FOTD -- October 15, 2005 (Rating 5) Fractal visionaries and enthusiasts: When sleep will not come, or sometimes just for fun, I like to imagine myself exploring the surface of a four-dimensional hyperplanet. I start at the zero-zero-zero point and travel west along the equator. These three coordinates are needed to mark the starting point because the surface of such a hyper- planet is a curved three-dimensional space. Since my imaginary self still has only three-dimensions, I can never see the entirety of the planet, but I can scan any chosen section, using time as the missing dimension. The equator of such a hyperplanet is a great circle around the planet, the same as it is on earth. But the equator does not separate the hyperplanet into two hemispheres as earth's equator does. It is merely a line around the surface, which may easily be bypassed completely by taking a route that does not intersect or even come near it. I move west along the equator until I have gone 90 degrees from the starting point. Then I stop. There is now a big decision to be made. I want to leave the equator to check the higher latitudes, but I must decide in which direction I wish to travel. On earth the decision would be simple. The surface is two-dimensional and, setting out at a right angle, I could go only directly north or south. But on the hyperplanet the surface is three-dimensional and there is a full 360-degree circle of directions in which I could leave the equator at a right angle and move directly toward the pole. Something is wrong. By definition, the pole is 90 degrees from the equator and if I have a choice of 360 degrees in which to move directly toward it from one point on the equator, the pole cannot be a point. This is true. On a four-dimensional planet, the pole is a great circle, which circles the planet exactly like the equator, and every point of it is 90 degrees from every point of the equator. Does this mean that frigid arctic conditions form a ring around the planet? Not necessarily. Planets rotate. The equator of our hyperplanet rotates on itself exactly as earth's equator does. But it rotates around the polar circle as its axis. As the planet rotates, every point of the polar circle remains fixed in position and rotates in place, exactly as the two points of earth's north and south poles do. Things get even more interesting. The polar circle does not just sit there, its points turning in place. It can rotate around the equatorial circle just as the equatorial circle can rotate around the polar circle. A four-dimensional planet can be subject to two independent rotations at the same time, and planets being planets, if rotation is possible, it will almost certainly be happening. In my mind, this raises the question of the path the sun would take in the three-dimensional firmament of the hyperplanet. The points of a 4-D hypersphere subject to double rotation trace out paths that are impossible to visualize in 3-D space, but some- what resemble a curved helix with its ends joined into a circle. This is where I now stand. Would the equator of our 4-D planet be a circle of heat, while the polar circle is a circle of cold? Maybe or maybe not. The sun would travel a kind of spiral path in the sky, which at first would seem chaotic, but after a while a pattern would become evident. At times, the sun's path might or might not take it away from the equator and high over the polar regions, warming them and giving the entire planet a temperate climate. Right now I am not sure, but I expect many more nights of good sleep trying to find the answer. Oh, and before I forget, we have another fractal for today. It was created with an old formula written back in 1998 by Jay Hill (I wonder what happened to him). Since the image is part of a distorted Mandelbrot set, I named it "Mandel Variation-4". Its extreme magnitude made the mathtolerance entry necessary in the parameter file. The image rather resembles a scene far out on the negative stem of the M-set when the bailout radius is set to 4. Due to its appearance of familiarity, I could rate the image no higher than a 5, but the render time of under one minute means that little time will be lost in rendering it. Those who do not render can find the finished image on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> An unexpected amount of sun and an unexpected high temperature of 73F 23C kept the fractal cats' mood in the good range on Friday. They enjoyed several hours in the holly thicket, watching the birds fly by. Today is starting even better. The cats will be overjoyed. I expect my day to be acceptable, but it will be even better if the next fractal turns out well. The next fractal will appear in 24 hours. Until then, take care, and listen to the message of the fractals. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER FILE======================================= Mandel_Variation-4 { ; time=0:00:56.08--SF5 on a P200 reset=2004 type=formula formulafile=jim.frm formulaname=MixUp passes=b center-mag=-1.963594365754721/0/1.044777e+013/0 params=1/0/2/0/0.0700000002/0 float=y maxiter=500 inside=255 outside=real symmetry=xaxis periodicity=10 mathtolerance=0.05/1 colors=000h9tFkmKnlOqkStjWviU`ZSGPQEOODOMBNKANJ6GJ\ 29O3FS4KW5Q_6Vc7`g8ek9ko9ppQ_peJogKnhLmiMljNkkOjlP\ imQceNYYLSQJMIHGAFJBKMCQPDWSE`VFfXGkSDjNAiI8iD5h82\ g40gD9dLHaTPZaYWieTqmRbqWPu`BxdGt_KqVOmRSjMWfI_cDc\ `9KB8gT7hP6jM5lI4mE3oA2p71jOGedV`tiarjapjanjaljajj\ ahjafjbVdcK_c9VVLTNWRFfPN_RUTSaMThGUfMWeRYdW_ba`af\ b`kd_peXfZVYTSRNQNHRKBQI5R78TBAVFCWJEZNH`RJcVLeYNf\ aPgdRhgTijVjmXmoSppOrrJusFwwBzzJpsRmrZjrfgrnUeVUcO\ U`HUYAUW4dWOoWfzWytYso_nj`iebd`c_WeVRfQZaPeXOlTOsO\ NzKNyTby`rrVnkPjdKfYEbR8ZK3WHDVEMUBVU8cT5lS2uS3rT4\ oT5lT6iU7fU7cUMpETqDZrDdsDjtDpuDvuDXaT8Jg7NZ6QQ6TI\ 6RH6QG5OF5NE5LD4KC4IB4HA3F93D82C72A629517416314203\ 1010KM5NV2gb3d_2aY2_W2XT2VR2SP1PM1NK1FRqB`q7kq4uqe\ XI_NXUDkP4zXFqcPhj__qiRxsJwtOwtSwuXwu`ttWqtRnsMlsH\ jcQhOYg8eZDhQHjHMl9Qn8Tl8Vk7Yj7_i6bh6dg6ffRpejzdPm\ j4apIVbWPPiJChHLhFUhDb000 } frm:MixUp {; (c) Jay Hill, 1998 ; make sure p1 <> p2 and p3 <> 0 u=p1, v=p2, w=p3, c=pixel/w, x=v-u, z=(-u/v/w)^(1/x): z=z^u + w*z^v + c |z| <= 1000 } END PARAMETER FILE=========================================