December 2004 - Fractint - mailman.xmission.com

Re: [Fractint] Can't reproduce WOW image
by Jim Muth 07 Dec '04

07 Dec '04

At 11:15 AM 11/26/04 -0500, Vortex Swirling wrote: >I tried the [WOW!] par with fractint v20 and it >does the same thing, the Polynomial Degree won't >stay at 100. I wonder how Jim was able to 100? It's a mystery. On my fractal dedicated machine running DOS 6.22 I can raise the degree to 237 without problems. But it defaults back to 16 at 238. I am using the most recent developer's release of Fractint. Jim M.

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FOTD 26-11-04 (Hourglass [7])
by Jim Muth 07 Dec '04

07 Dec '04

FOTD -- November 26, 2004 (Rating 7) Fractal visionaries and enthusiasts: With today's image we return to the 'phoenix' formula, this time to the Julia version. I will not try to describe in words how this complex formula works, but the full details can be seen by striking the 'Z' key when the image is on the screen. Today's image pictures the whole parent fractal, the only change being that I rotated it -90 degrees. I gave the image its name "Hourglass" when it was perpendicular to its present orientation and actually resembled an upright hourglass. As it now stands, the hourglass has tipped onto its side. To add a bit of interest to the image I departed from my usual outside=iter inside=0 routine and set the outside to 'real' and the inside to 'bof61'. The areas of chaotic detail are 'outside'; the hourglass shape itself is 'inside. I must have done something right, for this resulting fractal rates a 7. The rating of a 7 is perhaps slightly flattering, but I did put a bit of extra work into the coloring, and I feel the rating is earned. Despite the complexity of the formula, the image is a very fast one, rendering in only 1-1/4 minutes. Combined with the rating of 7, this speed gives an overall worth of 529. This entire confusion of render time, rating, and overall worth can be avoided by downloading the pre-rendered image from the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> With a temperature of 57F 14C, Thursday was mild enough here at Fractal Central, and the showers ended before noon, but the wind blew a gale all afternoon, making the fractal cats too skittish to stay out longer than 15 minutes. They spent most of the afternoon on the shelf by the window, watching the dried leaves blowing around, turning away from the window only occasionally to wash each other behind the ears. Surprisingly, they asked for only a small treat of tuna before bedtime. Today is start- ing sunny, but it is freezing cold. I guess the cats will have very clean ears by the time the day ends. I don't know what I'm going to do for the rest of the day -- I am simply not accustomed to days with totally free time. But I might need a lot of that free time; the next formula in line is a killer. To see what kind of fractal results, check back in 24 hours. Until then, take care, and get in the big time with the big-time fractalists. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Hourglass { ; time=0:01:19.42--SF5 on a P200 reset=2004 type=phoenix passes=1 float=y center-mag=0/0/0.7961192/1/-90/-1.233735336115e-014 params=0.2109862666707358/0.9854165166173284/-3 maxiter=250000 bailout=4 inside=bof61 outside=real logmap=yes symmetry=xyaxis periodicity=10 colors=000uhAodAj`AfXAaUAYRAUOARLANJAKGAGBAD9AC5AA\ 1A90A1AAdZAcjAcwA3sCcnCciCUdCK_CAWCCSCDOCFKCGGCICC\ J8CL4CM0CP2JS4PV6VY7a`9gcBnfDthEzbBkY8_T6RR8CQAAOC\ 8NE6MG4LJ8KMBJPEISIHVLGYOF`SEbVDeYChaBkdAng9qk8tn8\ vq5qS2l28h4Ee5Kb6Q_7WW9aTAgQBlNCkOFjOHiOJhOLgOOfOQ\ fPSePUdPXcPZbP`aPbaPd_T`YXXW`UUdQShNQlJPoGNkSLgbUL\ RPSMLZHHdCDk79q2Am8AiEBfKBbQCZWCW`TXfiXlyXrz`jzdbz\ hVzlNzqFzv7zz0zzcqf2jZ3bR4TJ5MB6G46IelrNUdHOSBIF5D\ b4Kz4RuCUpJXkQ_fXbaceXjhSqkUgdWYYYORZFLSEOLDRECTFA\ QG9OG7LH6JH5HL4EO4CS49V47UKFT_NSoUQm`OlfMkmKjsIiyH\ goGefFcXFbOVSJjHEy69vCBsHDpNFnSGkYIhbKfgLifOleQodT\ rcVubYwa_gbeTbkEbqBgm8li6pfFneNldVjdbhcjfcrdbzbbxa\ avaataar``p``n``l``jWdhRhfMleHocCsa7w`3z_CmZL`YUPX\ bCXj0_1DQKUHajLVcPPXSJRWDKZ7EV7LS7SP7ZM7eJ7lQ9dXBY\ cCQjEJqGBwH4kE1fI9aMGYQOTUVOYbKaiFeqBhxGfvKetPdsTc\ qYaoa`nf_ljZkkVddcDgbCjaC } END PARAMETER FILE=========================================

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Re: [Fractint] FOTD 01-12-04 (Spider-Arches [8])
by Fr. Jim Lacasse 06 Dec '04

