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: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> 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 20.0 PAR-FORMULA 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|<a } END 20.0 PAR-FORMULA FILE==================================