December 2003 - Fractint - mailman.xmission.com

FOTD 31-12-03 (Fiery Pastels [9])
by Jim Muth 31 Dec '03

31 Dec '03

FOTD -- December 31, 2003 (Rating 9) Fractal visionaries and enthusiasts: Today's discussion will once again be short, as there is still some work I must finish before the end of the day. I actually found today's image several weeks ago, but I have been saving it since then, to present it as a quality end to the FOTD year. It is perhaps the best image of the month, maybe even of the year, at least in my opinion. But I could still rate it no higher than an 9, which is excellent enough. The image shows a very well defined midget lying in the fractal that results when portions of Z^(-1.4) and Z^(-4.1) are combined and (1/C) is added. The parent appears as a bloated and indescribable Mandeloid with a smaller, far more conventional Mandelbrot set in its northern suburbs. Today's scene is located in a cut-off filament in the East Valley area of the smaller M-set. The cut-off nature of the filament is reflected in the eight cut-off 'elephants' surrounding the midget. I named the image "Fiery Pastels" because of the fiery red background and the pastel effect of the entire scene. The wait of 7 minutes while the image renders can be avoided by downloading the finished GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> The 52F 11C temperature on Tuesday here at Fractal Central was pleasant enough, but the wind was far too high for cat comfort. The duo spent less than one hour in the outdoors, and the rest of the day wishing the wind would die down. They were not all that unhappy when evening came, but I gave them a treat of tuna just to be sure. Today promises to be another repeat. Now if I can only find a fractal repeat . . . Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Fiery_Pastels { ; time=0:07:20.22--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix2 function=recip passes=1 center-mag=-0.43142567276170910/+3.130775435388886\ 00/3.507041e+009/1/20/5.71769986824899323e-006 params=0.41/-1.4/0.14/-4.1/0/-50/0/0 float=y maxiter=900 inside=0 logmap=141 periodicity=10 colors=000U3AX2AZ1A`0Ae0Af0Ak0Ak0An2Ao5Aq9AsDAuHAv\ KBvNDxSFyWIz_KzdNzhQzmTzpVxnWqnXknZenZakZZjZWhZSgZ\ OeZMdZNbZQbZTaZWbZZaZabZdaZgbZjaambdpagsbjuamvcpxa\ sybxzczzczzezzazzazy_xxXsvWpsVkpSgmQbjOZhNVdQQ_TMX\ WJTXIQ_HObFMdFKgEMjDMkBOnAOq9Ns9QpDSnHTmJVjNWhQXgV\ ZdX_baaadaVnaOxaJzdMzeQzhpzjpzkmzjkzhhzghzegzdezbd\ yabv_asZapXanWamVakTahSagQaeJadOadNabMaaznjzmhykhv\ kgsjgpjemhejhdggddedaebZdbWdaTbaQb_Na_0ID0JF0KH0MI\ 0NJ0OK1QM3SN5TQ8VS9WTBXVEZWH_XJaZ2z83yA4xB5vD7uE7s\ F8qH9pIAnJBmKBmNDkOEjQFhSHgTHeVIdWJbXKaZ1703924B45\ E77H99IDAKFBNIDQKFSOHVSIXVJ_XT3SS7TQ9VQDVOFWOIWNMX\ NOXMSZMWZKZ_8zFAyIBvKDqNEnQHkTIgWJdZbz7zzTzzVzzVyz\ VxzVuzWsyWpxWnvWkuXjsXgqXepXbpXasZZvZXyZVzZTz_Qz_O\ z_Mz_bzOWzTQzXAz9BzABzBDzDDzEDzFEzHEzIFzJFzKFzMHzN\ HzOIzQIzSIzTJzVJzWKzXKzZpz0_zHBzdDzbDzbEzbEzbFzbFz\ aHzaHzaIzaIzaJz_JzSMzNQzJ } frm:MandelbrotMix2 {; 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)+(p4)), 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 30-12-03 (Something Fishier [6])
by Jim Muth 30 Dec '03

