September 2003 - Fractint - mailman.xmission.com

Mandelbroat Set Chaotic Existence 2
by mdoctorow＠comcast.net 22 Sep '03

22 Sep '03

>From Osher Doctorow Ph.D. >From r = a(1 - cos(u)), we get alternatively to the last posting: 1) dr/du = asin(u) Since sin^2(u) = 1 - cos^(u) and from the first equation cos(u) = 1 - r/a, we get: 2) dr/du = asin(u) = sqrt(1 - (1 - r/a)^2) when we're in the first and third quadrants, and the minus sign goes in front of sqrt in the second and fourth quadrants. Since discontinuities occur for simple real-valued functions mainly of types y = 1/(polynomial function of x), y = log(x), and y = sqrt(x) at x = 0 or x = root of polynomial (x = 0 for polynomial = x, etc.), we see that the discontinuity in the Mandelbroit Cardioids of the previous posting are not simply "removable singularities" or irrelevant as they might be considered in complex analysis, quite important. The Mandelbroit Cardioids are characterized in terms of Existence among the various types of Existence discussed by infinities or discontinuities - not at all points, but at some points (often only one). Osher Doctorow Ph.D.

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Re: Centre de Gravité
by JackOTradez 22 Sep '03

22 Sep '03

Some good info on this at: http://www.mrob.com/pub/muency/centerofgravity.html Check it out. Nice site too, lotsa math and stuff. JoTz

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Mandelbrot Set Growth and Existence via Cardioids
by mdoctorow＠comcast.net 22 Sep '03

22 Sep '03

>From Osher Doctorow Ph.D. The Mandelbrot Set from a computational viewpoint involves following orbits of 0 like 0, c, c^2 + c, (c^2 + c)^2, ..., where c is a complex number of form c1 + ic2 where i = sqrt(-1) and only those c orbits which never exceed 2 in modulus or absolute value are in the Mandelbrot set. However, the Mandelbrot set is from another viewpoint a bunch of cardioids that are smaller and smaller, where a cardoid is a heart-shaped object which looks like a circle but has a sharp bend on "top" (or on its side depending on whether it is sideways) where the heart bends inward. The equation of a cardioid is simplest in polar coordinates, where any of the Mandelbrot cardioids in particular have equations of form: 1) r = a(1 - cos(u)) where r is the distance from the origin to the cardioid (the radius or length of radial vector) and u is the angle that r makes with the positive x axis or "pole". The pair (r, u), often written (r, theta) where theta looks like a zero with a horizontal slash through its center, is called the polar coordinates of a point on the cardioid, and there is also a Cartesian or rectangular coordinates given by: 2) (x, y) = (rcos(u), rsin(u)) Let's multiply both sides of (1) by r, obtaining: 3) r^2 = ar - arcos(u) It is always the case that r^2 = x^2 + y^2 and rcos(u) = x, so we can write this as: 4) r^2 = ar - ax Let's calculate the partial derivative of r with respect to x, Dx(r), with u held constant. Actually, there is no explicit u in (4), but just in case we use the notation Dx(r) which is the "derivative of r with respect to x holding u constant". Those who are familiar with calculus can follow the following derivation, while others can just accept the final result below. We get (by implicit differentiation): 5) 2rDx(r) = aDx(r) - a and algebraically this gives us the final result: 6) Dx(r) = a/(a - 2r) Although this is not one of the usual growth equations dy/dt = ky, dy/dt = ky (1 - y), dy/dz = f(z) + g(z)y + h(z)y^2, it has the form dy/dz = a/(a - 2y) if u is held constant. By comparison with the other equations, it is recognizable as a somewhat rarer type of Growth Equation. If we had 2r = a or r = a/2, the denominator of (6) would be 0, which is not allowed in mathematics, but which would signify infinitely large value of Dx(r). Thus, the Mandelbrot set's cardioid(s) are the first growth equations which we have examined which are potentially "explosive" in a sense. Of course, the presence of z^2 in the orbits is another clue, as is the fact that replacing r^2 by x^2 + y^2 in (4) results in: 7) x^2 + y^2 = ar - ax which can be used to solve for r as a quadratic or bilinear/affine polynomial in x and y. The type of Existence expressed by the right-hand-side of (6) can be distinguished from ordinary Geometric Existence by referring to it as Potentially Infinite/Explosive Geometric Existence, or more simply by Chaotic- Fractal Existence or an example of the latter. Osher Doctorow Ph.D.

