; Date: Thu, 15 Apr 2004 10:52:20 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 15-04-04 (Land of Fractals [5]) ; Id: <1.5.4.16.20040415105540.0d6f424c@pop.mindspring.com> ; --------- ; ; FOTD -- April 15, 2004 (Rating 5) ; ; Fractal visionaries and enthusiasts: ; ; Today's image takes us into the fourth dimension. I have always ; been fascinated by the fourth dimension. I was not yet a ; teen-ager when, having just finished a book, (I think its name ; was "One, Two, Three, Infinity..."), I asked my father where the ; fourth dimension is, and, thinking of relativity, he told me ; that it is in the past and the future. This was not a very ; satisfactory answer. The past and future are time, not space. ; I wanted to know where the fourth dimension of space is. He ; then explained to me that as far as we know, there is no fourth ; spatial dimension, and even if there were, we would perceive it ; in three-dimensional terms. ; ; This is the problem. We are brain-washed to perceive our ; surroundings as a three-dimensional world. We have not been ; wired to perceive a four-dimensional space, and it is too late ; to change that. If we were suddenly dropped into a four- ; dimensional world, we would not see the fourth dimension. We ; would see a three-dimensional world that obeyed different rules. ; Solid objects would change shape as we watched; objects would ; appear and vanish. We would think the world had gone crazy. We ; would probably go crazy ourselves, (as some say I already have), ; but we would be no closer to visualizing the fourth dimension. ; ; Even now, we are able to fully visualize and understand the ; shadows that would be cast by four-dimensional objects onto the ; three-dimensional surface of our space. A 4-D hypersphere would ; cast a spherical shadow if the ray of light fell perpendicularly ; onto our space. If the light struck our space at an angle, we ; would see an oblate spheroid, the eccentricity of which would ; vary with the angle of the ray of light. But even though we ; understand all of this, we are still totally unable to visualize ; the hypersphere that would be casting the shadow. ; ; We can also know and fully visualize the three-dimensional ; slices of four-dimensional objects. The 4-D hypercube may be ; sliced to give many curious prisms, as well as a tetrahedron and ; an octahedron, and of course the expected cube. But even this ; is of no help in our quest to realize the fourth dimension. ; ; Our lack of visualizing ability is curious. We can program our ; computers to 'think' in four dimensions almost as easily as in ; three. We could even give them three-dimensional screens on ; which to better display the 4-D worlds they would have no ; trouble in manipulating. Of course, we would then be unable to ; view the 3-D screens in the proper manner, from outside our ; space. We would of necessity need to view the screens ; 'edge-on'. We appear to be forever doomed, trapped in a world ; of three dimensions, while able to know of the far greater ; vistas that would lie before us if only we could rewire our ; minds. ; ; Since our minds are not about to be rewired, it's best we turn ; to today's image. ; ; Today's image is a two-dimensional slice of a four-dimensional ; object. In this case, the object is the 4-D Julibrot created by ; iterating the formula (1/Z)-Z+C. I have sliced the Julibrot in ; what I call the Oblate direction, which is determined by the ; imaginary C and real Z axes. ; ; The picture has a certain surreal quality about it, which ; inspired the above outburst about higher dimensions. It could ; almost be something we might see if we were suddenly dropped ; into a four-dimensional space. This is why I named the picture ; "Land of Fractals". ; ; Since the image is such a departure from my usual fractal fare, ; I was not sure how to rate it. I finally decided on a rating of ; an average 5. The short render time of 2-1/2 minutes raises the ; overall value to a mighty 202. ; ; An alternate way of seeing the other-worldly scene is to ; download it from the FOTD web site at: ; ; ; ; Light rain fell steadily all day Wednesday here at Fractal ; Central, keeping the fractal duo confined to the indoors. Even ; if it had not been raining, the temperature of 50F 10C would ; have been a bit chilly for their sensitive ears. Today is ; starting windy and chilly, but sunny. We'll see how the ; dynamic fractal cats take these conditions. ; ; For me it's work before play. The problem comes in separating ; the two. Until tomorrow's FOTD, take care, and most likely it's ; fractals all the way down. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE================================ Land_of_Fractals { ; time=0:02:28.69--SF5 on a P200 reset=2003 type=formula formulafile=allinone.frm formulaname=OblateMix2 function=ident passes=1 center-mag=-4.55271/3.10114/0.6584367/8.9718/-11.2\ 262659607652502/29.222244310548934 params=1/-1/-1/1/1/-0.2 float=y maxiter=1200 inside=255 logmap=yes periodicity=10 colors=000Di`Ei_FiZFhZGhYGhXHhWHhWHhWU_`fRfuHlsIkr\ JjpKjmLiUdkUckUbjUaiU`hU_gUZfUYeUXdUWcUWcUVbUVbUUa\ TUaST`QT`RS_PS_NRZMRZKQYJQYHPXGPXENWDLWBIVAGV8EU7C\ U8ET9MTANSBOSBPRCQRDRQETQFUPFWPGYOH_NIaMIcLJeKKgJL\ iIMkIMmINoIOqIPsINsHPtIQtIStITuJVuJWuJYvKZvK`vKawL\ cwLdwLfwLgxMixMjxMlyNmyNoyNpzOrzOszOtzOuvPusPupPum\ PvjPvgPvdPvaPubRtcTscVrdXqdZpe_oeanfcnfemgglgikhjj\ hliinhipgjrfhygjshlninhjpckrZltTmvOnxJozDpz8pz3lxE\ hrPdm_`gjYcvXbtXbsXbrXbqWbpWboWbnWbmVbkVbjVbiVbhUb\ gUbfUbeUbdUbcW`bY_bZZb`YbaWbcVbeUbfTbhRbiQbkPblObj\ NahNafM`dM`bL_`L_ZKZXKZVJYTJYRIXPIXNHWLHWJGVHGVFFU\ DFUBET9ET7DS5DS8GTBITELTGNTJQTMSTOVURXUU_UWaUZdUaf\ UgkVchU_fUXdTTbTQ`SMYSIWRFURBSQ2UJ8QQEMWKIbQEjPFhP\ GgPHfPIeOJdOKbOLaOL`NM_NNZNOYNPWNQVMRUMRTMSSMTQLUP\ LVOLWNLXMHUMLXLP_KTbJXeJ_gIcjHgmHkpGosFstCruFrvIrw\ LqxNqyQqzTqzWpzYpz`pzccwz } frm:OblateMix2 {; Jim Muth z=real(pixel)+flip(real(p3)), c=fn1(flip(imag(pixel)))+imag(p3), a=real(p1), b=imag(p1), d=real(p2), f=imag(p2): z=(a*(z^b))+(d*(z^f))+c, |z| <= 100 } ; END PARAMETER FILE================================== ; ;