C4 { x=real(pixel), y=imag(pixel) 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 } Recall that y=imag(pixel) is a real number (u da man Gerald) and so imaginary numbers don't come into play in this formula. One virtue of this approach is that it leads to interesting generalizations. A somewhat trivial example is offered in the formula below; 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 } -- On Fri, 31 Jan 2003 02:07:12 Guy Marson wrote: >At 12:20 29/01/03 -0800, you wrote: >>From Osher Doctorow (user in mdoctorow(a)comcast.net network). >> >>Let's examine the expression z^2 + c (where z^2 means z-squared) for z, >>c complex (c constant), that is to say z = x + iy, c = c1 + ic2, i = >>square root of -1. For the Mandelbrot set (M for short), we want /c/ < >>2 where /c/ = sqrt(c1^2 + c2^2), sqrt( ) meaning square root of >>whatever's inside ( ). We'll rewrite this in terms of x and y as >>follows. >> >>1) z^2 = (x + iy)(x + iy) = x^2 + iyx + ixy + (iy)^2 = [(x^2) - y^2] + >>2ixy >> >>2) z^2 + c = [x^2 - y^2 + c1] + i[2xy + c2] >> >>In Rare Event Theory (RET), the expression u + iv changes to real >>variables as follows. >> >>3) u + iv --> u - v >> >>where the arrow indicates the result of the change or transformation. >>So from (2) bove, z^ + c changes as follows. >> >>4) z^2 + c --> [x^2 - y^2 + c1] - [2xy + c2] >> >>Now we get a strange result. J. C. Sprott of Dept. Physics U. >>Wisconsin Madison, in "Mandelbrot Set Chaos," July 3, 1997, >>http://sprott.physics.wisc.edu/chaos/manchaos.htm computes the >>Mandelbrot set orbits in the c1-c2 plane, which he calls the a-b plane, >>respectively by the following where xn+1 means x with subscript n+1, >>etc. >> >>5) xn+1 = xn^2 - yn^2 + a >>6) yn+1 = 2xnyn + b >> >>These are precisely the right hand side expressions in the respective >>brackets in (4) with a = c1 and b = c2. >> >>Now, p1 is optimized at p1(x, y) = 1, where p1(x, y) = 1 + y - x. How >>do we write the right hand side of (4) in terms of p1? The right hand >>side of (4) has the form y - x for some y, x, so it is p1 - 1 for some >>p1. In fact, we have: >> >>7) x^2 - y^2 + c1 - [2xy + c2] = p1(2xy + c2, x^2 - y^2 + c1) - 1 >> >>Let's incorporate the -1 into c1 and call the result c1', so we can >>write (7) as follows. >> >>8) x^2 - y^2 + c1 - [2xy + c2] = p1(2xy + c2, x^2 - y^2 + c1') >> >>Now we can set p1 = 1, which says the following. >> >>9) x^2 - y^2 -(2xy + c2) + (c1' + 1) = 1 >> >>and therefore we have the following. >> >>10) x^2 - y^2 -(2xy + c2) + c1 = 0 >> >>or, expanding: >> >>11) x^2 - y^2 - 2xy + (c1 - c2) = 0 >> >>Except for possibly degenerate cases, it is proved in analytic geometry >>that this is part of a hyperbola (by proper rotation of axes and >>translations). >> >>We therefore can study the Mandelbrot set M as a part of a hyperbola in >>Rare Event Theory, which simplifies it considerably. We have c1^2 + >>c2^2 < = 4 from before. >> >>Osher Doctorow >> >> > > >ooooh, big stuffff..... please put it into a .frm (for 'normal people'), so >we pass it through the evolver and will see what happens.. > >thanks, > >Guy > > > > >_______________________________________________ >Fractint mailing list >Fractint(a)mailman.xmission.com >http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint > _____________________________________________________________ Get 25MB, POP3, Spam Filtering with LYCOS MAIL PLUS for $19.95/year. http://login.mail.lycos.com/brandPage.shtml?pageId=plus&ref=lmtplus

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Re: [Fractint] FOTD 28-01-03)
by Jim Muth 30 Jan '03

