FOTD -- October 19, 2004 (Rating 6) Fractal visionaries and enthusiasts: Today we turn to the Halley formula of Fractint, which is one of my favorite formulas. It converges on roots of polynomials much like the Newton's formulas, but with much more control over the appearance of the images. The 'epsilon' parameter controls how close to the actual roots the points must come before calcula- tion is stopped. The 'relaxation' parameter has nothing to do with relaxing after a hard day's work. It controls the amount of chaos in the image. A value above one allows more chaos, while less than one keeps the chaos under tighter control. An imaginary value, such as I used in today's image, twists the entire scene. The colors are determined by the number of iterations needed to reach the point where the change is less than the value assigned 'epsilon'. The smaller this value, the more iterations will be needed, and the greater the opportunity for chaos to sneak in. Today's image uses a value of an extremely small 10^(-15), which brings out all potential chaos. In addition to total chaos between roots, some settings of the Halley parameters produce total chaos along the borders between areas of different numbers of iterations to the same root. This is the case in the purplish areas of today's image, where magni- fication will reveal that no two adjoining areas actually touch. Regardless of the magnitude, there is always chaos along the 'borders'. Though it usually consists of only two colors, this chaos is interesting in itself. Sometimes, as in today's image, it is totally chaotic, while at other times, unexpected geome- tric patterns appear. I am not sure whether this chaos is part of the formula or an artifact of the math routine, but it is curious nevertheless. One of the early versions of today's image was colored with a turquoise theme, and resembled an undersea scene, with air bubbles rising toward the surface. I named it "The Rage of Poseidon". (If I remember my mythology correctly, Poseidon is the god of the sea, who really had it in for Ulysses for blinding his cyclops son.) I have since totally changed the color scheme to something more like a view down into the pit of hell, but I have kept the angry oceanic name. I would have rated the image at a 7, but I have a certain aversion to spirals, of which today's image is a perfect exam- ple, so I could rate it no higher than a 6. When the brief render time of just under 4 minutes is considered, the overall worth equals a humdrum 153. The image may be rendered from the included parameter file or downloaded, already completed, from the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> The weather remained uneventful here at Fractal Central on Monday. The partly sunny skies and temperature of 59F 15C were barely good enough to meet the quality standards of the fractal cats, who spent less than an hour lounging testily in the yard. Things will be no better for the duo today, since it is starting wet and chilly. They will not like this. The next FOTD is but 24 hours away. Until then, take care, and don't surrender to fractal terrorism. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER FILE======================================= Rage_of_Poseidon { ; time=0:03:54.98--SF5 on a P200 reset=2003 type=halley passes=1 center-mag=14.1471\ /8.14196/0.03333333 params=12/0.9/1e-015/0.59 float=y maxiter=95 bailout=9 symmetry=none inside=bof61 logmap=yes periodicity=0 colors=000pfC_lPZkPXjQWhQVgQUeRTdRJeWScR_aMg`IeYLd\ VNcSQbPSaMV`JX_G_ZDaYAdX7fX2jW4hW6gW8fWAeWCdWEcVGb\ VIaVK`VM_VOZVQYVRXXSWZTV`UUbVTdWSfXRhYQjZQl_Pn`Opa\ NrbMtcLvdKxeJzeJxgIvhHtiGsjFrkEwlAzmKznUzoczpmzqwz\ zzzzzzzzzzzzzzzzwzzmzzczwUzrLzmMzhMzcNzcNzcNwYOtKO\ qEPnAOkANhANcAM_1LX4OW6QW8SXBVXEXYHZZLaZOc_Re_Vh`Y\ kaanafrbjudnzbox`owZovXotWosUorSoqQooOonNomLokJojH\ oiEqiGohHngJmfKleMjdNicPhbQgbSfaTd`Vc_WbZYaYZ_X`ZW\ aYWcXVdWUfUTgTSiSRjRQkQQiRPgRPeRPdRPbRP`RO_ROYROWR\ OVROTRNRRNQRNORNMRNLRNOSQRSSUSVXSX_S_bSa_U`XV`VW`S\ X`QY`NZ`L``Ia`Gb`Dc`Bd`8e`6f`5f`4fa3fa2fa2fa5g`7g_\ AhZChYEiXHiWJjWMjVOjUQkTTkSVlRYlRyqH_mQamPdnOfnNin\ MkoMmoLppKrpJuqIwqHfW7gX8gX8gX8gX8gY8hY8hY8hY8hY8h\ Z8hZ9iZ9iZ9iZ9i_9i_9j_9j_9j_9j`9j`9j`Ak`Ak`AkaAkaA\ kaAkaAlaAlbAlbAlbAlbBmbBmcBmcBmcBmcBmdBndBndBndBnd\ CneCneCoeCoeCoeCofCofCpfC } END PARAMETER FILE=========================================