January 2003 - Fractint - mailman.xmission.com

Re: [Fractint] Triternions
by Russ Walsmith 29 Jan '03

29 Jan '03

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

2 1

Minibrot Chaos
by Raymond Filiatreault 28 Jan '03

28 Jan '03

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

1 0

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

27 Jan '03

1 0

FOTD 27-01-03 (Fractal Geometry [7])
by Jim Muth 27 Jan '03

27 Jan '03

FOTD -- January 27, 2003 (Rating 7) Fractal visionaries and enthusiasts: Today's pale, green-and-lilac image rates a 7. Only its somewhat washed-out appearance prevents a higher rating. I named it "Fractal Geometry" when I noticed the unusually regular geometric pattern of filaments surrounding the central midget, which in today's image takes second place to its surroundings. This pattern looks surprisingly like the arrangement of seeds in a sunflower, and can be brought out even more clearly with other color palettes. The parent fractal is a vaguely Mandel-shaped open-area, narrow on the east and filled with chains of debris. Some of this debris extends in a broad filamentary loop into the solid material west of the open area. Today's scene is located in a sub-loop on this loop. The parent came about when I subtracted 47 parts of minus- epsilon from 47 parts of minus-pi and added 1/C. The scene I found in the parent is fair enough in its normal version, but unlike yesterday's image, it is even better in its evaporated version, which, made visible by the bof61 inside fill, was used as today's FOTD. The render time of 2 minutes is a bargain for such a highly- rated image. Those who are not impressed by bargains may download the completed image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The weather started good enough here at Fractal Central on Sunday, with sun and a temperature just about freezing. But in the afternoon, clouds moved in and a brief but heavy snow shower passed over, ending the cats' chances of a pleasant time outdoors. They satisfied themselves eating and keeping warm. Today is starting much colder, making it unlikely they will have any time at all outdoors. For me it's another work day, and not surprisingly, I already have some work on the shelf. So until next time, take care, and brace yourselves for the imminent release of the FOTD CD. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Fractal_Geometry { ; time=0:02:00.34--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=-1.51658971637141400/-0.274992933413616\ 80/1.305493e+010/1/112.5/4.03474752740523579e-005 params=-47/-2.71828182845905/47/-3.14159265358979/\ 0/1e+100 float=y maxiter=1200 inside=bof61 periodicity=10 colors=000fG7gD7cJ6_P6XV6T`6Pf6Ml6Ir6Fw6Et9DrCCpEB\ nHBlKAjM9hP8fS7dU7bX6`_5Za4Xd4Vf7Wd9QcBXaDZ`G_ZI`Y\ KaWMaVOZTQWqSIbKNZklpjkpstzzzzXYeVMdTAcMKQPTMIa8G_\ AFYCEXEDVFCUHBSJARL9PM8OO7MQ6LR9NOBOMEPJGQHJRFLTCO\ UAQV8TW5VX3XY1aS7fMDkHIpBOu5Uz0Z2FqMPseYuyfvmdlbcc\ RbUGaL5`CXdcheZteUpdVmcWicXfbYbbZ_a_Xa`T`aQ_bM_cJZ\ dFZeCYf9Yf3_e5Zf7Yf9XfBXgDWgFVgHVgJUhLThNShPSiRRiT\ QiUQiSSgQUeOWcNXaLZ_J`YHbWGcUEeSCgQAiO9jM7lK5nI4oG\ 5mF5lF5kE5iE5hE5gD5fD6dC6cC6bC6aB6_B6ZA6YA6XA5VB5U\ B5TC4SC4RC4QD4PD3ND3ME3LE2KE2JF2IF2HF5O98V3AX5BZ6D\ _7Ea8Gb9HdAJfBKgCMiDNjEPlFQmGPiHOeINaJNZKMVLLRMKNN\ KKOQMSVNV_OYdP`iRdnSgsTjxUmsWkoYjk_ig`hcbgZdfVfeRg\ dNicJkbFlaMibTfc_dcfadl_dmZamZZnYWnYToXQoXNpWKpWHq\ VEqVBrU8rU5rU3dfGSsSRrTQqUPqVPpVOpWNoXMoYMnYLnZKm_\ Km_LoXLpVMqTMrRNsPNtNNuLOvJOwHPxFPyDPzBRvASrAUoAVk\ 9Wh9Yd9Za9_Y8aU8bR8cN7eK7 } 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==================================

