September 2003 - Fractint - mailman.xmission.com

A Mandelbrot-Feigenbaum-Similarity Insight Into Growth
by mdoctorow＠comcast.net 18 Sep '03

18 Sep '03

>From Osher Doctorow Ph.D. We are coming to a rather curious juncture in fractals and chaos research related to interdisciplinary growth processes. The good news is that we are making headway into interdisciplinary equations (logistic types especially), into dimensional analysis and its generalization Lie groups of point transformations and dimensionless ratios related to growth involving radiation (which expands from a source) and logistic and even exponential growth constants, and we are pushing our way into different phases including a more and more plausible superluminal phase. The bad news is that we still don't have a handle on growth in respect to expansion-contraction. One possible solution is to change our focus from expansion to contraction in an effort to understand similarity (e.g., self-similarity). Fractals have this tendency to create similar but smaller geometric objects on smaller and smaller scales, but it is difficult to find an equation form of this because of the difficulty of nth iterations or recursions or compositions. However, in geometry similar objects tend to have constant ratios at least in pairs of similar objects. The difficulty occurs in the fact that in a long sequence of similar objects, the ratios of sizes may changed for one constant to a different one and on and on. The Mandelbrot set give us a remarkable clue. It turns out that the sequence of disks as we proceed along the main direction of the Mandelbrot set (along the negative real axis from the origin) approaches a constant in their ratios of diameters, the Feigenbaum number, approximately 4.669 or 4.669201660910.... In mathematical language, with dn the diameter of the nthdisk (n = 1, 2, 3, ...), we have: 1) lim dn-1/dn = 4.669 approximately where the limit is as n approaches infinity (increases without bound). We can obtain a rate of contraction roughly analogous to the rates of expansion that we've been discussing logistically. This would be approximated by: 2) rate of contraction = lim (dn+1 - dn)/(dn+1) = lim (dn+1 - dn+1/4.7)/dn+1 = 1 - 1/4.7 where the Feigenbaum number 4.669 has been rounded off to approximately 4.7 Osher Doctorow Ph.D.

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Re: [Fractint] FOTD 16-09-03 (FMOD Mandeloid [5])
by Jim Muth 18 Sep '03

18 Sep '03

At 09:53 PM 9/16/03 -0700, John Wilson wrote: >Ah, now I understand! Jim, what you call a basement, >(which is perennially full of water), we other folk >call "a WELL"! > >John W. It was a paradoxical situation, being located 50 feet below the top of a 250-foot hill and having a wet basement. Fortunately, the problem has been solved for over a year. The culprit was, and probably still is, a partially blocked basement drain pipe. Normally, the pipe worked as it should and drained away any water that happened to get onto the floor. But in very heavy rain, the drain backed up into the basement, sometimes leaving 4 inches of water. I solved the problem by driving a wooden plug into the drain. I have no idea where the backed-up water goes nowadays. It's not my problem. BTW, the hurricane watch for the local area has just been changed to a storm watch, so the fractals should continue uninterrupted. Jim M.

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FOTD 17-09-03 (A Sparkling Midget [6])
by Jim Muth 17 Sep '03