06 Dec '04

Jim, this is for sure an 8. Outstanding!! Fr. Jim >>> jamth(a)mindspring.com 12/01/04 09:03AM >>> FOTD -- December 01, 2004 (Rating 8) Fractal visionaries and enthusiasts: Today's striking fractal takes us to the 'spider' formula. We are coming close to the end of the list, but, as today's image shows, the goodies are not yet quite exhausted. I did little exploring in today's formula. I did not need to, as the center of interest was obvious in the default image. The perturbed versions merely twist and distort the spidery arms that give this formula its name. The rating of an 8 is due almost entirely to the coloring. I could not resist the urge to turn the scene into another endless pathway to the infinite. The reason for the name Spider-Arches is obvious. The rating of a 6 may be a bit overdone, but I rather like the strong coloring. The arched pathway may be seen, but not traversed, by running the included parameter file, or by downloading the finished GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> The clouds increased as the day progressed here at Fractal Central on Tuesday, but the rain held off until after dark, which gave the fractal cat duo a pleasant afternoon in the holly thicket. The temperature of 52F 11C was a bit chilly, but they didn't seem to mind. When evening came, only a small treat was needed to keep them happy. This morning is starting with heavy rain and high wind -- two things the cats do not like. I fear they will not have another good day. My day will be within reason, which is all I could ask for. The next fractal is only 24 hours away. Until then, take care, and did fractals exist before the big bang? Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Spider-Arches { ; time=0:01:49.19--SF5 on a P200 reset=2004 type=spider passes=1 center-mag=-0.0086\ 7751/0/19.33333/1/-90/3.88578058618805e-016 float=y params=0/0 float=y maxiter=255 bailout=9 inside=bof60 outside=tdis logmap=yes periodicity=10 colors=000mtzafpSXfPPZLJRP8PU7LY6Ib5Eg4Bk38nB7pI6r\ Q5ua4vi3xu2zz2uz9puGkqNgnUbm`YlgUkmTjkSijRhhQhgPge\ OfdNfbMeaLd`KdZJcYIbWHbVGaTF`SE`RH_RKZROYSRXSnMcpW\ drfftphvziklaaZUSMNRNPRORQPSQPUQQWPRXPRZPS`OTaOUcO\ UeNVfNWhNWiPRlQNnSJqTFsVBvW7xX8wY9vZ9v_Au`BtaBtbCs\ cDrdDreEqfFpgFpdHqbIr`JrZLsXMtVNtTPuRQvPRvNTwLUxJV\ xSYv``tibrrepzgnxhmvhmthmrilpilnilljkjjkhjkjegk`dl\ XamSYnNVoJSkMSgPScRS_USWXSSZSOaSLcSLbVLaYLa_L`bL`e\ L_gL_jLZmLZoMbiMfcMiYMmSMpNPpLSpKVpJYpI`pHcpGfpEip\ DlpCopBrpAtp9uoAunBumBulCukCujDujDviEvhFvgFvfGveGv\ dHvdHnfIfhI_iISkILlIMkHMjHMiHMiHMhGMgGMgGMfGMeFMeF\ MdFMcFMcEMbEMaEMaEVZObWYkTgsQqoQskQujTvjVwiXwiZxh`\ xhbyhdyffxegwdhvcivbju`kt_lsZmsYnrXoqVppUqpTroSsnR\ tmQumOtlMtkKskIsjHsjFriDriBrh9qg8qg6qf4pf2pe1pe5lf\ 9hfDegGagKZhOVhSRhVOiZKibHjfDjiAjkCklElnGloImqKmrM\ nsOnrhOleXgbdb`lcbmdcnedo } END PARAMETER FILE========================================= _______________________________________________ Fractint mailing list Fractint(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint

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FOTD 05-12-04 (Bluebells [7])
by Jim Muth 05 Dec '04

05 Dec '04

FOTD -- December 05, 2004 (Rating 7) Fractal visionaries and enthusiasts: Today's fractal takes us on to the 'tim's error' formula, a formula that resulted when a coding error unintentionally produced an unexpectedly good result. The formula does not really come to life until the parameters are changed from 0,0, and the inside is set to an active fill. To create today's image, I set the function to 'abs', the parameters to 0.5,0.5, and the inside fill to 'bof61'. The book "Fractal Creations" tells that a pterodactyl is hiding somewhere in the formula. I found no prehistoric flying reptile, but I did find a lovely garden. This garden is growing fantastic flowers, which I have named "Bluebells". Yes, I know that they look more like tulips, but I have decided that they are bluebells. In addition to the flowers, the image shows a gloriously blue sky, some grass growing around the flower stems, some sandy garden soil, a puddle of water left over from a recent watering, and even a few tiny insects hovering in the grass. Such a lovely garden scene surely deserves an above-average rating. I gave it a rating of 7. The render time of 3-1/3 minutes raises this to an absolute worth of 206. Those who do not consider rendering fractals to be their cup of tea can find the image already rendered on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> It stayed quite chilly here at Fractal Central all day Saturday, but the sun was warm and the wind was light, giving the fractal cats enough time in the yard to put them in a good mood. Today is starting sunny and a bit milder. It looks like another good day for the duo. The day should be fairly good for me also. There is only one more escape-time formula to go before I decide what fractal project comes next. Until tomorrow, take care, and fractals are as real as we want them to be. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Bluebells { ; time=0:03:23.83--SF5 on a P200 reset=2004 type=tim's_error function=abs passes=1 center-mag=-1.28498/1.23687/10/1/-134.5/2.58335020\ 042466112e-014 params=0.5/0.5 float=y maxiter=255 bailout=9 inside=bof61 logmap=yes periodicity=none colors=000GRWGSWGTVHUVHVVHWVIXVIYVIZVI_VJ`UJaUJbUK\ mzKmzKmzKmzKmzKmzKmzKmzKmzKmzKmzKmzKmzKmzKmzKmzKmz\ KmzKmzKmzKmzKmzKmzKmzKmzKmzKmzKeQVfQ8ZPW`P3SPMUR1O\ Q1UP2QO2UN3SM3TL4UK4VJ5WI5XH6YG6ZF7_E7`D8aC8bB9cAS\ f98eA9dAAcABbABaAC`AD_AEUAFYAFXAGWAHVAIUAIUAJTAKSA\ LRAMQAMPANOAONAPMAQLAQKARJASIATHAUF9THATIBTJBTKCTL\ CTMDTODTPETQETRFTSFTTGTUGSWHSXISYISZJS_JS`KSbKScLS\ dLSeMSfMSgNQfHRgKShNThQUiTVjWWjZWk`XkcYlfZmi_ml`nm\ apm`nm`mm`lm_km_jm_im_hmZgmZfmZemZdmYcmXbmWamV`mT_\ mSZmSXmSWmRVmRUmRTmRSmQRmQQmQPmQOmPNmPMmPLmPKmOJmO\ ImOHmKEmMFmOGmQGmSHmUHmWImYImZJm`JmPKmPKmPLmPLmPMm\ PJmPLmPMmPNmPOmPPmPQmPRmPSmPUmPVmPWmPXmQYmQZmR_mR`\ mSalSclTdkTekUfkUgjUhjUiiUjiUkiXlgYlfZmd_mc`maan`b\ nZcoYdoWeoVfpTgpShpQiqPjqNkrMlrKmrJnsHosGpsFqrGqrH\ qqIqqJqpKqpLqpMqoNroOrnPrnPrnQrmRrmSrlTrlUsmVtmXum\ XvmXwmXwmWwmWwmWwmWwmWwmW } END PARAMETER FILE=========================================