1 0

FOTD 26-01-03 (Un-Named [3])
by JimMuth＠aol.com 26 Jan '03

26 Jan '03

FOTD -- January 26, 2003 (Rating 3) Fractal visionaries and enthusiasts: Today's image has no name. I studied it for several minutes, trying to find a name for such an unfocused image that rates a lowly 3, but came up with nothing. At one time I thought I saw a design for the cover of a hypothetical vampire novel, and considered naming the image "Dracula". Then I thought I would name it after one of those large but less repulsive fruit bats that inhabit the tropical Asian regions. Since the image has a metallic sheen about it, I also considered several metallic names. Then I remembered the invisible midget at the center, and considered a name such as "Enigma". But ultimately, all the potential names proved unsatisfactory, and I let the picture remain without a name. The generating formula takes Z^(-17), adds 10 times as much Z^(-1.7), then adds (1/C). The rendering was done with the outside set to < atan >, which is not one of my favorite options, though in this case, it works better than the basic equal-iteration bands. (I also tried evaporating the image, but found that this gives little more than chaos.) The render time of 18 minutes is a bit excessive for such a lowly image. But relief can be found on the WWW, which hope- fully is recovering from the latest hack attack. The completed GIF image can be accessed at Paul's site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal weather was milder on Saturday, but still rather cold, with a high temperature of 34F 1C. Since no one was at home most of the day, the cats passed the time sleeping by the heat. When we arrived at the door later in the evening, a loud scolding was heard from the inside. It was Thomas, who was irate because he had not been fed all afternoon. Today is a bit milder yet, and no unnecessary shopping trips are planned, so the dynamic duo should have a much-improved day. Until next FOTD, which will appear in 23 hours or so, take care, and go with the flow, never forgetting that those who make no waves accomplish nothing. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ An_UnNamed_Fractal { ; time=0:18:03.95--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+1.204807016015608/-0.01223600675442206\ /1.486447e+009/1/162.5/-1.62304540859345425e-005 params=10/-1.7/1/-17/0/525 float=y maxiter=1200 inside=0 outside=atan periodicity=10 colors=000KIJLIJLIJLHKMHKMHKNHKNHLOHLOHLNFKOGLOHLP\ HMPIMQJNQJNQKNRKORLOSMPSMPSNPTOQTOQUPRUPRVQSVRSVRS\ WSTWSTXTUXUUXUUYVVYWVZWWZXW_XX_YX_ZX`ZY`_Ya_Za`Zaa\ Zba_bb_cc`cc`ddaddadeaefbefbfgcfgcfhcgidgidhjehkei\ kfilfilfjmgjngknhkohlrimrjkohikfggdecbc_`aWZ_SXYOV\ XKSWLTWLTWLTWLTVLTVLTVMTVMTVMTUMTUMTUMTUNUTNUTNUTN\ UTNUTNUSOUSOUSOUSOUROUROURPVRPVRPVQPVQPVQPVQQVQQVP\ QVPQVPQVPQVORWORWORWORWORWNRWNSWNSWNSWMSWMSWMSWMTX\ MTXLTXLTXLTXLTXKUXKUXKUXKUXKUXJUXJVYJVYJVYJVYIVYIV\ YIWYIWYHWYHWYHWYHWYHXZGXZGXZGXZGXZFXZFYZFYZFYZFYZE\ YZEYZEZ_EZ_DZ_DZ_DZ_DZ_D__C__C__C__C__B`ZC__D__E__\ F__GZ`HZ`IZ`JZ`KZ`LYaMYaNYaOYaPYaQXbRXbSXbTXbTXbUW\ cVWcWWcXWcYVdZVd_Vd`VdaVdbUecUedUeeUefUegTfhTfiTfj\ TflUemVdjTffQhcOk_LmWIoTGrPDtLAvG8zI8xJ8vL8uM8sN8r\ P8pQ8oS8mT8kU8jW8hX8gZ8e_8d`8bb8`c8_e8Yf8Xg8Vi8Uj8\ Sl8Qm8Pn8Np8Mq8Kr8Jq9Kq9K } 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==================================