30 Dec '03

FOTD -- December 30, 2003 (Rating 6) Fractal visionaries and enthusiasts: Today is the next-to-last day of the year 2003 here at Fractal Central, (and everywhere else as well), and there is a bit of work I want to put behind me before the year changes. Today's discussion will therefore be short and to the point. The image is a scene in the fractal that results when 0.3 parts of Z^(-9.5) are combined with 3 parts of Z^(-1.5) and (1/C) is added. The parent fractal is one of those disorganized, rambling things that results when negative powers of Z are combined. Today's image is as much a demonstration of the program features as a discovery of a new fractal. I have rendered the image with the outside set to 'fmod' and the inside set to 'bof61'. I normally prefer to avoid program trickery, but in today's case, using the available features results in a more interesting image. The image as it is, rates a 6. When rendered in the normal manner, the image rates only a 4. The two extra rating points were IMO worth the departure from fractal purity. Something about the image reminds me of fish. I therefore named it "Something Fishier", though I can see really nothing in the picture that could be associated with fish. The render time of 27 minutes is slow, but can be avoided by downloading the finished GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Monday was quite pleasant here at Fractal Central. The sky was mostly sunny, the temperature reached 54F 12C, the wind was light, and the cats were happy. Today started wet, but the rain moved out quickly and the sun is now appearing. I expect the cats will be happy again. The next FOTD, the last for 2003, is already in the works, and will appear right here in about 24 hours. Until then, take care, and keep looking for fractals. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Something_Fishier { ; time=0:27:08.65--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix2 function=recip passes=1 center-mag=+1.72463778859888/+0.14466193464504/2.8\ 39504e+007/1/5/-5.201420565093e-008 params=3/-1.5/\ 0.3/-9.5/0/0/0/0 float=y maxiter=1200 inside=bof61 proximity=2 outside=fmod periodicity=10 colors=000BIGAIGAIGAIG9IG9IG9IG9IG8IG8IG8IG7IF8IG9\ IH9IIAIJBIKBILCIMDINDIOEIPFIQFIRGISHITHHURG_JEWJDX\ KCYLAXKBYKCZKC_KD`KDaKEbKEcKFdKFeKGfKGgKHhKHiJIiJK\ jJMkJOlJRmJTnJWoJYpJ`rJbtJevJgxJjyJlzKjyLhxMewNcvO\ `uPZtQWsRUrSRrTRrUQqVRqWRpXRpYRpYSoZSo_So`SnaTnbTm\ cTmdTmeTlfUlgUlhUkiUkjVjkVjlVjmVinWioWipWhqWhtVgqW\ hoWimWjkWjiXkfXldXlbXm`YnZYnWYoUYpSZpQZqOZrL_sMZrN\ YrOXrPWrQVrRUrSTrTSrURrVQrWPrXOrXNrYMrZLr_Kr`JraIr\ bHrcGrdFreErfDrgCrgBrhDsiFsiHsjIskKtkMtlNtlPtmRunT\ unUuoWupYvpZvq`vZbzrcvsewsgwtiwtjwulxvnxvoxwqxxsyx\ uyyvyyxyzzzzzzzzzzzzzzzzzxzzwzzuzztzzszzqzzpzyozxm\ zwlzvkzuizuhztfzsezrdzqbypayp`yoZynYymXylVykUyjSyj\ RxiQxhOxgNxfMxeKxeJxdIxcGxbFwaDw`Cw`Bw_9wZ8wY7wX5w\ W4wW3tY5rZ7p_8m`AkaCibDgcFddGbeI`fKYhLWiNUjPSkQPlS\ NmTLnVIoXGpYEq_BqaCr`Dr_ErZFsZGsYHsXIsXJtWKtVLtUMt\ UNuTOuSPuSQuRRvQSvPBXGBXG } frm:MandelbrotMix2 {; 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)+(p4)), 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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by mdoctorow＠comcast.net 30 Dec '03