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FOTD 13-09-03 (Elephant Spiral [7])
by Jim Muth 21 Sep '03

21 Sep '03

FOTD -- September 13, 2003 (Rating 7) Fractal visionaries and enthusiasts: Today we recreate the same parent Julibrot that held yesterday's image, but we slice it at one of its other two critical points. This point is entered as the real(p5) parameter. The Mandeloid that this point draws is shaped quite like the parent Mandeloid of yesterday's image, with the exception that, unlike yester- day's parent, today's has an East Valley. Today's image is located on the south shore of the East Valley of its parent, in a cut-off island located quite deep in an elephant-trunk spiral. The image consists of cut-off combina- tions of elephant-like features and spirals. These features inspired the name. Something about the image pleases me. Maybe the image is not as pretentious as some of my FOTD images, maybe it gives a stronger impression of being a real object, but whatever he reason, I have rated it at an above-average 7. The rating of 7 makes the slow render time of 54 minutes less extravagant. But those who would rather have no extravagance at all in their fractals may see the image by downloading it from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The rain set in just after noon here at Fractal Central on Friday, and the temperature reached only a chilly 64F 18C. The cats managed an hour or so of outdoor time before the rain began, so they were not too irate when they were forced onto the porch and then indoors. When they came inside, a tuna treat kept them happy. But it's raining again this morning, and due to continue almost all day. How will the dynamic duo adjust? Check back in 24 hours to see. Until then, take care, and those with fractals in their days live longer and happier lives. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Elephant_Spiral { ; time=0:54:36.47--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix3a function=ident passes=1 center-mag=+1.19654217940102500/-0.000107572576658\ 21/2.017302e+012/0.9999/-165.009901528464155/0.001\ 66372648813901958 params=-1/2/0.5/4/0.25/6/0.125/8\ /-0.78036013139/0 float=y maxiter=3600 inside=0 logmap=656 periodicity=10 mathtolerance=/1 colors=000lWAkY6j_9iaBhcDgeFfgHeiJdkLcmNbpPaqR`rT_\ sVZtXZuZVq_Sn_Pk_Mh_Id_Fa_CZ_9W_HYaO_cW`dbbfjcgqei\ xfjbhpIjvRplZubXr`Wp_UnZTlYRjWQhVPfUNdTMbRK`QJZPIX\ OHZMG_LFaKEbJDdICeHBgFAhE9jD8kC7mB6nABjDGfGKbIPZLT\ VNVWOWXOXYPYZP__Q``QaaQbbRdcRedSfeSgcSabQWYOQTMLPL\ FLJ9HH4DG4AI47J59I6AH7BG8CF8DE9EDAFCKGBPHB0U2eZidc\ cdhZckUcmPamS_mUZmWXmYWo_UqaTscRsePrhOqjMplLpnJopI\ nrKmtPhvUhVZjUcZThOSmCRn1Qm3Tm4Wl5Zl6ak7dk8gj9ijAl\ iBoiCrhDuhExhFzgDugCpgBkgAgg9bg8Yg7Ug6Pg5Kg4GcAH_F\ IWKJSPJOUKKZLGcMChMBgNBfOBeOBdPBcPBbQBaRB`RB_SBZSB\ YTBXUBWUBVVBVVETTHSSJRRMQQOPPROOTNMWMLYLK`KJbJIeIH\ gHGcIH`IIYIIUIJRIJOIKLIKHILEILBIM8IMGKJOMGWODcQBkS\ 8sU5zW3wY2rZ2o_2na2mb2lc1ke1jf1ig1hh1tjSqiRniQkiPh\ iOeiNbhM_hLXhKUhJRhIOgHLgGIgFFgEDgEMiEVkEcmEloEhpD\ dpCaqCYqBVqBRrAOrAKs9Hs9Ds8At8Fu7Kv7Pw8Ux8Zy8cz9hz\ 9mz9rz9wzAzzAzzAzzOzzJzzF } 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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FOTD 09-09-03 (Smokey-Joe Midget [6])
by Jim Muth 21 Sep '03