1 0

FOTD 25-01-03 (Spinning Off [5-1/2])
by Jim Muth 25 Jan '03

25 Jan '03

FOTD -- January 25, 2003 (Rating 5-1/2) Fractal visionaries and enthusiasts: In this world of multiplying triternion formulae, there is still room for the MandelbrotMix4. Today, this workhorse formula carries us to the virtual world that pops into existence when various portions of Z^(-1.618) and Z^(-11.48) are combined and (1/C) is added. This entire world is an oversized, fragmented fractal that is indescribable in shape, but has a small open area with numerous Mandeloid features. Today's scene is located in a sub-spiral of a spiral in the center of a feature that resembles a multiple Seahorse Valley more than anything else. I named the image "Spinning Off" because with its coppery fringes it reminds me of an old computer animation. (Is there any such thing as an *old* computer animation?) The original animation was of a spinning disk which disintegrates into coppery fractal dust as it spins. Upon seeing the final result of the day's effort, I stepped back and said, "this one deserves a rating of 6." Then I stepped a bit farther back and decided that the image rates only a 5. I finally decided on the rating of 5-1/2, which is infinitesimally above average. Today's image also can be evaporated and colored with the bof61 inside fill, but doing so results in a fractal a bit too chaotic to be a FOTD. The render time of 33 minutes is admittedly a bit excessive, but fractal salvation is at hand, for the completed GIF image may be downloaded 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 remained below freezing all day Friday, while the fractal cats remained by the heat sources all day. A tempera- ture of 30F -1C is too cold for fractal whiskers. The duo finally came to life in the evening, when their empty stomachs reminded them that food exists. Today is only slightly warmer, so I expect they will have a similar day. As for me, I will try to avoid an imminent shopping trip, most likely without success. Luckily, I have the patience to endure, and thoughts of fractals will carry me through. Until next FOTD, take care, and before we can find the answer, we must know the question. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Spinning_Off { ; time=0:33:53.28--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+1.34451506120471600/-0.107403741819450\ 00/5.433651e+011/1/155/-0.0660300637674467727 params=3.63/-1.618/5.94/-11.48/0/125 float=y maxiter=3600 inside=0 logmap=691 periodicity=10 colors=000UKfUIfTHeTFeSEdSCdRBcRAcUBeXBf_CgbCheDjh\ DkkElnEmmJkmOimTgmYembdnZfnWgoShoPioMjpIkpFlpCmlBj\ hAgdAda9bY8_U8XQ7UN6SJ6PF5MC5KODIZKHjRFzYEy_FxaGwc\ HveHugItiJskKrmKqoLpqMorMnqNmpOlpOkoPkoPjnQinQhmRh\ mRglSflSekTekTWhONaKEbF5YB6aA7cA8e99g9Aj8Bm8Cp7Ds7\ Ev6Ex6AaP6Lg25zB6rK7kT8da9YiARjCUkDXkEZlGamHdmIflL\ blNZkQVkSSkVOjXKj_HiaDid9if6fd5cb5``5YZ5WY5TW5QU5N\ S4KQ4IP4FN4CL49J46H44G4LJIaLWqNirMesLasKYtJUtJRuIN\ uHJvGFvGCnQRg_e`ht`Xo`Mk`BgWAZR9RM9JH8BC83EE7GKAIQ\ DKWHMaKOgNNlNQmQTnSWoVZpXaq_bv_cqadmcehefdgg`ihWki\ SmiLsiOniRiiUdiX`i_WibRieNihIpkDsr7nn9hkBgeDf`F_XG\ dZIoWK`TMbPNaMP`JR_GT_DU_EW_FY_G__Ha_Ic_Je_Jf_K`_L\ j_Mq_Nh_Oz_Pz_PsaUzcZmecjghgimdjrbgk_edXbaUNaHOaMP\ aRQaWVd__fbchehjilllqnouprnjkkkhhmeeobbq__sXYuUWwW\ UyXTzYRzZQz_Oz`NzaUzc`zdgzfnzguzhozbizXiz7czSYzMxz\ GrzHmzHhzHbzHYzHKz1Mz3Nz5 } 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==================================