30 Jan '03

The Programmer Dude wrote: <much cut> >Do you really doubt the, um, reality of reality? It all depends on how the word 'reality' is defined. And, yes, I sometimes do question the reality of at least the physical world, which in the world of science is considered the basis of reality. >As for answering your question, please re-read all of >philosophical and scientific readings from all of human history. It would be great fun, but unfortunately I haven't got the time. >After you finish that you'll realize that no one else knows, >either. I knew this even before you stated it. I myself stated that the answers to the questions are mysteries. >Not everyone feels [that consciousness is generated by the >brain]. The field of AI has a big division between those who >feel (human-like) consciousness is merely a matter of >sufficient complexity (If you build it, it will think.), while >others feel that the *mind* (opposed to the brain) is much more >than a big wet switchboard. A 'hard-AI' machine might behave as though it is thinking, but will it be aware that it is thinking? This is another debate that will never be settled, unless, that is, someone can devise a repeatable experiment that would objectively demonstrate consciousness itself rather than mere mechanistic behavior. Obviously, before this can be done, the field of metaphysics will need to be taken far more seriously. >(FWIW, I fall firmly into the latter category--I believe in the >soul.) I will take you at your word, but please describe this soul you believe in, and state a few of its attributes. I assume the soul of your belief is a bit more than an amorphous and undetect- able bit of vapor. If it is not, then what you have is hope rather than belief. >I recall a long ago short SF story about the world-wide phone >system achieving consciousness...I believe the last line was >about the phones starting to ring... everywhere at once! :-) I also recall the story. I think it was written by Ray Bradbury. >What you MEAN is that matter like rocks and tables [lack >consciousness]. It's not the presumably unconscious brain matter itself that is supposed to generate consciousness. It is the activity of the intricately connected material of the brain that creates, or some would say, *IS* consciousness. When a person is conscious, his brain matter is assumed to be doing the 'consciousness' activity; when that person is unconscious, the matter is doing something else. Here I wonder if the far more complex behavior of the entire universe might also be an aspect of a conscious- ness -- a far greater consciousness. >I take it you don't at all subscribe to the theories of hard AI? I'm actually neutral about the possibility. What I doubt is that we will ever devise an unambiguous way of determining whether hard (or strong) AI is present. Even the existence of human consciousness is a puzzling thing to determine. If you suppose that recognizing the existence of human consciousness is a simple task, ask a neurologist who has experience with patients in a deep coma. >NOTHING remains the same from day to day. NOTHING. Your >location is moving at huge speeds every second (relative to the >frame of the universe). Your cells are constantly changing. >The atoms in your body are constantly changing. So is your >mind and consciousness. Apparently, you wrote this without too much thought. Though the material and appearance of my body is changing slowly, and my thoughts are changing more rapidly; my individual identity, my particular awareness, most certainly has stayed the same. I am the same person today as I have been through my entire life. I assume this is also true with you. >I, for one, surely hope MY spirit evolves in step with the rest >of the universe. So do I. A spirit stagnated for all eternity would be one of the worst hells. >You continue to make what I consider a fundamental error in >thought: you think the immaterial can be illuminated by the >material. How can it? Are they not completely different >domains? You continue to mis-understand my view. I do not claim that the immaterial (spiritual) can be known through objective material means; I claim that it can be glimpsed through subjective means such as meditation. The problem is that knowledge gained through subjective means cannot be demonstrated scientifically, and therefore is not considered valid by those who put their faith in science. If there were no way at all of gaining knowledge of the immaterial, or if it did not exist, how would all the stories of its reality have come into being? (I am aware that primitive men are supposed to have invented the immaterial realm in an effort to escape the fear of death.) >**IF** what controls the brain and body is non-material, then >the universe is not, in fact, a closed system. And those who assume it is a closed system are mistaken. >How many of us really have an "intuition" of how DNA works? >Or, for that matter, a modern jet engine. Or any of a myriad >*physical* things. An important point. Everything I know about most scientific things I have learned by reading books written by the 'high priests' of science. And I must accept these things on faith, because I have personally observed very few of the experiments that demonstrate the truth of what is in the books. Science is like religion, where only the chosen few have had the mystical experience, (observed the experiment), that validates the belief (hypothesis). >As far as we know so far, these [quantum] phenomena are quite >material. They're just a bit beyond our ken. I would say they are quite far beyond our ken. Ten dimensions, time that flows backward, quantum entanglement, etc. are not things we observe every day. And if subatomic particles actually are tiny bits of materialized energy zipping around in an imaginary Hilbert space, is this not an interaction of the material and immaterial such as we have been discussing? Jim (now back to fractals) Muth