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General slicing formulae for quaternion based Mandelbrot and Julia sets-- and request for math validity check
by Hiram Berry 05 Dec '04

05 Dec '04

A little while ago Jim Muth commented on his disappointment in the results he had in the "quat" formula built in to Fractint, suggesting that not enough degrees of freedom were usable to produce all the 2-D unequivalent slicing views of the set in an R4 space, and pointing out that the analogous Julia formula had more degrees of freedom in choice of parameters. He went on to relate that he used his own formula for exploring the hypercomplex equivalent, which he later posted. I outlined a possible algorithm to exhaustively and uniquely specify all the quat slices of this nature, and have now attempted to implement it for quaternion Julia and Mandelbrot sets. I would appreciate it if those reading would check the procedure for these formulae and and see if I have made errors in writing them; there are settings which do what Jim talked about before, ie. looking either identical to the well known complex M-set or a slice through a rotation of it, there are other views very unlike such an object, and there is a strange behavior for a certain parameter arrangement. So please take a look at this and comment, and point out any obvious errors I've made in the implementation. The algorithm for producing the slices is this: choose an arbitrary vector in R4 specified by p1 and p2 (as four dimensional spherical coordinates); establish a basis in the 3-space orthogonal to the vector with the vector head as origin; read a 3-D spherical coordinate direction from p3; choose the plane normal to this direction passing through the 3-space origin as the slicing plane and figure 2 orthogonal unit vectors on that plane; then the formula generates on each pixel pass a quaternion defined as the origin plus a linear combination of the unit vectors with coefficients as in the parts of the "pixel" variable. That quaternion is either the value of C, for M-sets, or the starting value of z, for Julia sets. The strange behavior is that when a zero value is entered for the length of the vector rho in real(p1), inside points are plotted properly but when the length of rho is nonzero, even very small, an outside color is plotted in the very same region. It doesn't seem reasonable to me that a tiny change in the origin of the slicing plane would always cause the iterations to diverge, and only where the inside points used to be. So this is strange. Also I would appreciate it if someone would check that I used the correct coefficients in going from the 4-space to the 3-space. Thanks. Here are the formulae, and 3 pars for the QM-set which if valid seem to show that it is not simply a rotation of the complex M-set-- the stretching seems too complicated to me at small scale to be due to rotation alone, but that may simply be an artifact of the slicing. Comments are welcome. ---Hiram frm: MandelQuatAnySlice{ ; the quaternion M-set sliced by arbitrary plane in R4 space. Fractint z-value at each iteration is ; z calculated to be the projection of the quaternion value onto that plane (to allow orbital images to be viewed through the ; passes=o option). Plane specified by R4 vector spherical coordinates and ; a normal direction in the 3-space orthogonal to the vector: ; p1=(rho(real),omega(-90..90degrees,4-space orientation angle,"ternitude")), ; p2=(phi4(-90..90deg,"longitude"),theta4(-180..180deg),"latitude"), ; p3=(phi3(-90..90deg,normal long),theta3(-180..180deg,normal lat)), ; p4=first two components of the quaternion perturbation ; p5=last two components of the quaternion perturbation ; z-value each iteration is the projection of the quat onto the plane IF(initialized==0) initialized=1 bailout=64 degtorad=pi/180 pert_xy=p4,pert_zw=p5 rho=real(p1),omega4=imag(p1)*degtorad,phi4=real(p2)*degtorad,theta4=imag(p2)*degtorad phi3=real(p3)*degtorad,theta3=imag(p3)*degtorad Pnxy=cos(omega4)*cos(phi4)*(cos(theta4)+flip(sin(theta4))) ; unit vector in direction of point