1 0

Re: [Fractint] Triternions, dimensions and orbits of configurations
by Jim Muth 25 Jan '03

25 Jan '03

At 09:38 PM 1/24/03 +0100, Jos Hendriks wrote: >Dear reader > >I have read most of the things about triternions. And rendered >some of the frms/pars. A lot of things are coming into my head. >Some of them I try to write down here. I hope I will get some >comment. <snipped> These latest two triternion formulas come the closest yet to the ideal triternion M-set, which would have no foreshortening and skewing. I have a short formula that draws very similar images with far less calculating. Unfortunately, I have never figured exactly what the formula does, but I have included two parameter files for comparison with the triternion formulas. Hyper1 { reset=2002 type=formula formulafile=allinone.frm formulaname=HyperMandelbrot center-mag=0/0/0.666667 params=0/0/2/0/0.001/0 float=y maxiter=1200 inside=0 logmap=yes periodicity=0 colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\ zzz00zG0zV0zj0zz0zz0jz0Vz0Gz00zG0zV0zj0zz0jz0Vz0Gz\ 00z00zG0zV0zj0zz0jz0Vz0GzVVzbVzjVzrVzzVzzVrzVjzVbz\ VVzbVzjVzrVzzVrzVjzVbzVVzVVzbVzjVzrVzzVrzVjzVbzhhz\ lhzqhzuhzzhzzhuzhqzhlzhhzlhzqhzuhzzhuzhqzhlzhhzhhz\ lhzqhzuhzzhuzhqzhlz00S70SE0SL0SS0SS0LS0ES07S00S70S\ E0SL0SS0LS0ES07S00S00S70SE0SL0SS0LS0ES07SEESHESLES\ OESSESSEOSELSEHSEESHESLESOESSEOSELSEHSEESEESHESLES\ OESSEOSELSEHSKKSMKSOKSQKSSKSSKQSKOSKMSKKSMKSOKSQKS\ SKQSKOSKMSKKSKKSMKSOKSQKSSKQSKOSKMS00G40G80GC0GG0G\ G0CG08G04G00G40G80GC0GG0CG08G04G00G00G40G80GC0GG0C\ G08G04G88GA8GC8GE8GG8GG8EG8CG8AG88GA8GC8GE8GG8EG8C\ G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGDBGFBGGBGGBFGBD\ GBCGBBGCBGDBGFBGGBFGBDGBCGBBGBBGCBGDBGFBGGBFGBDGBC\ G000000000000000000000000 } Hyper2 { reset=2002 type=formula formulafile=allinone.frm formulaname=HyperMandelbrot center-mag=0/0/0.666667 params=0/0/0/0/0.25/0.25 float=y maxiter=1200 inside=0 logmap=yes periodicity=0 colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\ zzz00zG0zV0zj0zz0zz0jz0Vz0Gz00zG0zV0zj0zz0jz0Vz0Gz\ 00z00zG0zV0zj0zz0jz0Vz0GzVVzbVzjVzrVzzVzzVrzVjzVbz\ VVzbVzjVzrVzzVrzVjzVbzVVzVVzbVzjVzrVzzVrzVjzVbzhhz\ lhzqhzuhzzhzzhuzhqzhlzhhzlhzqhzuhzzhuzhqzhlzhhzhhz\ lhzqhzuhzzhuzhqzhlz00S70SE0SL0SS0SS0LS0ES07S00S70S\ E0SL0SS0LS0ES07S00S00S70SE0SL0SS0LS0ES07SEESHESLES\ OESSESSEOSELSEHSEESHESLESOESSEOSELSEHSEESEESHESLES\ OESSEOSELSEHSKKSMKSOKSQKSSKSSKQSKOSKMSKKSMKSOKSQKS\ SKQSKOSKMSKKSKKSMKSOKSQKSSKQSKOSKMS00G40G80GC0GG0G\ G0CG08G04G00G40G80GC0GG0CG08G04G00G00G40G80GC0GG0C\ G08G04G88GA8GC8GE8GG8GG8EG8CG8AG88GA8GC8GE8GG8EG8C\ G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGDBGFBGGBGGBFGBD\ GBCGBBGCBGDBGFBGGBFGBDGBCGBBGBBGCBGDBGFBGGBFGBDGBC\ G000000000000000000000000 } frm:HyperMandelbrot {; periodicity must be turned off a=(p1),b=(0,0): q=sqr(a)-sqr(b)+pixel, b=(p2+2)*a*b+p3, a=q, |a|+|b| <= 100 }