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Mandelbrot and Julia Sets Explored by Rare Event Theory
by MARLENE DOCTOROW 30 Jan '03

30 Jan '03

>From Osher Doctorow (user in mdoctorow(a)comcast.net network). Let's examine the expression z^2 + c (where z^2 means z-squared) for z, c complex (c constant), that is to say z = x + iy, c = c1 + ic2, i = square root of -1. For the Mandelbrot set (M for short), we want /c/ < 2 where /c/ = sqrt(c1^2 + c2^2), sqrt( ) meaning square root of whatever's inside ( ). We'll rewrite this in terms of x and y as follows. 1) z^2 = (x + iy)(x + iy) = x^2 + iyx + ixy + (iy)^2 = [(x^2) - y^2] + 2ixy 2) z^2 + c = [x^2 - y^2 + c1] + i[2xy + c2] In Rare Event Theory (RET), the expression u + iv changes to real variables as follows. 3) u + iv --> u - v where the arrow indicates the result of the change or transformation. So from (2) bove, z^ + c changes as follows. 4) z^2 + c --> [x^2 - y^2 + c1] - [2xy + c2] Now we get a strange result. J. C. Sprott of Dept. Physics U. Wisconsin Madison, in "Mandelbrot Set Chaos," July 3, 1997, http://sprott.physics.wisc.edu/chaos/manchaos.htm computes the Mandelbrot set orbits in the c1-c2 plane, which he calls the a-b plane, respectively by the following where xn+1 means x with subscript n+1, etc. 5) xn+1 = xn^2 - yn^2 + a 6) yn+1 = 2xnyn + b These are precisely the right hand side expressions in the respective brackets in (4) with a = c1 and b = c2. Now, p1 is optimized at p1(x, y) = 1, where p1(x, y) = 1 + y - x. How do we write the right hand side of (4) in terms of p1? The right hand side of (4) has the form y - x for some y, x, so it is p1 - 1 for some p1. In fact, we have: 7) x^2 - y^2 + c1 - [2xy + c2] = p1(2xy + c2, x^2 - y^2 + c1) - 1 Let's incorporate the -1 into c1 and call the result c1', so we can write (7) as follows. 8) x^2 - y^2 + c1 - [2xy + c2] = p1(2xy + c2, x^2 - y^2 + c1') Now we can set p1 = 1, which says the following. 9) x^2 - y^2 -(2xy + c2) + (c1' + 1) = 1 and therefore we have the following. 10) x^2 - y^2 -(2xy + c2) + c1 = 0 or, expanding: 11) x^2 - y^2 - 2xy + (c1 - c2) = 0 Except for possibly degenerate cases, it is proved in analytic geometry that this is part of a hyperbola (by proper rotation of axes and translations). We therefore can study the Mandelbrot set M as a part of a hyperbola in Rare Event Theory, which simplifies it considerably. We have c1^2 + c2^2 < = 4 from before. Osher Doctorow

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Re: [Fractint] Triternions
by Russ Walsmith 30 Jan '03

30 Jan '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: Wed, 29 Jan 2003 01:29:59 +0100 > >Russ Walsmith wrote: > > > 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: > >Russ, >this seems to be a small misunderstanding regarding >Fractint's parser. The imag()-function returns the >imaginary component of a number already as a real! >That is, from a + i*b you get b + i*0, not 0 + i*b. > >Regards, >Gerald > > Thanks Gerald for calling that to my attention... but I admit that I had to run about six versions of TEST { y=imag(pixel): y^2 > 1 } before I became convinced. Good thing I didn't realize that, I guess, else I never would have rectified the imagined imaginary value in Mr T's formula, and TMan would not have come to life. Then Jim Muth couldn't have turned him sideways to create the fraternal twin, TMan1, and if TMan1 didn't exist, he could not have been so swiftly dissected, his strange internal organs expertly laid bare in front of God and everybody on the Fractint list. And if that hadn't happened, from whence would have come the inspiration for the haunting, surreal video 'Mandelbrot Rising', now playing at http://www.sonic.net/~jot/ufvp.html So hey, it's all good, and to quote Morgan Owens, "...the important thing here are the visual results..." Indeed, those and other happy outcomes are accruing to the bottom line. But if the imag(pixel) theory doesn't hang, then what is going on? I must say that the situation is way more curious than I initally supposed. There is much to ponder in this realm; e.g., I've noticed that CP8 { c1=real(pixel),c2=imag(pixel)*p3,c3=p1,c4=p2 z1=z2=z3=z4=0: t1=z1*z1-z3*z3-2*z2*z4 t2=2*z1*z2-2*z3*z4 t3=2*z1*z3+z2*z2-z4*z4 t4=2*z1*z4+2*z2*z3 z1=t1+c1,z2=t2+c2,z3=t3+p1,z4=t4+p2 z=sqrt(t1^2+t2^2+t3^2+t4^2) z < 4 } seems to be completely indifferent as to whether the value chosen for p3 is real or imaginary. Do you have any insights here? Regards, Russell _________________________________________________________________ Tired of spam? Get advanced junk mail protection with MSN 8. http://join.msn.com/?page=features/junkmail