17 Sep '03

FOTD -- September 17, 2003 (Rating 6) Fractal visionaries and enthusiasts: This morning at Fractal Central features a plethora of junk e-mail and media hype. One of the junk e-mails actually had a subject line that got me to open it. The line was 'you left your umbrella'. Since I recently lost an umbrella, I thought the letter was from some good samaritan who had found it and wanted to return it. I never stopped to wonder how the sender had known not only the person who had left the 4-dollar umbrella, but his e-mail address as well. When I opened the e-mail, I saw the usual gibberish that results when an html document is opened in text mode. I have yet to fathom what a lost umbrella has to do with selling a low-interest loan. Then there is the current local weather forecast, which has a hurricane watch in effect for Thursday and calls for winds of 35-40mph. Hurricane winds start at 75mph. I could see a storm warning, but a hurricane watch . . .? I guess a hurricane situation is remotely possible. The local TV outlets have of course gone into overdrive. By Friday morning, when the storm has just passed its mini-peak, they will probably be frantically searching for a fallen tree and showing rain water flowing down the gutters. So much for the local nonsense. Today's image was created by combining decreasing portions of Z^2, Z^3, Z^4 and Z^5. The resulting Julibrot figure was then sliced at a minor critical point, which drew a tiny Mandelbrot set located well off to the west of the default screen. This M-set is rotated 180 degrees and is surrounded by tiny dots, each of which contains a universe of M-sets. These M-sets are unusual in that they have basically no patterns around them, but are surrounded instead by flat iteration bands containing sparse clouds of star-like points. I have named today's image "A Sparkling Midget" because it is surrounded by little other than widely scattered stars. The coloring was done using the usual equal-iteration bands method. The other outside coloring options give more detail to the image. In fact, some of them give too much detail, but I wanted to show the basic star-like blandness of today's scene, so I chose the standard outside=iter option. The image rates a 6, which is not bad for such a bland picture. It renders in 3 minutes, which is even better. Best of all, it downloads in one minute or so from . . . <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The weather remained perfect throughout the day here at Fractal Central on Tuesday. Not a single cloud was seen all day, the wind was light, and the temperature reached 77F 25C. The dynamic fractal cats made the most of the perfection by spending several hours in the yard. At the end of the day, no tuna was needed. Today is starting perfect once again, but that hurricane is still lurking in the vicinity. Stay tuned to see what develops in the next few days. Until tomorrow's FOTD, take care, and try to be at ease when nature misbehaves. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ A_Sparkling_Midget { ; time=0:03:19.93--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix3a function=ident passes=1 center-mag=-16.27242179554596000/-0.06293582629549\ 968/4.17399e+010/1/25/-0.000463342159198340542 params=1/2/-0.67/3/0.33/4/0.17/5/-2.73312824662175\ /0 float=y maxiter=255 inside=0 periodicity=10 colors=000clsaot`sn_viF5oH4iJ3cL2YN1SO0MS8RWFV_MZb\ Tbf`gjgknnoqus4REAVIFYML`QQcUWgY`jafmekphdijYckRYl\ LSmEMn7Go1Ap5Dj96MCCPGJPKMONOMRRMVUPYYSa`VedYki`rp\ czzemsfllfkggjagicghWhg_hf0hvJRfw0Cq6GkBKjGOiLShQW\ gVZf_U6AW4SY3hdJTzYEyXGeWIbVJ`ULYTMVSOTRPTKRTDTT6V\ T0Wa4ci7kqAroDswGzmJulMugWvcdvZmvVvvNloGbi9Tc2JYUM\ KtP7jV8``8Rf9Hl97r9JlIVfRf`_qWgWUoBSwARm9Qd8PW7ON6\ NE6N5CRDIVLOYTM`RLbPKdNJfLIhJHjHGlFFnEOcJWTNcISk8W\ fIKbS9SXDI`HKaMLaRNaWOa`_U`kN`wG`uKcsNeqQgoTimWkkZ\ mjaoacgTe`KgTBiM2kF5lJ7lN9mRBmVDnZFnbIm`Ll_OkZRjXU\ iWXhVZhUWdRUaPRZMPWKMSIKPFHMDFJBKIGPILTIPYIUbIZfIb\ cGZ`FWYDSVCPUEVTF`SGfRHkZKYfNKnQ7jUCgXHc_M`cRXfWUi\ `RldtNZsQ`rSbqUcpWeoYfL0rJ9sIItGRuF_uDhvCqwBywGvuK\ ssOpqSmoWjm_gkcdieefgfdigakh_lhY`WOPJFTOMXTT_YZcbe\ ggljlrimqinqhnphopgpogpoemidkcbhYafT_cNZaHXZBWX6WW\ 7WW7WW7WW8WW8WW8WW9WW9WW9 } frm:MandelbrotMix3a {; Jim Muth z=real(p5), c=fn1(pixel), a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=real(p3), h=imag(p3), j=real(p4), k=imag(p4), l=imag(p5)+100: z=(a*(z^b))+(d*(z^f))+(g*(z^h))+(j*(z^k))+c, |z| <=l } END 20.0 PAR-FORMULA FILE==================================

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Re: [Fractint] FTOD cd question
by RENRAD1＠aol.com 17 Sep '03

17 Sep '03

In a message dated 9/16/03 9:43:45 PM Pacific Daylight Time, jamth(a)mindspring.com writes: << As soon as I organize the files, the CD will be ready to go, but from the way things are going, I cannot say when that will be. Jim M. >> Great!! I'm very happy not to have simply missed it; this was never intended to "jog your elbow" but just a check to see if I had let it slip by me. Thank you for your quick reply! I learn something every time I get the chance to fall into the FOTDs for a bit, thank you for persevering in the explorations and publishing them Cheers, pete (renrad1)

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Re: [Fractint] FOTD 16-09-03 (FMOD Mandeloid [5])
by Jim Muth 17 Sep '03

17 Sep '03

At 07:14 PM 9/16/03 +0100, John Lewis wrote: >I suppose that if the media are over-zealous like >this, it stops them getting sued! Or else, if some >one sues them because the disaster has not happened, >it will be cheaper. True. And the over-zealous reporting also increases ratings. >I hope you and your family (including the cats) will >be alright. I think we will be. The current forecast is for winds of 40-60mph, which should do no more than scattered damage that is not too severe. We are near the top of a hill, so flooding should not be a problem. Jim M.