1 0

Fracternions
by Russell Walsmith 24 Jan '03

24 Jan '03

Im almost used to the routine by now, which is that every time I believe I have the Tset formula written down properly, I see a way to refine it. With this latest and last little lick though, I think it's done My first efforts to resolve the set in 2D, were mostly to demonstrate its existence, and hopefully inspire someone to render it in 3D. I used an HP SX48 calculator, and it was a simple matter to construct a program with the 3D vectors that are built-in. The skewed, truncated Msets that this method produced seemed a reasonable result, although not quite what Id hoped for. Then, a few weeks ago, Tim Wegner directed my attention to the Fractint list and the Morgan Owens formula posted on 12/11/02. At the time, I was still focused on a full 3D rendering, so I wasnt that excited about seeing a 2D set in Fractint. Besides, the Owens formula looked about right, but it produces images quite distinct from those on the HP, and I thus suspected that the Fractint parser wasnt up to the job. But just for drill I guess, I tweaked the formula to bring it a little closer to the HP version and voila the images Id found earlier were there. Well that must be right, I thought, reckoning that if one machine confirmed the other, how could it be otherwise but in a short while I saw that the imaginary value for c2 (which the HP's vectors had snuck in on me) was not appropriate in this case. The simple reason is that triternions are based on the order six cyclic group, and there are no 4th roots of unity in C6 (or for that matter, in the other order six group, D3). I saw that simply squaring c2 would rectify the imaginary term, and the images that this change produced were far more intricate and fun to work with. Moreover, Jim Muths flip of c1 produced a nice two-fold symmetry (and incidentally z1=t1-c1*c1 works well with that too), so it looked like wed arrived. Upon further reflection, however, it became clear that in c2*c2, the term also acts upon itself as a scalar, and that this disrupts its proportionality with c1. The proper solution then is c2*(0,1) or (better) c2=imag(pixel) *(0,1) in the definition line, to minimize calculations. There it is then: Weve watched the Tset go from embryo to fetus to now a newborn, and guess what Cuzzins, Its A Girl! You can go to p.12 of my paper, where shes posing for a picture, dressed all pretty in pink: http://fibonacci-arrays.com/Triternions.pdf Or you can render her yourself with TGirl { c1=real(pixel),c2=imag(pixel)*(0,-1),c3=p1 z1=z2=z3=0: t1=z1*z1+2*z2*z3, t2=z3*z3+2*z1*z2, t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2+c2,z3=t3+c3 z=z1+z2+z3 z < 64 } TGirl is easily persuaded to produce Julia sets: TJul { z1=real(pixel),z2=imag(pixel)*(0,1),z3=p3: t1=z1^p1+2*z2*z3, t2=z3^p1+2*z1*z2, t3=z2^p1+2*z3*z1 z1=t1+p1,z2=t2+p2,z3=t3+p3 z=z1*z2+z2*z3+z3*z1 z < 64 } Try, say, z1=-0.75 with z2 and/or z3 at or near 0.1. Looking forward to seeing where this goes Ciao, Russell _____________________________________________________________ Get 25MB, POP3, Spam Filtering with LYCOS MAIL PLUS for $19.95/year. http://login.mail.lycos.com/brandPage.shtml?pageId=plus&ref=lmtplus