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Rare Event Theory 'Replaces' Composition by Addition-Subtraction
by MARLENE DOCTOROW 30 Jan '03

30 Jan '03

>From Osher Doctorow Category Theory uses composition of morphisms as its main operation, while Rare Event Theory (RET) uses addition-subtraction as its main operation although multiplication-division are shown to be secondary. See my postings at http://www.mathforum.org/math-history-list. This does not mean that composition does not exist in RET, but it does mean that when examining an iterated function in fractals it is enough in the RET representation to examine the original function ordinarily. If a pathology is found in the original function, that is usually enough, and vice versa. I will simply add here that in my opinion an enormous weight of evidence indicates that RET plays the key role in GROWTH (expansion- contraction) processes in the mathematical and physical universes, and the superstringtheory.com site as well as my paper in B. N. Kursunuglu et al. (Eds.) "Quantum Gravity, Generalized Theory of Gravitation, and Superstring Theory-Based Unification," Kluwer: N.Y. 2000 89-97 should be looked at for this. Radiation from a source is also a growth process, by the way. I respect opposite or different views and encourage competition between them, since in my opinion there are many pictures and many theories that lead to knowledge, and competition is extremely valuable for quantitative sciences/mathematics and philosophy and engineering and medicine. Osher Doctorow

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FOTD 30-01-03 (UnNamed Fractal-IV [5])
by Jim Muth 30 Jan '03

30 Jan '03

FOTD -- January 30, 2003 (Rating 5) Fractal visionaries and enthusiasts: Surrounding every Mandel midget is a basin of attraction. The midget behaves as though it wants to hold on to points that start near it. The closer to a midget a point lies, the more iterations will be needed before that point escapes beyond the bailout radius on its ever-accelerating but futile trip to reach infinity. Today's image, with the arms around the midget appearing to descend into the purple haze of the depths, illustrates this basin effect quite clearly. I originally had named today's image "Fractal Basin". Then it struck me that the name raised images of a bathroom. I quickly changed the name to the boring one that the image now bears -- "UnNamed Fractal-IV". And why not. We've been having a lot of nameless images lately. What harm will one more do? When it came time to rate the image, I could attach a rating no higher than an average 5. But the image is really not all that bad. Most likely I have been spoiled by the recent spate of evaporated images in the 7 and 8 range. At one point, I actually tried to 'evaporate' today's image, but much to my surprise, found that it would not evaporate, even though both exponents of Z are negative. I can only assume that this is because the parent fractal is of immense size, with fractal detail extending out at least as far as one-googol. Today's scene lies near the edge of a most curious double-bayed midget that lies near the edge of a larger double-bayed lake that lies near the shore of an immense ocean in the hugely bloated parent fractal. The parent was generated by taking one part of Z^(-1.11) and adding 0.001 part of Z^(-1000) to it, then adding 1/C. The render time of the parameter file is in the 7-minute range, at least on my exhausted old machine, which BTW I have named "Betsy". The wait may be avoided 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 sleet and freezing rain of Wednesday morning changed to light snow by noon, with a temperature that never topped 30F -1C. Only 1/2in 1cm of snow accumulated, but that was enough to convince the dynamic duo that this was not to be a day for a romp in the yard. Realizing the inevitable, they endured their confinement stoically. Today is starting once again cloudy and chilly. And the snow is still around. The duo will likely have another bad day. At least the work is light, which will give me more time to find one of those elusive fractals that rate a 9. Check in 24 hours to see if I succeed. Until then, take care, and all is not lost. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ UnNamed_Fractal-IV { ; time=0:07:43.96--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+15.47911410363785/+71.86947768905489/8\ 190988/1/5/9.87604994590954299e-005 params=0.001/\ -1000/1/-1.11/0/0 float=y maxiter=1350 inside=0 logmap=145 periodicity=100 colors=000WYDWXDXVDYUEYTEZSE_QE_PF`OFaNFaMFeSJiXNl\ aRpfUskYwpaxqdxrfvohtljrilmfnhckc`hZ_cUYZPWUKUPFSK\ BQ1BT1CW1CZ1C`1Dc1Df1Di1El1En1Eq1Ft1Fw1Fy1Lw7QuCWs\ I`qNfoTkmYij_hhagecfcee`gdZicWkbUlaVkaVkaVkaVk`Wj`\ Wj`Wj`Wj`Wj_Xi_Xi_Xi_Xi_XiZYhZYhZYhZYhZYhXWgWVfVTf\ USeTQcSPaRO_QMZPLZOJZNIZMHZKFZJEZICZHBZG9YF8XE7WD5\ UC4UB2VA1W90XC3_F6bH9eKChNFlPIoSLrUOuSQsQSqOToMVmK\ WkIYiG_gE`eCbcAca8e_6gY4hW2jU1kS2iU2hV3gX3fY4e_4d`\ 4ca5bc5`d6_f6Zg6Yh7Xj7Wk8Vm8Un9Sp9Rq9QrAPtAOuBNwBM\ xBLyEMwHMuKMsNMqQMoTMmWMlZMjaMhdMfgMdjMbmN`oOZnPXm\ QVlRTkSRjTPiURhVUgWXfX_eWcdVfcVidUldToeUqcVsaWu_Xv\ YYvWZuU_uS`tQatObsMcsKbrIarG`qE_qCZpBYoAXo9Wn8Vn7U\ m6Tm5Sl4Rl3Qk2Pk1Oj0Nj2Pg3Re5Tc6Va7WZ9YXA_VCaTDbRE\ dOGfMHhKJjIKkFLmDNoBOq9Pr7`sNltamucmvcmwchxccycZzc\ UzcPzcKzcEzc9zc4zc7zdAzeCzfFz_HzTKzNMzGPz9Rz3VzEYz\ PazUdzUezUSzUTzUUzUUzUVzU } 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 use of very complex numbers .......
by Burnett, David 30 Jan '03