30 Dec '03

>From Osher Doctorow Ph.D. Here's a slightly more general Lemma. LEMMA 3. In the equation: 1) dy/dt = A1(exp(t) - y) + exp(-t) y^2 if y = k exp(t), then A1(t) = k. PROOF. dy/dt = k d(exp(t))/dt = k exp(t) from calculus, and y^2 = k^2 (e^t)^2 = k^2 e^(2t), so we get: 2) ke^t = A1(e^t - ke^t) + e(-t)k^2 e^(2t) which simplifies to: 3) ke^t = A1e^t(1 - k) + k^2 e^t since e(2t)e^(-t) = e^(2t - t) = e^t. Divide through by e^t: 4) k = A1(1 - k) + k^2 and solve for A1(t): 5) A1(t) = (k - k^2)/(1 - k) = k(1 - k)/(1 - k) = k. Q.E.D. REMARK. It follows that if y = (1/e)exp(t) (which is exp(t - 1)), then A1(t) = 1/e. LEMMA 4. If (dy/dt)(1) = 1/e, then there is a real positive value of A1(1) which satisfies equation (1) and such that 0 < = A(1) < = 2. PROOF. From (1) for (dy/dt)(1) = 1/e, we get a quadratic equation in y(1)^2 with A(1) in two of the coefficients, and from the square root in the quadratic equation this is always real if and only if 0 < = A(1) < = 2. Q.E.D. REMARK. It is interesting that the values of A(1) and y(1) are linked, especially when constrained by (dy/dt)(1) = 1/e. The linkage is quadratic, that is to say an algebraic quadratic equation in y(1) with A1(1) in two of the coefficients. The equation is in fact: 6) y(1)^2 - A1(1)ey(1) + (A1(1)e^2 - 1) = 0 This is a parabola in the A1(1)-y(1) plane across different pairs of values of (A1(1), y(1)). It is also interesting that "exponential Information," or as I refer to it in Rare Event Theory "Knowledge", of the form y(t) = kexp(t), is an inverse of the Shannon Information-Entropy kernel (in fact, the Hartley Information) log(t) to base 2 up to a constant. Osher Doctorow

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[Fractint] A Lemma or "Subtheorem" on Exponential Explosion Cosmology
by mdoctorow＠comcast.net 30 Dec '03

30 Dec '03

>From Osher Doctorow Ph.D. mdoctorow(a)comcast.net A Lemma is a "sub-theorem" or intermediate stage Theorem on the way to (hopefully) more important results. Here's one based on the previous ideas with a proof. LEMMA 1. In the differential equation model: 1) dy/dt = A1(exp(t) - y) + y^2 exp(-t), y(1) = 1, the condition (dy/dt)(1) = 1, that is to say dy/dt is 1 at time t = 1, holds if and only if: 2) A1(1) = 1/e which is to say A1(t) at time t = 1 is given by 1/e, which is the boundary of Exponential Explosions. PROOF. Set dy/dt = 1 in (1), in which case exp(1) is e^1 = e and y is y(t) = y(1) = 1, and exp(-t) = exp(-1), so we get: 3) 1 = A1(e - 1) + 1/e and solving for A1: 4) A1 = (1 - 1/e)/(e - 1) = [(e - 1)/e]/[e - 1] = 1/e. Q.E.D. REMARK. We can thus regard A1(t) as a function or curve which takes on the Exponential Explosion boundary value 1/e when t = 1. It is a sort of Generalized Exponential Explosion Boundary. LEMMA 2. In equation (1) except that y(1) = 1 is not assumed, if the solution is y(t) = (1/3)exp(t) (so that y(1) = e/3), then A(t) = 1/3. In that case, the Universe is accelerating at least in reference to variable y with respect to time t. PROOF. Substituting y(t) = (1/3)exp(t) into (1), we get: 5) (1/3)exp(t) = A(t)[(2/3)exp(t)] + (1/9)exp(2t)exp(-t) = exp(t)[(2/3)A(t) + 1/9] and therefore (dividing through by exp(t)) A(t) = 1/3. Then dy/dt = (1/3)exp(t) so the acceleration is d(dy/dt)/dt = d[(1/3)exp(t)]/dt and since d(c f(t))/dt = cdf(t)/dt for a constant c, this is (1/3)exp(t) which is always positive for all real t. Q.E.D. REMARK. In Lemma 2, no discontinuity at t = 1 is assumed. We could still define A(t) separately in [0, .05) without changing the Lemma basically. The relationship of y(1) and (dy/dt)(1) to 1/e is now proportional to the inverse of 1/e which is e. Osher Doctorow

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Revision of the "Explosive Exponential" Theorem-Proof
by mdoctorow＠comcast.net 30 Dec '03

30 Dec '03

>From Osher Doctorow Ph.D. The Big Crunch "Exponentially Exploding Julia Set Boundary" theorem and proof from my last posting has a very subtle error. If y(1) = 1, then the solution y = exp(t) or e^t (in other notation) does not hold unless there is a discontinuity at t = 1, since e^1 = e which is approximately 2.72. This would not be surprising at a "singularity" at t = 1, but it would need to be incorporated into the definition and theorem and the proof. Another difficulty is that the whole proof of equation (6) requires that y = exp(t) be one solution, but y = exp(t) exceeds 1 except at t = 0, since in Rare Event Theory y is between 0 and 1. I'll try to revise the theorem and proof and eliminate the "backwards- forwards" parts that were intuitively motivated to obtain a clearer logical order, although it is interesting that intuition played a more "exploratory" role here than logic. Osher Doctorow

