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: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> 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 20.0 PAR-FORMULA 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 20.0 PAR-FORMULA FILE==================================