2 1

FOTD 24-01-03 (Without Reason [7])
by Jim Muth 24 Jan '03

24 Jan '03

FOTD -- January 24, 2003 (Rating 7) Fractal visionaries and enthusiasts: Having checked the Fractal Central e-mail box and actually read 3 of 53 pieces of junk, I wonder why so many spammers waste their time telling me that I am receiving the information I requested, when I have requested nothing, or pretending that they are secret admirers, or that I have been approved for something I do not want, or that I have signed up to receive such junk, when only a fool would sign up to receive spam. And if I have requested information or signed up to receive spam, why don't I remember doing it? Why must the spammers tell me something I should already know? But it is not all bad. Those computerized pseudo-personalized letters are particularly amusing. I always get a chuckle when I read something like, "hey jamth, you have been specially selected to receive a free password." Maybe I should create a new e-mail address with a particularly offensive name such as "stupidpigeon(a)******.com". Then the subject lines would read, "hey stupidpigeon, you can get rich quick." Why can't e-mailers just be honest and say something such as, "you are receiving this e-mail because your address was on a list of addresses we purchased"? If someone were to say this, and the product was worthwhile, I might actually consider doing business. But now, having sounded off about the spam, I have fractal business to attend to. I named today's image "Without Reason", though I can give no logical reason why I did it. It is another of my 'evaporated' images, showing the ghost of a fractal that once was there. I put a bit of extra energy into the coloring, which earns the image a rating of 7. Why do images evaporate when their generating formulae contain negative powers of Z? They do it because no points will reach infinity when the exponent of Z is negative. But the points will still jump around quite a bit within a limited range, and when the value of the bailout radius is reasonably low, they will temporarily pass beyond it, giving the program the signal that they have escaped permanently, and stopping calculation. When the bailout is excessive, no points will ever reach it in their travels, and will therefore be seen as trapped. The screen will appear flat, consisting entirely of trapped 'inside' points. The bof61 active inside fill option works quite well in today's image. It colors the trapped points according to which iteration comes closest to the origin, and therefore displays the wanderings of the trapped points. The resulting images can be quite impressive. The other active inside fills can also produce interesting images from a normally flat blank inside. The bof60 option colors trapped points according to how close they come to the origin, and produces bubble-like images. Many of the active inside fills have quirks that limit their usefulness, but they are all worth a try. Setting the logmap can be tricky with images that consist totally of inside points. One method is to bypass the problem by setting the logmap option to zero. Another method that works fairly well is to set the logmap to 1, create a totally black palette and give the 001 register a brilliant color. Then rotate the palette forward until the color first appears on the screen. When it does, note the number of the position where the colored register is located and set the logmap to this value. Unfortunately, this method fails when the colored register passes beyond position 128. The rating of 7 for today's image seems logical. It is above average, at least in my opinion. The render time of 12 minutes borders on the excessive. The download from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> is far more reasonable. Thursday was the coldest day in 5 years here at Fractal Central. And the snow showers and high winds didn't help at all. When the 14F -10C air touched the noses of the fractal cats, they turned around and headed straight for the warmest radiator they could find. They slept peacefully until the time arrived for their evening meal. There is not much cat action around F.C. in this kind of weather. It's a little warmer this morning, but still well below freezing. I suppose the duo will be housebound again today. Being housebound is my normal situation, but it most certainly does not mean idleness. Since the time for work has arrived, the time for the FOTD has come to an end. In 24 hours, the next FOTD will appear. Until then, take care, and be faithful to your fractals. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Without_Reason { ; time=0:12:16.44--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+1.686817713361516/-0.6536317292615328/\ 9.601575e+009/1/-132.5/-5.15785767828125241e-006 params=11/-1.2/4.3/-6/0/1e+100 float=y maxiter=1200 inside=bof61 logmap=128 periodicity=10 colors=000OGRUGWMGNJGJHFLGEMFEODDPCEQBGSAIT8JU7LW6\ MX4MY3M_4N`6Oa8PbAObCN`EM`GLYIKVJJSKIPLHMMGKNFKOEK\ PDKQCKRCKSDHUFHWHHYJH_KHaMHcOHeQHhRHmTHrVHwXHyYHz_\ HzaHzcHzdHzbKyaNx_PxZSxYUyWXyV_yTaySdyRfzPizOlzMnz\ LqzKsxNqvQptTorVnpYmn`llcjjeihhhfkgdlfcmeXo_RmVKkQ\ EiLHgJKeHNcFPaDS_BVY9YW7_W6RUCIUI9TNDSQGRSJQUNOWQP\ YTQ`WRb_TdbUfeVhhWjaYdWZ_Q_UJaPDbJ7cEFcFNcFVcGbcGe\ cKgcNicQkcUmcXoc_pcaqcbrccscdtbeuZfuVgvRhwNixJjyFk\ zBlz7mwBkuEirHgpKfnNdkRbiU`fX_d_YbbW_fUYeTVdRTcPRc\ OQcRQcUPcXPc_OcbObeN`hNYkMUnMSqLPtLMwLKzKMvJNsJOoI\ PlIRhHSeHTaGUZFVVFXSEYOEZLD_HD`EEXKEZ69X0DW4GV8JUB\ MUFQTITSMWSPZRTaQXeP_hPckOfnNjqNmoMlmLlkKljJlhIlfH\ leGlcFlaEk`DkZCkXBkWAkU9kS8kR7kS8eS9`SAVSAQSBKSCFS\ D9SD4XE7`E9eFBiFDmGFrGHvHJzHLrQKjZKbfKefJhgHkhFniD\ qjBsjAtgLudWubfsdcrf`qgYpiWojTnlQmmOloLkpIjrGisDhu\ Agv8doJahU_bdXWoeGfaGbYGZ } 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==================================