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FOTD 29-12-03 (Fractoid [7])
by Jim Muth 29 Dec '03

29 Dec '03

FOTD -- December 29, 2003 (Rating 7) Fractal visionaries and enthusiasts: Greetings once again from Fractal Central, the half-fantasy, half-real home of the world's most over-and-under rated fractals, and the totally-real home of the world's laziest, craziest cats. Today's fractal offering rates a very good 7, a rating that seems honest, at least at the moment. In a day or two it may seem far too high or low, but I will not know until the time comes and the rating has been fixed in stone. The name fractoid is a combination of the words fractal and android, though I can see nothing human-like in the image with its lilac swirls and beige background. The stripes in the background were created by rendering the scene with the outside set to 'tdis', an option that colors the points according to the distance they peregrinate (I like the word) before passing beyond the escape radius. The parent fractal is a distorted version of the Mandelbrot set, with the buds of the main bay rotated far around the shoreline in a CCW direction. Today's scene is located in a spiralling elephant trunk on the south shore of East Valley. The image takes 17 minutes to render from the included parameter file. That time can be cut drastically by downloading the completed image from Paul's FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> The past several days were acceptable here at F.C., with no extremes of weather, no extremes of cat behavior and no further word from Ken. Today is starting as though it will turn into yet another perfect day. But such perfection cannot last long before something goes wrong, so until next time, take care, and don't bet your bippy. You might lose it, and bippies are hard to replace. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Fractoid { ; time=0:17:26.93--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+0.54306023277118430/+1.408216019104502\ 00/3.663052e+007/1/-94.9999918776349119/-3.9243277\ 7156226376e-007 params=1/1.1/-1/-1.3/0/0 float=y maxiter=3000 inside=0 outside=tdis periodicity=10 colors=000SZc`UWiOQrJJzEDyHGwJIuMLrONpQP3ep0hq0js0\ nu0pv0qs4rr8spCuoGumJvkOwiSxhVxfZyeazcezahz`izcjzg\ jyjefezLzWxzzzzrzrzjWhopfjkcfgkcc_ZZXVVVQQTMMQIHOD\ CM87J43PCDVIM`OUfUak_iqeqzozxjzvgzscyq`xpZvoWvmTuk\ QujOsiMshJrgHrfErgHmgJhgMchN_hPVhSQiTMiVHiXCiZ7mVE\ pSMsOTvL_yHfzDmzAszHpyMnwQjuVhr`epecni_knXisUgxSez\ OczMezvczuazs`zs_zrZzq3Gp7IpAJpDMpHOpLPpNSpQUoUVoX\ Xo__oc`ofcohcoffjehgcicck_`nW_oSZqOXrLXpMXoMXnMXkN\ XjNXiNXgOXfOXeOXcOWeSVfUUgXTg_ShcQiePigOjjNkmMmpLm\ rJnvIoxIozHhrHakGVeGOXEHQE9JE3CACH7JM3SQ0ZU8U`HPfP\ LkXGqeCwgHriLokOkmShoVeqZasaZueVwhSykOzoLuaPoPToVS\ oaQphOpnNquLqzJqzIszNvzQxzUzzZzzazzezzhzzkzxnxupvq\ rrnupjwngykcziazgcyeevcgq`hn_jiXkfVnaToZQqUPrQNoVL\ m_IicGghEcmCaq9Zv7Wz5Uz8TzATyDTwETsHTqJToLTmNSiPSg\ SSeTSaVS_XSXZSV`TXaT_cTacTceTffThhTiiTkjTnkTpmTrnT\ uoTvgWu`ZsU`rMcqEfp7hoIch } 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] Could the Big Crunch Be An Exponentially Exploding Julia Set Boundary?
by mdoctorow＠comcast.net 29 Dec '03