21 Sep '03

FOTD -- September 09, 2003 (Rating 6) Fractal visionaries and enthusiasts: Long ago, there was a coal-burning ferry that plied the Chesapeake Bay, ferrying cars from Baltimore on the western shore to Love Point on the eastern shore. The boat was named 'Smokey Joe' because its whereabouts could always be determined by the plume of smoke that constantly rose from its smokestack. The ferry was in operation many years, but went out of service when the bridge across the bay opened. I came on the scene too late to ride the ferry, but my father often told of the times he had ridden it on his way to a special fishing spot, the location of which he never made clear. He described how the plume of smoke from its stack could be seen coming long before the boat itself became visible. I never saw the boat in operation, but in my imagination I often tried to picture the smoky ferry moving across the water, trailing its smoke behind it. For some reason, today's image reminds me of the plume of smoke that I never saw in person. The image is named "Smokey-Joe Midget" in honor of that never- seen but much-imagined ferry. The formula combines Z^(-8.8) with 8 times as much Z^(-0.88) and adds (1/C) instead of C. The resulting parent fractal is one of those indescribable things that result when two negative powers of Z are mixed -- a fractal with more valleys than a penguin has feathers. Today's scene is located at the point where one of these valleys splits into two separate valleys. The basic scene with outside=iter was acceptable enough, but I tried the other outside fills before deciding to evaporate the entire scene. None of the other outside options made an image that was worth the effort. Then I evaporated the scene by raising the escape radius to one-googol, and applied the various inside options to the blank screen that resulted. The 'bof61' option worked best, so that's the one I used. The end result rates a 6 -- not outstanding, but at least a little above average. The render time of 8-3/4 minutes is acceptable, but the image may be seen far more quickly by downloading it, already rendered, from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Monday ended in a satisfactory manner here at Fractal Central, with contented cats and up-to-date work. True, the clouds thickened as the afternoon progressed, but no rain fell, and the temperature of 79F 26C kept things comfortable for the cats. Today is starting the same as yesterday. I hope for another good day. Until 24 hours from now, when the next FOTD appears, take care, and don't get spammed. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Smokey-Joe_Midget { ; time=0:08:46.90--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+2.52482220888604100/-0.244198577126244\ 40/794648.8/1/-5/-9.42602412651316612e-008 params=1/-8.800000000000001/8/-0.88/-0.93/1e+100 float=y maxiter=750 inside=bof61 periodicity=10 colors=000i6si6sg7rg7pf7oi6pf7oc9m`AlYCjXCjTDiQFgO\ GfMGfJIdGJcDLaAM`9M`6O_3PY0QX0QX0SV0TT0VS0XQ0VS0VS\ 0VT0VT0VV0VV0VX0VX0VY1VY3V_4V_6V_7V`9V`AVaCVaDVcFV\ cGVdIVdJVfLVfMVfOTgPTgQTiSTiTTjVTjXTlYTl_Tm`TmaToc\ TodTofTpgTpiTrjTrlTsmTsoTupTurTvsTvuTvvXsvYpv_mv`j\ vaivcfvdcvf`vgYviXvjTvlQvmOvoLvpJvrGvsDvuAvv9po6lg\ 3g`0cT0_O0VG0Q90M10I00G00G00F10F40F60D70D91CC3CD6C\ F7AGAAJC9LF9MG9OICPLDQOFSQGTTITXJV_LXaMYcOYfP_iQ`l\ SaoTcrVcuXdvYfy_gz`gzaizchzdlzZjzcizagz`fz_dvYarX`\ mT_iSYdQX`PTVOSQMQMJPIIMDGL9FJ4DI0CG0PDDaCToAjz9zy\ CzvDzsFzpGzmJzjLzgMzdOzaPz_SzXTzTVzQXzOYzL`zIazFcz\ CdzAfzOdm`cYmaIz`1z`0zc0zf0zi0zl0yo0vr0ss0pv0my0jz\ 0gz0dz1cz1jzIrzXyzlzzzzzvzzsyvoxplujisddp_aoTYlOVj\ ISiGTgFTfDVdCVcAXa9X`7Y_6YY4_X4_V3`T1`S0aQ0aP0cO0c\ M0dL0dJ0fJ0fP0gLYCVVGcTLlQPuPTv0zu0zs0zs0zr1zr1zp1\ zp3zo3zo3zm3yl4xl4xj4vj6u } 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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Existence, Change, and Mandelbrot Set Area
by mdoctorow＠comcast.net 21 Sep '03