30 Jan '03

This got me thinking (a rare occurance and not without its dangers). Our fractals are mostly based on the micro-interactions of the real and imaginary components of a complex number. Has anyone explored using very imaginary numbers (ie n^4 = -1) or where a very complex number has the form ax + biy + cjz (where j^4 = -1). Or further extensions into any power of 2 = -1. too hot here for anything but such meandering musings. db > -----Original Message----- > From: Jim Muth [mailto:jamth@mindspring.com] > > 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?

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the use of very complex numbers ............
by Burnett, David 30 Jan '03

30 Jan '03

This got me thinking (a rare occurance and not without its dangers). Our fractals are mostly based on the micro-interactions of the real and imaginary components of a complex number. Has anyone explored using very imaginary numbers (ie n^4 = -1) or where a very complex number has the form ax + biy + cjz (where j^4 = -1). Or further extensions into any power of 2 = -1. probably been done (hasn't it all) - but you never know........ too hot here for anything but such meandering musings. db > -----Original Message----- > From: Jim Muth [mailto:jamth@mindspring.com] > > 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?

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The chaotic Mandelbrot region of the real line via Rare Events
by MARLENE DOCTOROW 29 Jan '03

29 Jan '03

For c2 = b = 0, we get the following for optimality: 1) x^2 - y^2 - 2xy + c1 = p1(2xy, x^2 - y^2 + c1 - 1) = 1 and therefore: 2) x^2 - y^2 - 2xy = 1 - c1 But the chaotic Mandelbrot real line region is most of -2 < = c1 < = - 1.4011, from which 1 - c1 > = 2.4011, but x^2 - y^2 - 2xy < = x^2 < = 1. This is a contradiction (recall that x, y are in [0, 1] X [0, 1] as arguments of p1). So the chaotic region is pathological in the sense that optimizing p1 there leads to a contradiction. It might be asked whether there are any non-chaotic regions which are pathological, and if so what does that mean. Formally, for c between 0 and 1, the above contradiction disappears. However, for c in [-1, 0], we still get a contradiction. We already know that there are pathologies other than based on contradictions, from earlier examples, and perhaps there is another explanation for c in [-1, 0]. Osher Doctorow

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