; Date: Fri, 17 Jan 2003 10:43:47 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 17-01-03 (Four_Esses [5]) ; Id: <1.5.4.16.20030117104635.2a07d376@pop.mindspring.com> ; --------- ; FOTD -- January 17, 2003 (Rating 5) ; ; Fractal visionaries and enthusiasts: ; ; Though it appears quite weird, today's image was not created ; with the M-Mix4 formula. It is actually part of the Z^1.93+C ; fractal, which, when the multi-valued nature of the complex ; logarithm is taken into account, is infinite in surface area. ; In today's case, we have examined the fractal as it appears 57 ; levels down the log spiral. ; ; In fractals like today's, the strangest and most interesting ; things are found along the negative X-axis, where the major ; split usually exists. This is especially true in the parent of ; today's image, where things that I would never expect are ; happening along the X-axis split. ; ; There is a fairly normal almost-quadratic midget at the center ; of today's image, though it lies beyond the limit of resolution. ; In fact the image itself is so close to the break-up point that ; I included a mathtolerance=/1 entry in the parameter file to be ; sure the image renders at the correct magnitude. ; ; The image consists of a mixture of inside and outside material. ; It was rendered with the inside set to < fmod >. It is this ; inside coloring that creates the effect of concentric rings. ; ; I named the image "Four Eesses". There are four almost-perfect ; letter 'S's in the image. I rated it at a 5. It takes more ; than a few embedded letters to make an above-average image. ; ; The render time of 17 minutes can be avoided by downloading the ; pre-rendered GIF image from: ; ; ; ; or from: ; ; ; ; Now let's get to constructing that 4-dimensional 24-cell. To ; keep things simple, we'll start at the two-dimensional level. ; It's best to have a glass-topped light table, straight edge, and ; X-acto knife for this kind of dissection and construction. I ; did a lot of this kind of play stuff back in the slow times of ; the good old days of the 70's and early 80's, when I worked at a ; light table, cutting and pasting by hand. But since light ; tables are expensive and not everyone has one, the same thing ; can be done with paper and scissors. ; ; As I said in yesterday's FOTD, the 24-cell is the only regular ; 4-dimensional polytope that has no analog in 3-space. It does ; however have a corresponding figure in 3-space that is similarly ; constructed -- the rhombic dodecahedron. ; ; To understand the construction of the rhombic dodecahedron we ; can start in 2-space, with two equal paper squares. Lay the two ; squares side by side on the table top. Then take one square and ; cut it along its diagonals, resulting in four right triangles, ; whose bases are the edges of the square and apexes the center. ; Now take the four triangles and attach their bases to the four ; edges of the other square. The result is a larger square, ; rotated 45 degrees, with twice the area and 1.4142 times the ; width. Not a very impressive construction, but it is a start. ; ; The next step will be a three-dimensional one -- more difficult ; to perform, but relatively simple to visualize. Lay two equal ; cubes on the table. Take one cube and slice it into six square ; pyramids, whose bases are the faces of the cube and apexes the ; center. Then take the six pyramids and attach their bases to ; the six faces of the other cube. The resulting polyhedron is a ; rhombic dodecahedron. It is not a regular figure because its ; rhombic faces are not regular, and more importantly, because it ; has two different kinds of vertices -- those where 3 edges meet ; and those where 4 edges meet. ; ; The final step -- the construction of the 24-cell -- cannot be ; visualized, but the analogy is so clear that it can be followed. ; To begin, we need an extra dimension. Assuming we have found ; the extra dimension, we can get started. Lay two equal 4-D ; hypercubes on the hypertable. Take one hypercube and slice it ; into eight cubical hyperpyramids, whose bases are the cells of ; the hypercube and apexes the center. Then take the eight hyper- ; pyramids and attach their bases to the eight cells of the other ; hypercube. The resulting figure is the 24-cell. 4-space is the ; only space higher than 2 in which this construction produces a ; regular figure. In the higher spaces, there are too many lower- ; dimensional parts to the figure, which must be shaped and fit ; together just right. ; ; It appears I'm getting so involved in hyperspace that I almost ; forgot about the Fractal Central weather. ; ; The weather was cold again here at Fractal Central on Thursday, ; with a temperature of 28F -2C and only a hazy sun. The cats ; chose comfort over adventure. They decided to spend the day ; indoors. This morning is once again cold, and to make matters ; worse, about 3cm of fresh snow fell overnight. The cats, who ; dislike cold wet paws, will not be happy about this. ; ; I have a minor pile of work sitting on the shelf to my right, ; but unlike the fractal duo, I can do more than be unhappy about ; the situation. So until the next FOTD appears almost by magic ; in 24 hours, take care, and be moderately happy. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE================================ Four_Esses { ; time=0:17:30.18--SF5 on a p200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotBC1 function=floor passes=1 center-mag=-0.4282456346546068/+0.0113573879900267\ 6/5.48016e+012/1/-132.5/0.00936043190115933704 params=1.93/0/-57/0 float=y mathtolerance=/1 maxiter=1200 inside=fmod periodicity=10 colors=000_U2bS2cR2eP2hR2gP2eN2eM2cM2cL2bJ2bI2`I2_\ G2_F2YD2YD2XC2XB2VB2U92U83S74S74R56R47P28N28N1AM0B\ M0BL0CL0EJ0FI0FI0HG0IG0KF0KF0LD0MD0MD0TD5ZF9fDCdDD\ bCGbCIaBLaBM_BPZ9RZ9UX8VX8YW7_W7bU7cT5gT5hR4kR4mQ2\ qQ2rO2vM1wM1zL0zL0zK0zK0zK7z25z25w25t25r25o25k24j2\ 4g24c24b24_24X22V22S22P22N32L42I62G71G70F70F60D60D\ 60C60C20C20C600A00B00C00E00F00F00H01I01K01L02K02K0\ 2L04M04M04L05M05M05N05M07N07P07Q08S08S08T19V19X19Y\ 2BY2B_2BY4CW4CY4Cd5Df5Dg5Di5Di4Ck4Ck4Ck2Ck2Au2Ak2C\ k1Ck1Bl1Bl1Bl0Bl0Bl0Bl0Bl0Ba0Bl09n09n0Sn09n0Sv0Qn0\ 9n09n09n08d08k08o08o08o08o08o08o07o08p09n09n0Bk0Ck\ 1Ch4Dg5Fe7Fe8GbBIaCIZDJXGLUILTJMQMNMNNLV14U24S44R5\ 4P54N73M83L93J93IB3IC2GD2FD2DF2CT27E29I28J27L25M20\ M25T2S`FSlWbqUloUqmTrkTtjTthRvgRveQwcQybQy`Oz_OzYO\ zXMzVMzULzSLzRLzPKzNKzLMzMKzMIzNHzNEwNCtPBqP8oP7kR\ 6hR4eR2cS2RY3SY2VX2XV2YU2 } frm:MandelbrotBC1 { ; by several Fractint users e=p1, a=imag(p2)+100 p=real(p2)+PI q=2*PI*fn1(p/(2*PI)) r=real(p2)-q Z=C=Pixel: Z=log(Z) IF(imag(Z)>r) Z=Z+flip(2*PI) ENDIF Z=exp(e*(Z+flip(q)))+C |Z|