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Mathematical Dimensionality and Fractals

    Fractal geometry is the study of self-similarity, which is represented by iterative or recursive equations. The term "fractal" was first used by Benoit Mandelbrot in 1975.

"Politics, religion, philosophy, science, mathematics, economics, war and peace, and dozens of other subjects that, in the past, had been poured into his mind and into his soul, were swirling and twirling around in there like some fractal design from Chaos theory." (pg 3)

    Most people have seen some images of fractal designs even if they don't know the mathematics behind them.  Those of us who are more familiar with the term really can "see them" everywhere.

"Actually he was searching for patterns everywhere. Nature was full of symmetry and fractals and tilings and other interesting mathematics." (pg 72)

"Sam sat down on the steps of his back stoop under the American flag and caught his breath. Some leaves had already fallen and were covering the herringbone pavers with fractals. " (pg  449)

    I have given the reader a literary introduction to the subject in the section entitled "Fractals." Describing the images in words, without images and without resorting to pure mathematics, was a challenge. Hopefully, I can add the images on this webpage, but first, here is some of the section at issue without the entire story line.



... Like a bearded ĎSimon Saysí master, he instructed them to, "Look at your watch. Look to the left. Look to the right. Look up at the ceiling. Look down at the floor. Look straight ahead. Look behind you. Look at your watch." When they finished and were loosened up a bit, he continued, "Thatís about all any of you see of the universe, Samís (x, y, z, t), and unless someone has trouble with his watch, you can only travel in one direction along the t axis. Are you ready yet?" he snapped at Silvio.

Silvio was quite nervous speaking in front of this group of people, ... he couldnít stall any longer. "Iím ready, Ari."

Silvio put up his title slide, "Fractal Geometry and Hyperspace." He began nervously to repeat an obviously memorized talk, "Consider a line segment of unit length. Like this one, in figure (a). Then, in figure (b), we subdivide it into three equal parts. Next, in figure (c), we use the middle subsegment as the base of an equilateral triangle, and erase the base. You now have ... in figure (d), a polygonal line, a polyline, consisting of four segments, each with a length of 1/3. So the total length of the polyline is ... 4/3."


... "Repeat the subdivision process on each of these four segments. You now have a polyline with 16 subsegments, each of length 1/9. It looks like a little star rising from some surface, with two little bumps on either side." Sam knew Ari would skewer Silvio for that description.

"Let the diagrams speak for themselves. Leave the poetry out!" Sam heard Ariís voice.

Silvio said, "Letís start over. Go back to the beginning ... and consider the original unit length line segment. Take three of these and form an equilateral triangle. Subdivide each of the sides of the triangle, as we did before with the line segment. You should have ...", and he switched slides again, "... a six pointed star, with twelve edges. Repeat the process of subdivision, ..." ... "... and you get a much more complicated looking object. Like a snowflake. If we repeat the process indefinitely, we get what mathematicians call a Koch snowflake." ... "If you look at the flat version, it sort of looks like clouds on the horizon or vegetation."


Silvio took a deep breath. ... "One interesting property of this curve is derived from considering the length of the curve. Let the function, L( ), represent the length of the curve. Then L(1) = 1, means that the length of the curve with no subdivisions is 1 unit. L(1/3) = (1/3) * 4, calculates the length of the curve with a single subdivision. In general, L(1/3) = (1/3)* 1/ (1/3)D ... D is the dimension of the curve, and in this case it is 1.2618... . That is, the curve is of fractional dimension! It is also infinite in length, continuous, and nowhere smooth, so it canít be drawn. But it exists! And if you think of the snowflake version, we have just described an infinitely long, folded string, fitting inside a finite circle of area Pi/3."

"Now shrink the original radius ... you have a string inside every point! A very complicated universe indeed!" said the Asian man. Ari turned the lights back up, and Silvio, sweating slightly, packed up the overhead projector and collected his slides.

...  (pgs 114-117)



    Dr. Abu BinBob has his apprentices, scattered around the world, trying to decipher the alien message.  Obviously, they must be competent in fractal geometry to do so.

""Dad turned him on to Escher and fractal designs, now all he sends relates to fractals," Julie said." (pg 128)

Just then Abu returned from his elsewhere, and showed he was at least partly here. "They use addition modulo two, and a pseudo random-number generator based on a numerical feedback loop just as fractals are based on a geometric feedback loop." And then he was gone again, gazing at the honey he was dripping onto a biscuit. The rest of them looked at each other and laughed. (pg 129)

    Dr. Jennifer Beseret's lecture is about Catastrophe Theory, and I don't need to include more description of fractals and Chaos Theory at this point.  Assume you were required to have taken that course before you signed up for her class.

"Jen responded politely but firmly, "You are confused. Chaos and Catastrophe may sound alike to you, and both fields of mathematics are theories about dynamic systems, but Chaos deals with self-similarity and fractal geometry. Catastrophe is topological. You are familiar with Topology of course?" she moved to stand directly in front of him as she asked. " (pg 139)

    Dr. Ari Bosronov, however, never holds back on any mathematics.

"Fractals and hypercubes were flying all over the board, developing into chaos theory and catastrophe theory." (pg 121)

    After all the genetic engineering, I thought it only appropriate to place some real-life fractal plants into my landscape. Fractals are often used to depict foliage in computer graphics applications, so I bio-engineered them for planting by Oyun in Hotan near the Golden Jade.

"The fractalies plants were rippling in the wind all down the right side of the road. They were twice as tall as when they were planted last year, so Jenís experiment must have worked. They had genetically modified one of the desert bushes to improve its ability to absorb what little moisture there was in this climate, and to help the roots get some nutrients out of the sand. These were a gray-blue, but the ones out by the two smaller hostels in the mountains were pinkish." (pg 153)

"In the morning Oyun had the weavers, the guards, the farmers, the bike riders, and the others lined up next to the fractalies hedge, along the dirt road sloping down from the Golden Jade to the clinic door. " (pg 369)

"The old Golden Jade was visible in the distance, greener than before, as were the farms up river due to Jenís genetically engineered drought resistant trees and fractalie shrubs. " (pg 431)

    I only mention it once, but I claim that the efficient generation of energy by the hyperthreads requires the proper orientation of those threads with respect to each other.  What better patterns to use than fractals?

"In the end, it was all about patterns and mathematics. Single sheets of hyperfabric began with simple grids of hyperthreads, then evolved into space filling curves, fractals, and chaos theory. That was easy." (pg 406)

    Wolfram, Stephen. A New Kind Of Science. Champaign: Wolfram Media Inc., 2002. ISBN: 1-57955-008-8