; Date: Tue, 03 Dec 2002 09:03:30 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 03-12-02 (Training Oblates [5]) ; Id: <1.5.4.16.20021203090542.3907452c@pop.mindspring.com> ; --------- ; FOTD -- December 03, 2002 (Rating 5) ; ; Fractal visionaries and enthusiasts: ; ; I am sometimes asked, 'where is the fourth dimension located?' ; It is not an easy question to answer. The first answer that ; comes to mind is the intuitive common-sense reply that the ; fourth dimension does not exist. It cannot exist because our ; space has only three dimensions, with no room in it for an extra ; dimension. But science has long since passed beyond the realm ; of common-sense, and such an answer can no longer be considered ; the final word. ; ; Almost 100 years ago the theory of relativity appeared on the ; scene, which treats time as a fourth dimension. But taking time ; as a fourth dimension is not very satisfying. We want an extra ; spatial dimension, a space in which time would be the fifth ; dimension. The fact that time is handled differently mathemati- ; cally is an admission that time is not the new dimension we seek. ; ; When studying the world of the very tiny, Physicists sometimes ; work with what is known as 'Hilbert Space'. This space can have ; as many dimensions as necessary, but it is more a mathematical ; abstraction than a physical reality. Hilbert space cannot serve ; as the extra dimension. ; ; In the 1980's the idea of superstrings burst upon the scene. ; These infinitesimal things are supposed to lead the way to a ; theory of almost everything. They are assumed to exist in 10 or ; 11 dimensions, with their vibrations manifesting in our space as ; the various sub-atomic particles. But the extra dimensions are ; now so tiny that they lie far beyond the range of observation. ; And even if a way were found to actually observe the hidden ; dimensions, we could do nothing more than take a measurement ; that told us that the dimensions were really there. With our ; two-dimensional retinas and three-dimensional-trained minds, we ; could never observe the extra dimensions. ; ; Yet four-dimensional objects do exist mathematically. The 4-D ; Julibrot is really there in math-space, appearing objectively ; identical to all who study it. With our computers we can slice ; it in any direction desired and study the slices. We can study ; the six perpendicular planes and the four perpendicular three- ; spaces that exist in 4-D space. Yet, despite this ability, we ; still cannot visualize the fourth dimension nor a 4-D object as ; simple as a hypercube. ; ; The limitation is apparently in our visual apparatus and minds. ; Living in a universe of perhaps infinite dimensions, we can ; visually observe only 3 of them. And this limitation assures ; that regardless of how hard we try, we shall never visualize ; even one extra spatial dimension. But if the fourth dimension ; actually did exist, in which direction would it lie? ; ; When asked this question, I reply that the way to the fourth ; dimension is sideways to the insides of solid objects. If a ; point were to start at the center of a sphere and move perpendi- ; cularly into the fourth dimension, the point would not move ; toward any point on the surface of the sphere. It would dis- ; appear at once from the center of the sphere, and move away from ; it, remaining at the same distance from all points on the ; sphere's surface. ; ; Of course, to our 3-D-trained minds living in 3-D space, this ; seems impossible, and in three-space it is impossible, but in ; four-space it is only one of the many apparently impossible ; things that can happen. I'll have much more to say about four- ; dimensional curiosities in the upcoming FOTD's. ; ; And before I forget, we do have a fractal. Today's average- ; rated fractal is a scene in the Oblate direction of the Julibrot ; created by the formula Z^(-2.5)+Z+C. I have named it "Training ; Oblates" because of the train of indescribable open areas lined ; in a straight diagonal line. Once again the straight theme ; appears in an odd slice. I will explain this curiosity also in ; a soon-to-appear FOTD. ; ; The render time of just under 4 minutes is slightly slow for an ; average image. Downloading the completed GIF file from the ; internet at: ; ; ; ; or: ; ; ; ; will be a bit more convenient. ; ; Monday was cloudy and chilly here at F.C. The temperature of ; 45F 7C was not too extreme, but for some reason the dynamic duo ; of fractal cats chose to remain indoors by the heat source. Not ; one to interrupt a good thing, I let them sleep most of the ; afternoon. ; ; Today promises to be sunnier, but also blustery and much colder. ; It will likely be another indoor day for the intrepid ones. For ; me it will be another work day, and if I hope to finish, I must ; begin. Until tomorrow, same time same place, take care, and ; always look to the fractal side. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE================================ Training_Oblates { ; time=0:03:52.72--SF5 on a P200 reset=2002 type=formula formulafile=slices.frm formulaname=OblateMix passes=1 center-mag=0.531869\ /2.53209e-014/14.49969/0.6895/-69.8080439126262604\ /59.5866633049899761 params=1/-2.5/1/1/0/-1 float=y maxiter=2500 inside=0 logmap=6 periodicity=10 colors=000U8QR7PO6OO5NO4MO3LO3ON6RN8UMAXMC_LEbLHeK\ JhKLkJNnJPqJRtRQwYPzdPzlOzsNzzNzmRm`UqcQoeNmgJliGj\ lDhn9gp6er3dp6co9bnBamEalH`kJ_jM_iPZhRYgUXfXXeZWda\ VccVadQ_eMYfIWgEVhAQnPLtcQodVkdZfdcbegYelUepPfuLfy\ HfkNbZT_LZW8dT8eV7gW7iX6kY6mZ5o_5q`5sa4ub4wc3yd3ze\ 2zf2zg2zhIzVYzHlz4hz7ezAbzD_zGXzJUzMRzPOzSLzUJzUIz\ UGzUFzUEzUCzUBzUAzU8zU7zU6zUczO`zMYzKVzISzGQzENzCK\ zAHz8Ez6Cz5Fz6Hz7Jz7Lz8Oz9Qz9SzAUzAXzBZzC`zCbzDdzD\ ezDezDezDezDezDezDezDezDezDezDezDgzHhzKizOjzRlzVmz\ YnzaozdqzhrzkszotzrezmRzhDzcCzcCzdBzdBzdAzeAze9ze9\ zeJzcTzabz`_z_XzZUzYSzXPzXMzWKzVHzUEzTBzT9zS6zR3zQ\ 1zQ7zNCzLIzINzGTzEYzBcz9hz7nz4sz2xz0wz1vz1uz1uz2tz\ 2sz2rz3rz3qz3pz4oz4oz4nz5mz5mz5iz7ez8az9ZzBVzCPzEQ\ zERzETzFUzFVzFUzETzDSzCRzBQzBPzAOz9Nz8Mz8Pz8Rz8Uz8\ Wz8Zz8`z8cz8ez8fzAgzChzEizFjzHkzJlzLmzMnzOozQpzSpz\ TqzSrzRszQszQNzVNzUNzTNzS } frm:OblateMix {; Jim Muth z=real(pixel)+flip(real(p3)), c=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================================== ; ;