21 Sep '03

>From Osher Doctorow Ph.D. mdoctorow(a)comcast.net I posted yesterday to geometry-research in the Math Forum (which can be accessed by keywords Math Forum or geometry-research or geometry- research(a)forum.swarthmore.edu) the posting "Geometric Existence" which basically strongly indicates that for growth-expansion-contraction processes and events, including optimally controlled processes and events, Geometric Existence is the Inverse or Opposite (in a certain sense) of Geometric Change. This explains the exponential bacteria growth and radioactive decay equations, the logistic population growth equation in a limited-supply environment, and the Riccati optimal controlled growth equations, and much more. For those who know Calculus (not required to roughly understand these things) or Differential Equations (similarly), Geometric Change is the Derivative, while Geometric Existence is the Integral, under growth scenarios. For those who don't know Calculus, it turns out that lengths, areas, and volumes are direct expressions of Geometric Existence under growth situations. Hence, the area of the Mandelbrot Set becomes of critical importance. Robert P. Munafo, 2000 Aug 31 in "Center of Gravity, Mu-Ency at MROB," www.mrob.com/pub/muency/centerofgravity.html, also stated to be from Munafo's Mandelbrot Set Glossary and Encyclopedia, 1987-2003, points out that the Center of Gravity of the Mandelbrot Set is obtained similarly to calculating the area of the Mandelbrot Set, and is very closely approximated by: 1) (ln(3) - 1/3)^Feig1 where Feig1 is the first Feigenbaum constant, equal to .28676 82633 82935 02685 29586... However, the first Feigenbaum constant is the (limiting) ratio between the areas of each circular region (main regions in a sense) and the next smaller one in moving to the left from the origin in the Mandelbrot set. Although it may be premature of conclude that we understand this, it seems as though Geometric Existence is central to Chaos and thereby to Fractals. I would guess that the Universe in its formation was on the borderline of Chaos if not in Chaos, and that this was an expression of the nature of its early Geometric Existence and also Geometric Change (expansion, etc.). Osher Doctorow Ph.D.

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Re: [Fractint] [Fwd: Undelivered Mail Returned to Sender]
by Jim Muth 20 Sep '03

20 Sep '03

At 11:23 PM 9/17/03 -0700, Kevin Sexton wrote: >hmm why would I get this, after my message went through? >Return-Path: <MAILER-DAEMON(a)arkion.es> <rest of bounced letter snipped> It's a mystery. I've been getting similar returns for about 2 weeks. It rather looks like one of the list addresses is no longer valid and the mail is bouncing back to the individual senders rather than the list itself. Jim (now watch this bounce back) Muth

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[Fractint] Growth Equations from Lorentz Contraction of Special Relativity
by mdoctorow＠comcast.net 18 Sep '03

18 Sep '03

>From Osher Doctorow Ph.D. The gamma or beta factor of Special Relativity or its inverse is given by: 1) b = sqrt(1 - v^2/c^2) Since 1 = 1^2, this motivates the idea of an ellipse via: 2) (r/ro)^2 = (definition) b^2 = 1 - v^2/c^2 which is the ellipse: 3) r^2/ro^2 + v^2/c^2 = 1 The simplest differential equation which can be derived from this is obtained by setting: 4) v^2 = 1 +/- d(r^2)/dt The positive sign results in the solution: 5) r^2 = exp(k1)exp(-t/ro^2) where k1 is the constant of integration. The negative sign results in the solution: 6) r^2 = exp(k1)exp(t/ro^2) Thus, we obtain either exponential growth/expansion or exponential contraction with time. Since v is more or less unrestricted, r could be related to a macroscopic or microscopic radius or length including a principal radius of the Universe. Interestingly, (6) appears to permit acceleration of the Universe by implicit differentiation, although it doesn't require it. Osher Doctorow Ph.D.