Pnzw=cos(omega4)*sin(phi4)+flip(sin(omega4)) x4=real(Pnxy),y4=imag(Pnxy),z4=real(Pnzw),w4=imag(Pnzw) IF( w4==0 && z4==0) ; then 4-comp expression of basis in the 3-space must use x4 and y4 instead Bx4uxy= flip(conj(Pnxy)), Bx4uzw= 0 ; unnormalized basis x-unit vector in the orthogonal 3-space By4uxy= w4*Pnxy, By4uzw= flip( -|Pnxy| ) Bz4uxy= |Pnxy|*(z4*Pnxy), Bz4uzw= |Pnxy|*(-sqr(w4)-|Pnxy|+flip(z4*w4)) ELSE ; use w4 and z4 for the basis Bx4uxy= 0, Bx4uzw= flip(conj(Pnzw)), By4uxy= flip( -|Pnzw| ), By4uzw= y4*Pnzw Bz4uxy= |Pnzw|*(-sqr(y4)-|Pnzw|+flip(x4*y4)), Bz4uzw= |Pnzw|*(x4*Pnzw) ENDIF x3=cos(phi3)*cos(theta3),y3=cos(phi3)*sin(theta3),z3=sin(phi3) ; direction cosines in the 3-space IF( z3==0 && y3==0) ; set plane's basis in the 3-space Bx3uxy=z3, Bx3uz=-x3 By3uxy=x3*y3-flip(sqr(x3)+sqr(z3)),By3uz=y3*z3 ELSE Bx3uxy=flip(-z3),Bx3uz=y3 By3uxy= -sqr(y3) -sqr(z3) + flip(x3*y3), By3uz=x3*z3 ENDIF len_x4=sqrt(|Bx4uxy|+|Bx4uzw|),len_y4=sqrt(|By4uxy|+|By4uzw|),len_z4=sqrt(|Bz4uxy|+|Bz4uzw|) len_x3=sqrt(|Bx3uxy|+|Bx3uz|),len_y3=sqrt(|By3uxy|+|By3uz|) Bx4xy=Bx4uxy/len_x4,Bx4zw=Bx4uzw/len_x4 ; normalization By4xy=By4uxy/len_y4,By4zw=By4uzw/len_y4 Bz4xy=Bz4uxy/len_z4,Bz4zw=Bz4uzw/len_z4 Bx3xy=Bx3uxy/len_x3,Bx3z=Bx3uz/len_x3 By3xy=By3uxy/len_y3,By3z=By3uz/len_y3 ; sum the contributions of each 3-space component to each 4-space component of plane's unit axes: Bxx=real(Bx3xy)*real(Bx4xy)+imag(Bx3xy)*real(By4xy)+Bx3z*real(Bz4xy) ; quat components of Bx unit vector of Bxy=real(Bx3xy)*imag(Bx4xy)+imag(Bx3xy)*imag(By4xy)+Bx3z*imag(Bz4xy) ; planar variation Bxz=real(Bx3xy)*real(Bx4zw)+imag(Bx3xy)*real(By4zw)+Bx3z*real(Bz4zw) Bxw=real(Bx3xy)*imag(Bx4zw)+imag(Bx3xy)*imag(By4zw)+Bx3z*imag(Bz4zw) Byx=real(By3xy)*real(Bx4xy)+imag(By3xy)*real(By4xy)+By3z*real(Bz4xy) ; quat components of By unit vector of Byy=real(By3xy)*imag(Bx4xy)+imag(By3xy)*imag(By4xy)+By3z*imag(Bz4xy) ; planar variation Byz=real(By3xy)*real(Bx4zw)+imag(By3xy)*real(By4zw)+By3z*real(Bz4zw) Byw=real(By3xy)*imag(Bx4zw)+imag(By3xy)*imag(By4zw)+By3z*imag(Bz4zw) Pxy=rho*Pnxy, Pzw=rho*Pnzw ; quat components of the planar origin ENDIF Q_xy=pert_xy,Q_zw=pert_zw ; corresponds to z in usual Mandelbrot image C_xy=Pxy+real(pixel)*Bxx+imag(pixel)*Byx+flip(real(pixel)*Bxy+imag(pixel)*Byy) ; corresponds to usual C C_zw=Pzw+real(pixel)*Bxz+imag(pixel)*Byz+flip(real(pixel)*Bxw+imag(pixel)*Byw) : Q_xy=sqr(Q_xy)-|Q_zw|+C_xy, Q_zw=2*real(Q_xy)*Q_zw+C_zw ; iteration Q->Q^2+C z=real(Q_xy)*Bxx+imag(Q_xy)*Bxy+real(Q_zw)*Bxz+imag(Q_zw)*Bxw ; dot product with planar x-direction z=z+flip(real(Q_xy)*Byx+imag(X_xy)*Byy+real(Q_zw)*Byz+imag(Q_zw)*Byw) ; dot product with planar y-direction |Q_xy|+|Q_zw| <= bailout } frm: JuliaQuatAnySlice{ ; the quaternion J-set sliced by arbitrary plane in R4 space. Fractint z-value at each iteration is ; z calculated to be the projection of the quaternion value onto that plane (to allow orbital images to be viewed through the ; passes=o option). Plane specified by R4 vector spherical coordinates and ; a normal direction in the 3-space orthogonal to the vector: ; p1=(rho(real),omega(-90..90degrees,4-space orientation angle,"ternitude")), ; p2=(phi4(-90..90deg,"longitude"),theta4(-180..180deg),"latitude"), ; p3=(phi3(-90..90deg,normal long),theta3(-180..180deg,normal lat)), ; p4=first two components of the "C" quaternion for the set ; p5=last two components of the "C" quaternion for the set ; z-value each iteration is the projection of the quat onto the plane IF(initialized==0) initialized=1 IF(real(p4)==0) bailout=64 ELSE bailout=real(p4) ENDIF degtorad=pi/180 rho=real(p1),omega4=imag(p1)*degtorad,phi4=real(p2)*degtorad,theta4=imag(p2)*degtorad phi3=real(p3)*degtorad,theta3=imag(p3)*degtorad Pnxy=cos(omega4)*cos(phi4)*(cos(theta4)+flip(sin(theta4))) ; unit vector in direction of point Pnzw=cos(omega4)*sin(phi4)+flip(sin(omega4)) x4=real(Pnxy),y4=imag(Pnxy),z4=real(Pnzw),w4=imag(Pnzw) IF( w4==0 && z4==0) ; then 4-comp expression of basis in the 3-space must use x4 and y4 instead Bx4uxy= flip(conj(Pnxy)), Bx4uzw= 0 ; unnormalized basis x-unit vector in the orthogonal 3-space By4uxy= w4*Pnxy, By4uzw= flip( -|Pnxy| ) Bz4uxy= |Pnxy|*(z4*Pnxy), Bz4uzw= |Pnxy|*(-sqr(w4)-|Pnxy|+flip(z4*w4)) ELSE ; use w4 and z4 for the basis Bx4uxy= 0, Bx4uzw= flip(conj(Pnzw)), By4uxy= flip( -|Pnzw| ), By4uzw= y4*Pnzw Bz4uxy= |Pnzw|*(-sqr(y4)-|Pnzw|+flip(x4*y4)), Bz4uzw= |Pnzw|*(x4*Pnzw) ENDIF x3=cos(phi3)*cos(theta3),y3=cos(phi3)*sin(theta3),z3=sin(phi3) ; direction cosines in the 3-space IF( z3==0 && y3==0) ; set plane's basis in the 3-space Bx3uxy=z3, Bx3uz=-x3 By3uxy=x3*y3-flip(sqr(x3)+sqr(z3)),By3uz=y3*z3 ELSE Bx3uxy=flip(-z3),Bx3uz=y3 By3uxy= -sqr(y3) -sqr(z3) + flip(x3*y3), By3uz=x3*z3 ENDIF len_x4=sqrt(|Bx4uxy|+|Bx4uzw|),len_y4=sqrt(|By4uxy|+|By4uzw|),len_z4=sqrt(|Bz4uxy|+|Bz4uzw|) len_x3=sqrt(|Bx3uxy|+|Bx3uz|),len_y3=sqrt(|By3uxy|+|By3uz|) Bx4xy=Bx4uxy/len_x4,Bx4zw=Bx4uzw/len_x4 ; normalization By4xy=By4uxy/len_y4,By4zw=By4uzw/len_y4 Bz4xy=Bz4uxy/len_z4,Bz4zw=Bz4uzw/len_z4 Bx3xy=Bx3uxy/len_x3,Bx3z=Bx3uz/len_x3 By3xy=By3uxy/len_y3,By3z=By3uz/len_y3 ; sum the contributions of each 3-space component to each 4-space component of plane's unit axes: Bxx=real(Bx3xy)*real(Bx4xy)+imag(Bx3xy)*real(By4xy)+Bx3z*real(Bz4xy) ; quat components of Bx unit vector of Bxy=real(Bx3xy)*imag(Bx4xy)+imag(Bx3xy)*imag(By4xy)+Bx3z*imag(Bz4xy) ; planar variation Bxz=real(Bx3xy)*real(Bx4zw)+imag(Bx3xy)*real(By4zw)+Bx3z*real(Bz4zw) Bxw=real(Bx3xy)*imag(Bx4zw)+imag(Bx3xy)*imag(By4zw)+Bx3z*imag(Bz4zw) Byx=real(By3xy)*real(Bx4xy)+imag(By3xy)*real(By4xy)+By3z*real(Bz4xy) ; quat components of By unit vector of Byy=real(By3xy)*imag(Bx4xy)+imag(By3xy)*imag(By4xy)+By3z*imag(Bz4xy) ; planar variation Byz=real(By3xy)*real(Bx4zw)+imag(By3xy)*real(By4zw)+By3z*real(Bz4zw) Byw=real(By3xy)*imag(Bx4zw)+imag(By3xy)*imag(By4zw)+By3z*imag(Bz4zw) Pxy=rho*Pnxy, Pzw=rho*Pnzw ; quat components of the planar origin C_xy=p4,C_zw=p5 ENDIF Q_xy=Pxy+real(pixel)*Bxx+imag(pixel)*Byx+flip(real(pixel)*Bxy+imag(pixel)*Byy) ; corresponds to usual C Q_zw=Pzw+real(pixel)*Bxz+imag(pixel)*Byz+flip(real(pixel)*Bxw+imag(pixel)*Byw) : Q_xy=sqr(Q_xy)-|Q_zw|+C_xy, Q_zw=2*real(Q_xy)*Q_zw+C_zw ; iteration Q->Q^2+C z=real(Q_xy)*Bxx+imag(Q_xy)*Bxy+real(Q_zw)*Bxz+imag(Q_zw)*Bxw ; dot product with planar x-direction z=z+flip(real(Q_xy)*Byx+imag(X_xy)*Byy+real(Q_zw)*Byz+imag(Q_zw)*Byw) ; dot product with planar y-direction |Q_xy|+|Q_zw| <= bailout } par: NotAMinibrot { reset=2004 type=formula formulafile=quat.frm formulaname=mandelquatanyslice passes=1 center-mag=-0.85477129473239550/+0.40709301723382000/266.6654 params=0/0/0/90/15/-60/0/0/0/0 float=y maxiter=15000 inside=0 outside=summ periodicity=0 rseed=28455 cyclerange=2/253 colors=000Gfrww0<3>wY0wR0wK0<3>UCU<3>Y6jZ4n_2ra0w<14>P8tO8tN9t<3>JBtHBtG\ CsECsCCs<2>CIsDJsDJs<44>girgirhjrhjrikr<9>qqtqrtrrt<3>uuuvvvvwv<2>yyvzzw\ zyq<3>xuRwtKwtK<3>wsKwsKwsKvrK<3>uiHugGudFtbE<6>sOCsMCsKC<2>sECsCCqCC<2>\ kCCiCCiCC<39>fClfCmfCnfCofCp<4>cFrbGraHr`Ir_Jr_Js<17>JasIbsHcsGdsFes<3>D\ ksCmsCosCqsCss<10>CsY } par: AParticularView { ; of the QM-set, using outside=summ to show the probabilit ; y that the distortion of the M-set in the slice is far ; more complicated than likely achievable with a simple ; rotation. reset=2004 type=formula formulafile=quat.frm formulaname=MandelQuatAnySlice passes=1 center-mag=0/0/0.6666667 params=0/0/75/135/-135/-24/0/0 float=y maxiter=600 inside=0 outside=summ logmap=yes periodicity=0 rseed=28455 cyclerange=2/253 colors=zzz000ww0<3>wY0wR0wK0<3>UCU<3>Y6jZ4n_2ra0w<16>N9tM9tLAtKAtJBt<2>E\ CsCCsCEsCGsCIs<41>cfrdfregregrfhrgirgir<3>jkrjlsklp<3>ppaqpZsqVtrSusOwtK\ <17>wtKwsKwsK<2>wsKvrKvpK<3>ugGudFtbEt`D<5>sOCsMCsKC<2>sECsCCqCC<2>kCCiC\ CiCC<40>fCmfCnfCofCpfCq<10>YLsXMsWNs<14>IbsHcsGdsFesEfs<3>CmsCosCqsCss<1\ 0>CsY } par: ExtremeStretching { ; happens in parts of this slice of the Quaternion M-set reset=2004 type=formula formulafile=quat.frm formulaname=mandelquatanyslice passes=1 center-mag=-0.484191/-0.633079/0.3625693 params=0/0/0/90/15/-60/0/0/0/0 float=y maxiter=1200 inside=0 outside=summ periodicity=0 rseed=28455 cyclerange=2/253 colors=000GfrVXe<2>B5O<12>9_b9ac9de<2>9kh8nj8nj<42>gGIhGHiFH<2>lCElCElDE\ <54>bsNbtNbuNbvNawO<50>rffsegseg<2>tdhtdhsdg<31>kjZkjZkjZ<3>ikXd_P<3>LjR\ GmSGmS<3>HmTHmUHmUImVlte<5>nwwhnqaekCs_CsY }