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Re: [Fractint] FTOD cd question
by Jim Muth 17 Sep '03

17 Sep '03

At 07:56 PM 9/16/03 EDT, Renrad wrote: >Hallo, Listers, >i miss so many of the emails that i'm afraid i may >have missed something about the status of the >Fractal of the Day cd that was in progress. Is that >still in the works? You have not missed the CD. It is still 'in the works'. The delay is because I have been very busy lately with my regular work and the ongoing FOTD. The problem is that the almost 2000 images are not connected to their discussions and even I have forgotten some of the cryptic 8-letter DOS names I gave to the image files. As soon as I organize the files, the CD will be ready to go, but from the way things are going, I cannot say when that will be. Jim M.

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FTOD cd question
by RENRAD1＠aol.com 16 Sep '03

16 Sep '03

Hallo, Listers, i miss so many of the emails that i'm afraid i may have missed something about the status of the Fractal of the Day cd that was in progress. Is that still in the works? thank you for your time, ~ren

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FOTD 16-09-03 (FMOD Mandeloid [5])
by Jim Muth 16 Sep '03

16 Sep '03

FOTD -- September 16, 2003 (Rating 5) Fractal visionaries and enthusiasts: Well, here we go again. Thanks to the over-zealous media, the local area is near a state of panic about the catastrophic hurricane that is supposedly headed directly this way and due to strike Thursday or Friday. Both the Maryland and Baltimore emergency centers have been activated. The local stores are totally sold out of batteries as people buy 20 and 30 at a time to keep their flashlights lit during the week without power that is probably not coming. Panicky residents are emptying the home-supply stores of plywood to board up their windows against the hurricane winds that are probably not coming. Canned goods and toilet tissue are flying off the grocery store shelves, but I am doing little but watching. I have seen these false alarms often enough to feel reasonably certain that we will have no hurricane and very possibly not even a storm. But the devilish side of me still takes wicked delight in seeing the weather people, TV news people, emergency officials, and the general public make fools of themselves preparing for something that will not happen. One of these days I may be wrong, one of these days I may be sorry that I did not prepare, but in over 40 years that day has yet to arrive, and I have saved probably thousands of dollars. (BTW, I do not recommend my reckless approach to others.) Being involved with the media hype, I spent little time in fractal land yesterday, but I did manage to find a fractal in the figure that results when negative portions of Z and Z^3 are combined. This parent fractal resembles an angel, much as the parent of yesterday's image did, though today's angel has fewer wings and a far less impressive halo. Today's image is located at the edge of one of the many needle- like valleys in its parent fractal. I named it "FMOD Mandeloid" as a description and also because I put no effort into finding a better name. As the name implies, the image was colored with the outside=fmod option. When the time came to rate the image, I could give it no more than a 5. A fractal that has so little effort put into it can hardly be considered above average. The render time of 21 minutes is a bit slow for an image of average quality. 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> is recommended. The weather Monday here at Fractal Central featured showers and heavy thunder-storms. The heaviest rains missed F.C., but with near an inch (2cm) of rain, it was wet enough to keep the dynamic fractal cats confined to the indoors for the day, unable to enjoy the temperature of 79F 26C. They eased their frustra- tion by burying their noses in the tuna that appeared several times in their food dishes. Today is starting very sunny and pleasantly cool. It looks like a perfect day coming. I must finish a bit of design work and finish thinking about the non-preparations for the big hurricane that most likely will never arrive. Until next time, take care, and doctors might bury their mistakes, but weather people put them on the front page. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ FMOD_Mandeloid { ; time=0:21:37.56--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=ident passes=1 center-mag=-0.1575059159887838/+0.1602162344426152\ /46574.45/1/35/-1.0622669411e-011 params=-1/1/-1/3\ /0/0 float=y maxiter=12000 inside=0 proximity=0.5 outside=fmod periodicity=10 colors=000G`7I`8Ka9Ma9ObAQcBScCUdCWdDYeE`fFbfFdgGf\ gHhhIjiIliJnjKpjLrkLtlMvlNxmOzmOylPylQylQykRykRykS\ yjSyjTyjTyiUyiUyiVyhVyhWyhWyhXygXygYygYxfZxfZxf_xe\ _xe`xe`xdaxdaxdbxdbxccxcdxcdxbexbexbfxafxagxagw`hw\ `hw`iw_iw_jw_jw_kwZkwZlwZlwYmwYmwYnwXnwXowXowWpwWp\ wWqzYtxXrwWqvWouVntVmsUkrTjqThpSgoSfnRdlQckQbjP`iP\ _hOYgOXfNWeMUdMTcLRbLQ`KP_JNZJMYILXIJWHIVGGUGFTFES\ FCREBQF8QEAQEBQEDQEEQEFQEHQEIQEJQELQEMQENQEPQEQQER\ QETQEUQEVQDXQDYQDZQD`QDaQDbQDdQDeQDfQDhQDiQDjQDlQD\ mQDnQDpQDqT1xR7uQDrPJoOPlNViM`fLfcKl`JrYGxUIwWJvYL\ uZMt`OsaPrcRqdSpfUogVniXmjYll_km`jobipdhrehtehsdgs\ cfrbepado`cn_blZakY`jX_hWZgVYfUXdTWcSVbRU`QT_PSZOR\ XNQWNPWMOVMNVMMVMLVLLVLKVLJVLJUKJUKIUKIUKIUJHUJHUJ\ HTJGTJGTIGTIFTIFTIFTHESHESHESHDSGDSGDSGCSGCSFCRFBR\ FBRFBRFAREAREARE9Q5bU4dZ5cW5bU5bR6aP6aN6`K7`I7_F7_\ D8ZB8Z88Y65X38Y4AZ5CZ6E_6 } 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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[Fractint] A Superluminal-Subluminal-Growth Tie-In With NASA Results
by mdoctorow＠comcast.net 15 Sep '03