29 Dec '03

>From Osher Doctorow Ph.D. mdoctorow(a)comcast.net Indications accumulate that the Riccati Equation is the equation of Expansion-Contraction-Growth (ECG) across biology and physics and other fields, and that this is a type of motion distinct from the usual one-direction-at-a-time Curvilinear Motion. ECG involves motion simultaneously in 2 or more directions (often infinitely many). Since the Universe expands, it seems like an ideal testing case for the Riccati Equation. The general Riccati Equation is: 1) dy/dt = A(t) + B(t)y + C(t)y^2 where dy/dt is the speed of expansion or contraction of y at time t and y^2 is y-squared (y times y) and A(t), B(t), C(t) depend on time t and may or may not be constant real numbers. It turns out that we can prove the following, although the words "certain plausible conditions" are somewhat restrictive. THEOREM. Under certain plausible simple conditions, the Universe goes from a Big Bang in which Rare Event Theory applies and dy/dt attains its maximum ( = 1) to a Big Crunch in which dy/dt = 1/e where e is the exponential constant approximately equal to 2.72 and 1/e is the boundary of exploding Julia sets for the exponential functions c exp(x) (that is, c = 1/e). When c < 1/e, the Julia set is small, and when c > 1/e the Julia set is the whole complex plane (the latter statement can be seen from Robert L. Devaney (1990) Chaos, Fractals, and Dynamics, Addison-Wesley: Menlo Park, CA). PROOF. Since the Riccati Equation and its special cases the Logistic Equation and Simple Exponential Growth/Decay Equations often involve solutions equal to exp(kt) or rational expressions in exp(kt), put exp(t) (k = 1 for simplicity here) into the general Riccati Equation. Since d(exp(t))/dt = exp(t) from calculus, and exp(t) is extremely difficult to "cancel" from an equation usually unless all the terms involve exp(t), let us write A(t) = A1(t)exp(t), C(t) = C1(t)exp(-t). We don't have to change B(t) as will be seen below. Then we obtain: 2) dy/dt = exp(t) = A1(t)exp(t) + B(t)exp(t) + C1(t)exp(-t)exp(2t) since exp(t) squared is exp(2t) from elementary algebra. Since exp(t) divides out in (2) on both sides, we are left with: 3) dy/dt = A1(t)exp(t) + B(t)y + C1(t)exp(-t)y^2 But dividing through by exp(t) on both sides of (2) also leads to: 4) C1(t) = 1 - A1(t) - B(t) This in turn suggests the simplification: 5) B(t) = - A1(t) which also nicely simplifies C1(t) from (4) to C1(t) = 1 Now equation (3) becomes: 6) dy/dt = A1(t)(exp(t) - y) + exp(-t)y^2 When t = 0, dy/dt = A1(0)(1 - y(0)) + (y(0))^2. When t = 1, dy/dt = A1(1)(e - y(1)) + exp(-1)y(1)^2. Since the Big Bang was "explosive" in various senses, we can set dy/dt at t = 0 equal to 1, in which case 1 = A1(0)(1 - y(0)) + (y(0))^2. This is perfectly satisfied if y(0) = 1. Although this does not hold for y(t) as the radius of the Universe, it does hold if y in a neighborhood of 0 is the Rare Event Probable Influence since the latter is 1 at 0. Choose the neighborhood of 0 as [0, .05), that is to say t goes from 0 to less than .05, in accordance with the typical statistical .05 "level of significance". Choose A1(t) = 1 - t + t^2 in the Rare Event neighbor- hood of 0 and A1(t) = 1 - t in [.05, 1]. Then dy/dt at t = 1 equals exp(-1)y^2(1). This already implicates exp(-1) unless y^2(1) = 0 which needn't hold in general. If y(1) = 1, then dy/dt = exp(-1) = 1/e at t = 1. But y(1) = 1 and y(0) = 1 for continuous but non- constant y describes a cyclic universe, and 1/e is the explosive boundary. It should be noted that y(t) and t are between 0 and 1 in the Rare Event scale and that this scale should be restored in the neighborhood of 1 since 1 in the Rare Event scale corresponds to infinite time or to the supremum of finite times, which is a Rare Event (although A1(t) needn't change). Q.E.D. The words "Big Crunch" may not be precisely what we think of as a second "Big Bang", since for example at present the Universe is accelerating in size, but in Rare Events things may not quite be what they seem. Osher Doctorow

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All is well at F.C.
by JimMuth＠aol.com 27 Dec '03

27 Dec '03

Fractal Enthusiasts: The major part of the holidays have passed here at Fractal Central and all is still well. The cats have destroyed their catnip mice, the food is gone, there has been no further word from Ken with the blivot, but the fractals are still alive and well. The next FOTD will appear on schedule in 2 days. Until then, take care, and best wishes to all. Jim M.

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Important Message for Paul and Scott
by Jim Muth 23 Dec '03

23 Dec '03

At 08:28 PM 12/22/03 -0800, John W. wrote: >Whilst pondering this thread, I took a look at that original 1999 Blivot and >was astounded. This must be one of the all-time uglies of the fractal >world! I wonder how Jim would now rate it at, using his present, (ahem), >method? :-) Looking at the old fractal today, I also wonder how I could have chosen it. I must have been having a really bad day. P.S. Methinks that Ken should promote his commercial endeavors with words that are not in such common usage. Jim M.