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FOTD 25-11-04 (Analemma [7])
by Jim Muth 05 Dec '04

05 Dec '04

FOTD -- November 25, 2004 (Rating 7) Fractal visionaries and enthusiasts: Today's image takes us to the next formula in line, the 'newton' formula, which works the same as the 'newtbasin' formula except that it colors the images according to the number of iterations it takes for the iterated points to reach their roots. For some arbitrary reason, I named the image "Analemma". An analemma is the figure-8 graph usually seen in the Pacific Ocean areas of world globes. At a single glance it shows the sun's declination and the equation of time for each day of the year. But there is little in today's image that resembles an analemma. The word simply seemed to fit the mood of the moment when I thought of it. The image itself resembles a solar eclipse more than anything else. Or some might see a steering wheel at the center. It is an unusual picture when compared to my standard routine of an unending string of images with decorated Mandelbrot midgets at the center. But since starting the tour of the Fractint formulas, I have done many images with no midgets, so maybe today's image is not so unusual after all. I created the picture by setting the single parameter to a rather high value of 240 and rendering the scene with the outside set to 'mult'. Apparently just for the fun of it, the 'mult' option colors the pixels by multiplying the iteration count of escaping points by their real divided by maginary values at the time they escape. The render time on my ancient worn-out machine is a nominal 12-3/4 minutes. This joins with the rating of 7 to give an overall worth of 55. For those who do not render, the image is available for download on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Those who download from the web site can ignore the overall worth. An off-and-on light rain fell all day Wednesday here at Fractal Central. The rain did not amount to much, but it was enough to keep things wet and keep the fractal cats confined indoors. Luckily we were well stocked with tuna. Today is starting partly cloudy and quite mild, but showers are in the area, and the wind is rising. The cats have yet to pass judgement on the conditions. In my department the work is caught up and I have little to do for the rest of the day but take it easy. So far I have heard nothing about being invited to any dinners. Unless something changes, this will leave me plenty of time to find the next fractal. Until that next fractal appears, which will happen right here in 24 hours, take care, and then take further care. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Analemma { ; time=0:12:47.53--SF5 on a P200 reset=2004 type=newton passes=1 center-mag=0/0/0.1\ 261867 params=240 float=y maxiter=7500 bailout=9 inside=0 outside=mult logmap=yes symmetry=origin periodicity=10 colors=000000zzzzzzzm4zm6zm7zm8tKgrMhqNhpPhoQhnShl\ TikVijWiiYihZif`jeajdcjcdjbfj`gk_ikZjkYlkXmkYljZkj\ Zjj_ij_hj`gjagjafjbejbdjccjcbjdajeaje`jf_jfZigYigX\ ihXiiWiiVijUijTikSikRilRimQimPinOinNioMisKjqLipLio\ MinMhmNhlNhkOgjOgiPghPggQffQfeRfdRecSebSe`Te_TdZUd\ YUdXVcWVcVWcUWcTXbSXbRYbQYaPZaOZaN_`M_`K``J``Ia_Ha\ _Gb_FbZEcZDcZCdZBdYAeY9eY8fX7fX6gX3iW4hX4hX5gX5gX6\ fY6fY7eY7eY8dY8dZ9cZ9cZAbZAb_Aa_Ba_B`_C`_C_`D_`DZ`\ EZ`EYaFYaFXaGXaGWaGWbHVbHVbIUbIUcJTcJTcKScKScLRdLR\ dMQdMQdKSbLRcMQdMQeNPeNPfOOgOOhPNhPNiQMjQMkRLkRLlS\ KmSKnTJnTJoUIpXHtWIrVIqUIpTInSImRIlQIjPIiOIhNIfMIe\ LIdKIbKIaJI`IIZHIYGIXFIVEIUDITCIRBIQAIP9IN8IM8ILAH\ NCGPEFRGETIDVJCWLBYNA_P9aR8cP4fS7dUAbWC`YF__HYbKWd\ NVfPThSRkTSjURjURjUQjUQjUQjUPjUPjVOjVOjVOjVNjVNjVN\ jVMjWMjWLjWLjWLjWKjWKjWKjXJjXJkYKlZKm_Kn`KoaKpbKqc\ KrdKseKtfKugKvhKwiKxjKykK } END PARAMETER FILE=========================================

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FOTD 04-12-04 (Hidden Helmet [5])
by Jim Muth 04 Dec '04