15 Sep '03

>From Osher Doctorow Ph.D. NASA at http://science.nasa.gov/headlines/y2003/09sep_blackholesounds.htm reports in "Black hole sound waves," Sept. 9, 2003 (the story can probably be accessed by the title as keywords if you have any problem with the http) that the following (in my words) seems to be roughly occurring in galaxy clusters to explain why they have so much hot gas and so little cool gas. Heat and light appears to be transformed into sound waves in vast cavities extending away from central black holes (in particular the Perseus cluster 250 million light years from Earth studied by the Institute of Astronomy in Cambridge, England, in coordination with NASA), and the sound waves then dissipate in the cluster gas to keep the gas warm. In fact, the deepest sound "note" ever detected in the Universe (57 octaves lower than middle-C in music, as compared with a typical piano that only contains about 7 octaves) has been detected by them coming from the Perseus cluster and its supermassive black hole. Readers who've been following my development of logistic growth equations here and on researchmathematics(a)yahoogroups.com (a moderated group) may be aware of my recent posting isolating the "momentum kernels" mv and v of the superluminal and subluminal logistic models respectively: 1) d(mv)/dt = kmv(1 - mv) 2) d(v)/dt = k1v(1 - v) Alternatively to (2), velocity or speed v could be replaced by kv where k1 is a constant usually having magnitude 1 but dimension that of mass which might be approximated by sqrt(1 - v1^2/c^) times a constant very close to the speed of light c and where v1 is a constant speed near but below the speed of light. In the second site above I isolated the superluminal/subluminal ratios: 3) mv/v 4) mv/(k1v) where the first has dimension of mass and the second is dimensionless since the mass and speed dimensions cancel. Now consider this equation, where speed v is written u and s is the speed of sound and m is mass: 5) (u/s)(s/c)(c/(mu))(mu/(uk1)) = 1/k1 This is a simple algebaic identity, but (u/s) is the well known Mach Number (Mach for short here) in dimensional analysis, mu/(k1u) is my superluminal/subluminal ratio which I abbreviate as SupSub here, and I will refer to s/c as PERSEUS because of the NASA results that relate sound to heat and light. So (5) becomes: 6) Mach * PERSEUS * (c/(mu))* SupSub = 1/k1 and where * is just multiplication of real numbers. Surely c/(mu) can't be important? On the contrary, it is the partial "reflection" of mu/u (ratio of superluminal momentum to subluminal "momentum" with constant mass) into the ratio of light speed ("light momentum" with the mass replaced by 1 since photons are supposed to have mass zero) to superluminal momentum. We could credit the NASA results (although not their theories, which didn't notice this) with Mach and PERSEUS dimensionless ratios, but logistic growth definitely gets credit for SupSub and c/(mu) (the first dimensionless, the second dimensional with dimension of mass to the negative 1 exponent). Equation (6) can be used by itself even though it is only an identity, but it also has a second use: it tells us the importance of the four ratios which can be used either separately or in combination in astrophysical and cosmological and various mathematical physics or theoretical physics problems. Since growth via the logistic equations and fractals and chaos are so interrelated, we seem to have entered a new "dimension" of applications of fractals and chaos. Osher Doctorow Ph.D.

