|
Fractal of the Day
by Jim Muth
|
|
Fractal visionaries and enthusiasts: The day turned out surprisingly slow, giving me a chance to ponder away most of the afternoon. The subject of my pontifications was, not surprisingly, mathematical instead of metaphysical -- the fourth dimension in fact. (To avoid the pontification, heaven knows why, skip to the description of the fractal, which follows the four-dimensional stuff.) The fourth dimension is not an absurdity, but a useful mathematical concept with a well-developed geometry involving no contradictions. However, to gain even a partial understanding of its nature, we must resort to analogy with dimensions of a lower order. An aggregate is said to be one, two, or three-dimensional according to whether one, two, or three numbers are necessary to determine the position of any one of its elements. Considering space as an aggregate of points, a line is a one-dimensional space, because to determine the position of any point on it, one number, giving its distance from some fixed point, is sufficient. Similarly, a plane is a two-dimensional space, and the point aggregate of ordinary space is three-dimensional. Now, if we have four variable, related quantities, each capable of assuming, independently of the others, every possible numerical value, we obtain a four-dimensional aggregate. Such an aggregate, if of points, constitutes a four-dimensional space. If we connect all points of our 3-space with an assumed point outside of it, then the aggregate of all the points of the connecting lines constitutes a 4-space, or hyperspace. Again, just as a point moving generates a line, a line moving outside itself generates a plane, and a plane moving outside itself generates a solid, so a solid moving outside itself generates a hypersolid. Any space is that which forms the boundary between two portions of the next higher space, and just as a plane divides our 3-space into two separate portions, so a 3-space divides 4-space into two separate portions between which that 3-space forms an infinitely thin boundary in the fourth dimension. Just as objects in our 3-space are bounded by flat or curved two-dimensional surfaces, so objects of hyperspace are bounded by hypersurfaces, that is flat or curved three-dimensional spaces. Hyperspace contains not only an infinity of flat 3-spaces like ours, but also an infinity of curved 3-spaces or hypersurfaces of different types. A hypersphere, for instance, is a closed hypersurface, all the points of which are equally distant from its center. Freedom of movement is greater in hyperspace than in our space. The degrees of freedom of a rigid body in our space are 6, namely, 3 translations along and 3 rotations about 3 axes, while the fixing of 3 of its points can prevent all movement. In hyperspace however, even with 3 of its points fixed, the body could still rotate about the plane passing through those 3 points. In hyperspace, a rigid body has 10 possible movements, namely, 4 translations along 4 axes, and 6 rotations about 6 planes, while at least 4 of its points, not all in the same plane, must be fixed to prevent all movement. In our 3-space two movements of rotation will combine into a single resultant rotation, similar to its component rotations, except that the direction of the axis is different. In hyperspace however there is no resultant for two rotations, and a body subject to two rotations is in a totally different condition from that which it is in when subject to only one rotation. When subject to one rotation, a whole plane of the rotating body is stationary, the points of the plane turning on themselves while remaining in position. When subject to a double rotation, no part of the body is stationary except the single point containing the two planes of rotation. If the two rotations are equal, every point in the body except that one describes a circle. If the rotations are unequal, the points of the body describe something like a donut. In hyperspace, a sphere if flexible could without tearing and with only slight stretching be turned inside out. Two rings of a chain would fall apart at once, but a chain could be made of alternate rings and hollow spheres. Our 3-space knots would be useless. Just as in our space a point can pass in and out of a circle without touching its circumference, so in hyperspace a body could pass in and out of a sphere or other closed space without going through its surface. In short, all of our space, including the interior of the densest solids is open to inspection and manipulation from the fourth dimension, which extends in unimaginable directions from every point of our space. If one were to ask the direction of hyperspace, the best answer could be that it is sideways to our insides. All this is curious, but of what use is the conception of 4-dimensional space? For one thing, it gives a deeper insight into geometry. Thus, a circle, considered as a one-dimensional aggregate of points, has very few properties, while in a plane it has a center, radii, tangents, etc., and in 3-space has further geometrical relations with the sphere, cone, etc. Similarly, the properties of any given line or surface increase in number when investigated in hyperspace. Also, just as it requires a 3-space to include certain one-dimensional aggregates such as the helix, so in hyperspace previously unknown lines and surfaces such as the pseudosphere become possible. Just as the comprehension of plane geometry is enlarged by viewing plane figures from 3-space, so is solid geometry illuminated by the geometry of hyperspace. Finally, the concept of four-dimensional hyperspace effects a complete divorce between geometric space and real space, no longer considered necessarily identical, and in other ways also enlarges our mental horizon. Now that our horizon has been enlarged, it's time to get to today's fractal, which shows a midget in the Z^1.618034+C Mandeloid. The midget is surrounded by so many discontinuities that I named the picture "Discontinuities". Also, to add a bit of tension to the scene, I moved the midget off-center. Since I used most of my energy writing the discussion, the fractal suffered. But with its frenetic filaments and bizarre breaks it still rates a 4, which is only slightly below average. The 7-minute render time of the parameter file is slow enough to make most fractaliers go to: The fractal weather today was sunny and chilly, with a temperature of 53F (11.5C) that was cold enough to keep the fractal cats confined to the sunshine of the back porch. Although my mathematical philosophy flourished today, my metaphysical philosophy still slept. I'll try again tomorrow to dredge up some wisdom for the deprived masses of fractosophy fans, and one of these days I'll actually have something notable to say. Right now, all I can say is that it's time to shut down the fractal shoppe, settle into my TV chair, and try to avoid getting sucked into the election confusion. I'll put on a junky old sci-fi movie to keep me distracted. Until tomorrow, when I'll return with another fractal and more words, take care, and the less one thinks about the fourth dimension, the happier they'll be. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Discontinuities { ; time=0:07:14.46 -- SF5 on a P200
reset=2001 type=formula formulafile=branchct.frm
formulaname=MandelbrotBC passes=1 center-mag=-0.8602\
1090723480100/-0.09223104404366109/84935.56/1/-75
params=1.618034/0/-3/0 float=y maxiter=25000 inside=0
logmap=120 periodicity=10
colors=000M65<3>J64I64H63<2>E63D63C42<3>8848947A4<49\
>ZpH_qH_rH<2>auIauIbtJ<40>oNgpMgpLh<2>qIjqIjrHmrGosF\
qvDx<10>gG`fGZdGX<3>_HPaJP<3>XGLWFKVEJ<2>SCGQBERCF<6\
9>rqisrjssj<3>vvmwwmwwm<21>wwm
}
frm:MandelbrotBC = { ; Z = Z^E + C
e=p1
p=real(p2)+PI
q=2*PI*trunc(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| < 100
}
END 20.0 PAR-FORMULA FILE==================================
|
times.
