; Date: Sat, 15 Oct 2005 10:09:49 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 15-10-05 (Mandel Variation-4 [5]) ; Id: <1.5.4.16.20051015101109.29cf52f6@pop.mindspring.com> ; --------- ; ; 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: ; ; ; ; 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========================================= ; ;