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FOTD 15-09-03 (Slinky [8])
by Jim Muth 15 Sep '03

15 Sep '03

FOTD -- September 15, 2003 (Rating 8) Fractal visionaries and enthusiasts: All the talk today in the F.C. area today is about a hurricane that appears headed directly this way. Nothing has happened yet, but by Thursday it could be a different story. We'll see how things work out. The parent fractal of today's 8-rated image was created by combining one negative portion of Z with one negative portion of Z^4. It resembles an angel, complete with wings and a halo that is divided into two parts. Today's image lies in the narrow isthmus that separates the two halves of the halo. Much of the image consists of 'inside' material made visible by the 'fmod' inside fill option. This effect of this inside fill reminds me of those toys which consist of nothing more than a large floppy spring that can do things like walk down stairs. I named the image "Slinky" accordingly. The render time of 24 minutes is a bit slow, but the image is worth it. The fastest to see the image is to download it in completed form from one of the FOTD web sites at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The weather Sunday was actually quite pleasant here at Fractal Central, with partly cloudy skies and a temperature of 81F 27C. The cats made up their differences overnight and spent a pleasant afternoon in the yard, enjoying the pleasant condi- tions. It remains to be seen how they can handle a hurricane. I've got a bit of work to finish before I can enjoy pleasant conditions, and the way to finish is to start. So until next time, take care, and don't get caught with your fractals down. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Slinky { ; time=0:24:19.09--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=ident passes=1 center-mag=-0.20033578028784250/+0.346991742483397\ 9/1139769 params=-1/1/-1/4/0/0 float=y maxiter=1200 inside=fmod periodicity=10 colors=000RUKRTKQRKPQLOOLONLNLLMKLLILKHLKFLJEMICMH\ BMH9MG8MF6ME5ME4MD5LD6LD7KD8KD8JD9JDAJDBIDCIDCHDDH\ DEHDFGDFGDGFDHFDIEDJEDJEDKDDLDDMCDMCCNCCOBCPBCQACQ\ ACRACS9CT9CU8CU8CV7CW7CX7CX6CY6CZ5C_5C`5C`4Ca4Cb3C\ c3Cc3Fd5Hd6Jd7Ld9NdAPdBReDTeEVeFXeHZeI`eJadIacIbcI\ bbIcbIcaIdaId`Ie`Ie_If_IfZIgZIgYIgXHhXHhWHiWHiVHjV\ HjUHkUHkTHlTHlSHmSHmRHnRHnQGnPGoPGoOGpOGpNGqNGqMGr\ MGrLGsLGsKGtKGtJGuIFtJGtKHtLHtMItNIsOJsPKsQKsRLsSL\ rTMrUNrVNrWOrXOqYPqZPq_Qq_Rq`RpaSpbSpcTpdUpeUofVog\ VohWoiWojXnkYnlYnmZnnZno_mp`mq`mramsamtbmsbkhckgck\ fdkedkdekcekbekafk`gk_gkZgkYgkXhkWhkVikUikUhkThjTh\ iTghTfgTefTdeTcdTbcTabT`aT_`TZ_TYZTXYTWXTVWTUVTTUT\ STTRSTQRTOQTNPTMOTLNTKMTJLTIKTHJSGIRFHQEGQDFQCEQBD\ QACQ9BQ8AQ79Q68Q56P28Q49R6BT8CV9EXBFZDH`EIbGKdILfJ\ NhLOjNPlOPmOPmOOnOOoPNoPNpPMqPMrPMrPLsPLtPKtPKuPKv\ PJwPJwPIxPIyPHzOIyPIyQIyR } 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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