04 Dec '04

FOTD -- December 04, 2004 (Rating 5) Fractal visionaries and enthusiasts: Today's not-so-glorious image was created by the 'tetrate' formula, a formula even more simple (if possible) than the Mandelbrot formula. The iterated expression is simply C^Z, but even this simple expression creates a fractal. The fractals it creates consist mostly of large flat blocks, so to add interest I rendered today's image with the inside set to 'atan' and the outside set to 'real'. In the resulting fractal I found a scene that vaguely reminded me of the top part of a helmet or ceremonial headdress peeking over a hill. Since the bottom of the helmet, along with the head wearing it, is hidden, I named the picture "Hidden Helmet". There is really nothing in the image that could be rated above average. The mathematical interest is lacking, the coloring is smooth but unexceptional, the composition is nothing out of the ordinary. The rating of 5 is as good as I could do with the image. Luckily, the calculation time of a brief 1-1/2 minutes means that not much time will be spent rendering it. The overall worth is 314. The rendering can be avoided entirely by downloading the image from the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Mostly sunny but cold weather here at Fractal Central on Friday, with a temperature of 46F 8C, limited the fractal cats' outdoor time to under an hour -- not nearly enough to keep them conten- ted. An extra treat of tuna was needed to make up for the yard time they missed. Today is starting much the same. More tuna may be needed before the day ends. I expect a relatively calm day -- that is unless something unex- pected happens. Until the next FOTD fractal appears here in 24 hours, take care, and try to prove that the world is real. If you fail, try to prove that it is unreal. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Hidden_Helmet { ; time=0:01:35.62--SF5 on a P200 reset=2004 type=tetrate passes=1 periodicity=10 center-mag=2.8193/-1.30897/1.577268/1/-98.75/1.310\ 75705844807544e-014 params=0/0 float=y maxiter=2500 bailout=1600 inside=atan outside=real logmap=yes colors=000L5KN8MRBOVEQYHS`KVcNTdQTdSTeTXfVYhXZjZ_l\ ``nbbqdcrfdshctjaskaqkbpjboibmhblgbjfchecgdcfccebc\ dadc`da_d`ZcZYbYXaWWaVV`TU`ST_RSZPRZNQZMPZKOYJNXHM\ WGLVEKUDJSBIRAIQ9ST8aJ7jAOwKNxMMxOLxQKxSJxTIxVHxXG\ xZFy_EyaDycCyeByfAyh9yj8yl7ym9vjBthDqfFocHmaJj_LhY\ MeVOcTQaRSZPUXMWUKYSIZQGYPIYOJXNLXMMXLOWKPWJRWISVH\ UVGVVGXUFYUE_TD`TCbTBcSAeS9fS8hR7iR6kR6lP8jN9iLAgJ\ BfICdGEcEFaCG`AHZ9IYAKZBLZBMZCNZDPZDQZERZFSZFUZGVZ\ HWZHXZIZZI_ZJ`ZKaZKcZLdZMeZMfZNhZOiZOjZPkZQmZQnZRo\ ZRpZTo_Uo_Vo_Xo`Yn`Zn``naanabnadmaembfmbhmbilcjlcl\ lcmldnldpkdqkdrketkeujevjfxjfyjfzjftfdocbj`adX__UY\ VRXPNVKKTFHSGJRGKQGLQHMPHNPHOOHPNIQNIRMITMJULJVKJW\ KJXJKYJKZIK_IK`HLaGLcGLdFMeFMfEMgDMhDNiCNjCNkBNlBM\ hBKeAJaAIZ9HV9GS8FP8ERDETHEVMEXQEZVE`ZEbbDdgDfkDhp\ DjtDlyDnzDpzCozCozBozBozBozAoyAowAov9nu9ns8nr8nq8n\ o7nn7nm7nlS_akMRjNUjNWjNY } END PARAMETER FILE=========================================

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FOTD 03-12-04 (Flutterby Fractal [5])
by Jim Muth 03 Dec '04