1 0

Re: [Fractint] Triternions
by Russell Walsmith 24 Jan '03

24 Jan '03

-- On Tue, 21 Jan 2003 16:36:20 Morgan L. Owens wrote: >...though I still say TMan and TJul need a little prophylactic code to stop them from spilling out into six dimensions - unless that's desirable; mainly from the line z=z1+z2+z3. Right on, Morgan, you da Man! That statement and your Hilbert space reference of a while back helped me recall that a metric space of n dimensions is defined when (a^2+b^2+...n^2)^.5 measures the distance from a point in that space to the orgin. Clearly, the appropriate statement is z=(z1^2+z2^2+z3^2)^.5 I tried this in the formulas and it sharpens the images nicely... so much so that the so-called 'TGirl' now looks more like the TMan in drag... (Hey, we knew from the git-go that he might be different...) What to do? I guess we can't very well call him/her/it 'TIt', so it's back to TMan (XAXIS) { c1=real(pixel),c2=imag(pixel)*(0,1),c3=p1 z1=z2=z3=0: t1=z1*z1+2*z2*z3 t2=z3*z3+2*z1*z2 t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2+c2,z3=t3+c3 z=(z1^2+z2^2+z3^2)^.5 z < 16 } TJul (XAXIS) {;Try z1=-0.75 z1=real(pixel),z2=imag(pixel)*(0,1),z3=p3: t1=z1*z1+2*z2*z3 t2=z3*z3+2*z1*z2 t3=z2*z2+2*z3*z1 z1=t1+p1,z2=t2+p2,z3=t3+p3 z=(z1^2+z2^2+z3^2)^.5 z < 64 } Since you've already summoned the Dobiasovsky genie, Morgan, maybe you can wish him on to these formulas too... BTW, I don't mean to diss your D3Man when I tell you this (he was, after all, a sort of midwife/godfather to TMan), but he's not really based on D3. The reason is that D3, and in fact a dihedral group of any order, is non-Abelian: Therefore, for some term(s) in the number triplet, say j, the outcome of j^2 will not be equal to -j^2. A fractal formula based on D3 will thus require conditional statements in the initialization line... Now at the risk of being reviled by the list members (big deal...), I must admit that I've actually been using Winfract for most of the past 10 years. Indeed, it's only been within the last year or so that I even installed a real DOS version of Fractint, within just the past few months that I've tried diligently to work with it, and within this latest week bygone that I find a bit of a feel for it at last. There are still things I like about Winfract; e.g., with an frm file in a small window, the alt/tab toggle puts it over the image for editing. But every time I ever tried to load an imported par file, this relatively liteweight program simply choked. Eventually I despaired of the effort, flat gave it up. But in this venue (the List), par files fly around like cards at a poker table and if you can't read 'em, Maynerd, it's really hard to stay in the game. So far, in Fractint, I've actually gotten these files to render, but in Lee Skinner's "Fiery Pit' for example, the evidently crucial color info was lost in the shuffle somehow. In general, the process is still far from intuitive... Suggestions anyone? Thanks, Russ _____________________________________________________________ Get 25MB, POP3, Spam Filtering with LYCOS MAIL PLUS for $19.95/year. http://login.mail.lycos.com/brandPage.shtml?pageId=plus&ref=lmtplus

3 2