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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's formula, the 'hypercomplexj', draws the Julia sets that correspond to the Mandelbrot sets drawn by the 'hypercomplex' formula. These formulas use 4-part hypercomplex numbers, which draw four-dimensional M-sets and J-sets. These two four-dimensional collections of sets are oriented so that they intersect in single points, with an entire 4-D Julia set corresponding to every individual point of the 4-D M-set. For this relation to be possible, fully eight dimensions are needed. So when we explore with the 'hypercomplexj' formula, we are actually exploring an eight-dimensional Julibrot figure. In eight-dimensional space, there are six mutually-perpendicular directions that are at the same time perpendicular to a two-dimensional plane, such as that of the screen. All six directions can be explored with the six variable parameters of the 'hypercomplexj' formula. (For some reason, the 'hypercomplex' formula ignores the 'warped' M-sets.) My dream formula would be one that draws every possible orientation through this 8-D Julibrot. Unfortunately, my math ability is not up to the task of writing such a formula, so the formula will remain a dream. When I first saw today's image, it reminded me of something familiar that I could not quite recall. Then I recognized a resemblance to the continent of Antarctica, and named the image "Antarctic Fractal". The image is one of the Julia sets that corresponds to yesterday's 'Elephant-Seahorse' Mandelbrot image. In the 8-D Julibrot not all Julia set slices are symmetrical, as they are in the more familiar 4-D Julibrot. In today's image, which shows the entire Julia set, I emphasized this lack of symmetry. As was the case in yesterday's image, today's image has places where different elements overlap, and the lower-iteration elements are obliterating the higher-iteration ones. What we have is two distinct Julia sets fighting to be displayed on the screen, with the lower parts of each set winning. It strikes me as much like human nature, where we know what is right but are prevented from doing it by our baser instincts. With so much calculating to do in each cycle and a maxiter of 27000, today's image is a slow one, taking an hour on my tired old workhorse 200mhz machine. (On the new 2200mhz machine, a test-rendering took 8 minutes, but that unit is for real work, not fun.) Today's image has much mathematical interest but lacks artistic beauty. I rated it at a 6, mostly for its math interest. When the one-hour render time is figured in, the overall worth equals a humble 10. My recommendation is to avoid such excessive humility by downloading the completed GIF image from the FOTD web site at: But if your machine is a fast one and is Fractint compatible, be humble. Take a few minutes to enjoy the coming-into-existence of a tiny part of an unimaginable 8-dimensional world. And unless I am mistaken, our universe is currently theorized to contain 11 dimensions. A chilly, dreary, misty day here at Fractal Central on Wednesday kept the fractal cats confined to their shelf by the window. For some unknown reason, their moods were not bad however. As a reward, they received extra tuna for their evening dinner. Today is again starting dreary and chilly. More tuna will be needed. In my department, the commercial work is still caught up, and the FOTD fractals will keep coming every day. The next one will appear, not quite by magic, at this same time and place 24 hours from now. Until that ecstatic moment, take care, and don't disturb the fractal peace with excess exuberance. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START PARAMETER FILE=======================================
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END PARAMETER FILE=========================================
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