03 Dec '04

FOTD -- December 03, 2004 (Rating 5) Fractal visionaries and enthusiasts: Moving onward from yesterday's 'sqr(1/fn)' formula, we reach the 'sqr(fn)' formula, which is a slightly altered version of the first formula. Actually, both formulas seem a bit redundant. A formula such as 'sqr(fn(fn))' would have done everything both formulas do, and far more. But the 'sqr(fn)' formula is there, it is an escape-time formula, and it does create fractals. Like yesterday's formula, the only variable is the function. The heavy exploring must be done by tweaking things such as the inside and outside coloring options and the bailout radius. For today's image I set the function to tangent, the inside to 'bof61', the outside to 'imag', and the bailout to a very large 10000. The result is an oversized fractal filled with things that remind me of butterflies. But they are not exactly butterflies, so I thought of them as 'flutterbys' and named the image "Flutterby Fractal". As its name would imply, the image features a large butterfly- like feature that fills the frame. To intensify the effect of butterfly wings, the background colors have been reduced to pastel shades, and the edges of the wings colored a brilliant yellow. When all this is done, the resulting image is average. Like all average images, it rates an average 5. The fast render time of 53 seconds gives the image an overall worth of 566. And as always, those who do not render can find the image finished and waiting on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> But before going there, give Paul a chance to post the image file. Thursday was a far more pleasant day here at Fractal Central. The warm sun and temperature of 50F 10C were just what the duo wanted. And the wind, which calmed to a mere 10mph 16kph, made it all the more pleasant. The duo spent over three midday hours in the yard, doing little but sitting still, watching the world go by. Today is starting much the same. I expect to have happy cats when the day ends. Work is waiting, so I had best get busy. Until next time and next fractal, take care, and win with a top fractal. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Flutterby_Fractal { ; time=0:00:53.01--SF5 on a P200 reset=2004 type=sqr(fn) function=tan passes=1 center-mag=-1.05105000119388900/+1.605975348682925\ 00/1800.071 float=y maxiter=255 bailout=10000 inside=bof61 outside=imag logmap=yes periodicity=10 colors=000AFE1U2DZ3Oc4_h5jm5ik9hjDhiHggKgfOfeSecVe\ bZdabd`eOJOQMRROTSQVTSXmUZmW`mYbm_dmafmchmejmglain\ ZdpW_qzuXQQtNLvLHwURia_XihKqq7liHgbQbV_ZOheS`lVUsZ\ NyaGyXHzSHzOHzJIzEIzAIzDJzGKzJKzMLsPMrSMqVNpYOo`On\ cPmfQliQklRjoRHXHFSKDNMBIPADRBFPCGNDILEJJFLHGMFHOD\ IPCHSDHUEHXFHZGHaHGcHGfIGhJGkKGmLFpLFrMFuNFwOFwPFw\ PDwVCu`Bqe9nk8jq7gvA`mDUeGNYJGPM9HP39I6EC9J6CO9GMB\ KLDOKvWGZw0PgAhLnZNcPPTb_kSwVFlHFaI`VkYXlW_mTbnRdo\ OgpMjqKlrVifegVpeJp_GpVEpQCpLApG8pB6p64kA6gE7cI9_M\ AWQCRUDNYFJaGFeIBiJ_YIxNHwQLvSOvVRuXUtZXta_scbrefr\ hiqjlploporoquosxnoemkNlg4kh6ji8jiAijChkEhkFjjGliG\ niHphHrhHriGrjGrkFrkFrlFrmErnErnEroDrpDrpDgnFXlGMj\ IBhJ1gK4iL7jL9lMCmMFoNHpNKqONsOPtPSvPVwQXwQ_sUboYe\ j`hfdkahnYkqTosPrnQoiRldRi_SfWScZSbaSacS`fS`iS_kSZ\ nSYpSYhcXaoXZmYWkZTiZQg_Ne_Kc`Hb`E`aBZa8Xb5Vb2Tc0S\ c1Rd1Qd2Pd2Od3Nd3Md4Ld4Kd } END PARAMETER FILE=========================================

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Can't reproduce WOW image
by Vortex Swirling 03 Dec '04

03 Dec '04

I've tried to generate the WOW ftod image and don't seem to get the same image as shown on the ftod page. What is even weirder is that at SF5 it generates one image and at SF6 it generates another. I'm using use Fractint 20.04 under XP. I placed the images on the web for comparison. At SF5: http://rogerkaufman.mystarband.net/WOW.SF5.gif At SF6: http://rogerkaufman.mystarband.net/WOW.SF6.gif -- No virus found in this outgoing message. Checked by AVG Anti-Virus. Version: 7.0.289 / Virus Database: 265.4.2 - Release Date: 11/24/2004

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orbits.par file
by Jonathan Osuch 02 Dec '04

02 Dec '04

The orbits.par file has some examples of what the new passes=o feature can do. For those of you missing the orbits.par file, which would be anybody that didn't download the dif file, here it is: demonic_moose { ; Jonathan Osuch josuch(a)fractint.org reset=2004 type=tetrate passes=o center-mag=0.325467/4.44089e-16/0.5788618/1/-90/3.88578058618804789e-16 params=0/0 float=y inside=0 outside=summ orbitdelay=30 orbitinterval=5 colors=000FFF<24>p33r33s22<2>x11z00z10<24>zn0zp0zr0<2>zx0zz0zz1<24>zznzz\ pzzr<2>zzxzzzzzz<57>zX5zX4zW3zW2zV1<27>z50z40z30z20z10z00<28>f00e00d00<3\ 1>FFF } space_debris { ; Jonathan Osuch josuch(a)fractint.org reset=2004 type=mandelfn function=sin passes=o center-mag=-3.9968e-15/3.10862e-15/0.3360215 params=0/0 float=y inside=0 outside=summ orbitdelay=40 colors=000FFF<24>p33r33s22<2>x11z00z10<24>zn0zp0zr0<2>zx0zz0zz1<24>zznzz\ pzzr<2>zzxzzzzzz<57>zX5zX4zW3zW2zV1<27>z50z40z30z20z10z00<28>f00e00d00<3\ 1>FFF } solar_flares { ; Jonathan Osuch josuch(a)fractint.org reset=2004 type=fn(z)+fn(pix) function=recip/sqr passes=o center-mag=-1.55431e-15/1.11022e-15/0.613196 params=0/0/1/0 float=y inside=0 outside=summ orbitdelay=50 orbitinterval=7 colors=000FFF<24>p33r33s22<2>x11z00z10<24>zn0zp0zr0<2>zx0zz0zz1<24>zznzz\ pzzr<2>zzxzzzzzz<57>zX5zX4zW3zW2zV1<27>z50z40z30z20z10z00<28>f00e00d00<3\ 1>FFF } spaghetti { ; Jonathan Osuch josuch(a)fractint.org reset=2004 type=mandel passes=o center-mag=-0.116813/0.0273794/2.73224 params=0/0 float=y inside=0 colors=00000z<3>00i00a<3>00L00H00D<2>000<10>d0Lg0Nk0P<3>z0Xv0Vr0Tp0Sn0R<\ 6>O0CK0AG08<3>000<9>X0a_0dc0h<3>p0w<9>K0NH0JD0F<3>000<9>bbbfffjjj<3>zzz<\ 9>OOOKKKGGG<3>000L00<8>k00<4>z00<2>p00l00h00d00a00<3>N00J00F00<3>000<9>b\ L0fN0jP0<3>zX0<9>OC0KA0G80<3>000<9>__0cb0gf0<3>ut0<9>ML0II0FE0<3>000GA4<\ 7>7c26g25k2<2>2v10z00w0<8>0O00K00G0<3>000<14>00v }

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