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FOTD 23-12-03 (Concentric Circles [6])
by Jim Muth 23 Dec '03

23 Dec '03

FOTD -- December 23, 2003 (Rating 6) Fractal visionaries and enthusiasts: This will be the last FOTD until December 29, but have no fear, it is just the seasonal rush that is putting the damper on the fractals, and there is still today's image to discuss. The image is a scene in the fractal created by combining 8.6 negative parts of Z^(-6.6) with 9.6 negative parts of Z^(-1.1) and adding (1/C). It was calculated by the MandelbrotMix2 formula, which works the same as the MandelbrotMix4 formula, with the added option of manually adjusting initial Z. The parent fractal of today's image is an oversized thing consisting of a central Mandeloid with a large arm extending north and terminating in a large ring. Today's scene is located on a filament inside this ring. The location is reflected in the concentric rings surrounding the midget. It is these rings that inspired the name "Concentric Rings". When I found the scene, I rendered it first with the standard equal-iteration-band method. When this method produced a thoroughly ho-hum image, I tried the other outside options. I also tried evaporating the image and rendering it with the various inside fills. The best option I could find was to render the scene un-evaporated, with the outside=tdis, which is the 6-rated version that appears as today's FOTD. The render time of 9 minutes can be cut to a minute or two by downloading the finished image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> A milder temperature of 54F 12C and as much sun as could be mustered on the shortest day of the year gave the fractal cats a good afternoon in the yard. Their greatest excitement came when a starling lighted on a nearby twig, causing Tippy to chatter in anticipation and Thomas to twitch his tail furiously. Luckily for the bird, the cats' bird catching days are pretty much behind them. Today looks to be a repeat. For me, it's work before fun, and the time to start working is now. Until next FOTD, which will be posted on December 29th, Take care, and see you in 6 days. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Concentric_Circles { ; time=0:09:34.85--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix2 function=recip passes=1 center-mag=-1.65813346453185800/+30.92368628273601\ 000/1.849903e+009/1/-5/0.0091790130718738816 params=-8.6/-6.6/-9.6/-1.1/0/0/0/0 float=y maxiter=1400 inside=0 outside=tdis periodicity=10 colors=000jszgiRfiSeiSeiTdiTdiUciUbiVbiVaiWaiW`iW_\ iX_iXZiYYiYYiZXiZXi_Wi_Vi`Vi`Ui`UiaTiaSibSibRicRic\ QidPidPieOieNjfOiePiePhdQhdRgdRgcSfcTfbTebUebUdaVd\ aWd`Wc`Xc`Yb_Yb_ZaZ_aZ_`Z``Y`_Ya_XbZXbZXcZWdYWdYVe\ XVfXVfWUgWUgVThVTiUTiUSjTSkTRkTRlSRmSQmRQnRPnQPoQP\ pPOpPOqONrONrNNsNMtNMtMLuMLuLLvLKuKKsKJrJJqJJrIItI\ IuIItHHsGHrFGrEGqDFoCFnBElAEjADh9Dg8Ce7Cc6Bb5B`4AZ\ 3AX19Y3AZ5AZ6A_8A`9A`BAaDAbEAbGAcHAdJAdLAeMAfOAfPA\ gRAhTBhUBiWBjXBjZBk`BlaBlcBmdBnfBnhBoiBpkBplBqnBrp\ CqoBpnBomBnmBnlBmkBlkBkjBjiBjiBihBhgAggAffAfeAeeAd\ dAccAbcAbbAaaA`aA_`AZ_9Z_9YZ9XY9WY9VX9VW9UV9TV9SU9\ RT9RT9QS8PR8OR8NQ8NP8MP8LO8KN8JN8JM8IL8HL8GK7FJ7FJ\ 7EI7DH7CH7BG7BF7AF79E78D78D77E87E97EA7EB7EC7ED7EE7\ EF7EF7EG7EH7EI7EJ7EK7EL7EM7EM7EN7EO6FP6FQ6FR6FS6FT\ 6FT6FU6FV6FW6FX6FY6FZ6F_6F`6F`6Fa6Fb6Fc6Fd5Ge5Gf5G\ g5Gg5Gh5Gi5Gj5Gk5Gl5Gm5Gn } frm:MandelbrotMix2 {; 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)+(p4)), 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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