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FOTD 18-09-03 (Image of Staghorns [5])
by Jim Muth 18 Sep '03

18 Sep '03

FOTD -- September 18, 2003 (Rating 5) Fractal visionaries and enthusiasts: I have very little time for a fractal discussion today. I want to get as far ahead as possible in my regular work in case a power failure happens. Power failures are the one predicted thing that usually does happen here at F.C. when the weather acts up. As for that weather, it is not looking too bad. We can survive 35 to 40mph winds with gusts over 50mph (80kph), and 3 inches (7.5cm) of rain. The threat of tornadoes has been removed from the local forecast, though it still exists for areas well south and east of here. For today's image I turned to the manzpower formula and the Mandeloid that results when the formula Z^(2.002)+C is iterated. I went to the area just east of the largest midget and examined the rail-like chaos that lies between the two main branches of the negative stem, which has split into fragments. Midgets are very numerous in this area, though they all look pretty much the same. I named today's image "Image of Staghorns". At one time, the image reminded me of staghorns, or stagantlers, (but 'horns' sounds better). I can no longer see much of a resemblance, but the name remains. Since the image is a routine one, found and colored with almost no effort, I could rate it only a 5. But it is very fast, and worth the two minutes required to render it from the parameter file. Those who would rather not render may download the finished GIF image from the internet at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Perfect weather here at Fractal Central once again produced perfect cats on Wednesday. Sunny skies, light winds, and a temperature of 79F 26C kept them so happy that their customary evening treat of tuna was not needed. Today however is starting far more threatening, with leaden skies, winds already picking up, and a tropical storm on the way. To see how the dynamic duo pulls through the mini-tempest, check back here tomorrow. As for me, I have finished all the preparations I am going to make, which involved 5-minutes of work to take in the lawn furniture. So until next time, take care, and I see no cause for alarm. (Of course, I could always be wrong.) Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Image_of_Staghorns { ; time=0:02:07.65--SF5 on a P200 reset=2002 type=manzpower passes=1 center-mag=-1.6\ 71974101237228/+0.00065516117700869/700556.1/1/-87\ .5/2.35157879280301074e-010 params=0/0/2.002/0 float=y maxiter=600 bailout=9 inside=0 logmap=49 periodicity=10 colors=00078S8AV8BX8D_9Ea9Gd9IgAJiA\ LlAMnBOqBPsBRvBSxBLlBEaB7RB1GI2OP2WW3cb3ki3rj6nj9j\ jCgjEcjH_jKXjMTkPPkSMkVIkXEk_Bkb7kd4ohErkOunYxqfwr\ fwsfvsfvtfvufuufuvfuwftwftxftyfsyfszfszftwctuats_t\ qXtoVtlTtjQthOtfMtdKrfJqhJpjIolImnHlpHkrGjtGhiEgZD\ fOCeDBd2A`ELXPVU_dT`fT`gT`hS`iSajSalRamRanRboQbpQb\ rQbsPctPcuPcvPcwNcvLcvKcvIcuGcuFcuDctBctAct8cs6cs5\ cs6`t7Zt8Xt9VtASuBQuCOuDMuEKuFHvGFvHDvIBvJ9vL8uM7t\ N7tO6sP6sQ5rS4qT4qU3pV3pW2oX2ogISqX4rU3sS2tP2uM1uK\ 1rO6oSAlVEjZIgaMdeQahU_lYXoaUseSviUuhWthYsg_rg`qfb\ pfdpefoehndimdkmcmmcombpmblm`im_fmZcmY`mXXmWUmVRmU\ OmSLmRHmQEmPBmO8mN5mM2mLDmLNmLYmLgmLqmLomOmmRkmTim\ WgmYen`dobbpe`qhZrjXsmVtoTurSvthwnxxhuydrzapzZmzVk\ zShzPezLczI`zFZzCXzEVzGTzIRzKPzMUzQZzUczYhzamzerzi\ wzmrzinzeizaezY`zUXzQTzMdzCpz2MzrJzjGzbEzVBzN8zF6z\ 78zB9zEAzHCzKDzNEzQGzTHzWIzZKzaLzdMzg7zQ } END PARAMETER FILE=========================================

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[Fwd: Undelivered Mail Returned to Sender]
by Kevin Sexton 18 Sep '03

18 Sep '03

hmm why would I get this, after my message went through?

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