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: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> 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 20.0 PAR-FORMULA 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 20.0 PAR-FORMULA FILE==================================