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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: This is a corrected version of the already-posted FOTD. I inadvertently left the semicolon out of the comment line of the formula in the first version. A few days ago, Tim posted a question about three-part numbers, or triternions. Shortly thereafter, Morgan posted a basic formula that draws the triternion Mandelbrot set, which, as drawn by the formula, consists of a skewed M-set with its northern part chopped off. When I ran the formula, I was expecting a somewhat different figure, but I'll take whatever the numbers give. (Maybe several varieties of three-part numbers are possible, and one of the other varieties would draw a different M-set.) Adding a few simple variable parameters to the formula, and giving it a new name, I did a little experimenting, and found that the triternion M-set behaves much like the hypercomplex set. It seems to consist of overlapping M-sets with straight 'bridges' added. Unlike the bridges in the odd planes of the Z^2+C Julibrot, which consist of stretched elements and can be unstretched into recognizable figures, the bridges in the triternion set consist of truly one-dimensional elements, which no amount of stretching will make into recognizable figures. In the triternion set, as in the hypercomplex set, different elements often overlap, and when this happens, the element with the lowest iteration count will prevail. Today's image shows triternion bridges overlapping conventional M-set elements, with a Minibrot on the left and a spiral on the right. The resulting interplay produces some interesting 3-D effects, especially around the spiral. I named the image "Super-Bridges" on a whim. Not knowing quite what to make of the image, I rated it at an average 5. Regardless of the rating, the image is a fast one, requiring only 2 minutes to render from the attached parameter file. The image can also be seen by downloading it from: Partly cloudy skies on Wednesday could not make up for a raw east wind and a temperature that barely rose above freezing. The fractal cat Tippy expressed his displeasure by sulking. Thomas was too busy sleeping by the heat. And if I don't soon get busy, a few customers will be sulking. So until next time, which will arrive on schedule in 24 hours, take care, and keep careful watch for the spirit of the season. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START 20.0 PAR-FORMULA FILE================================
Super-Bridges { ; time=0:02:16.99--SF5 on a P200
reset=2002 type=formula formulafile=test.frm
formulaname=triternion passes=1 center-mag=-0.8647\
1861343083230/-0.33029117615557820/1220.083/1.6718\
/-135.15/1.65891648238705725 params=0/-0.07/0/-0.2\
3/0/0.14 float=y maxiter=475 inside=255 logmap=39
colors=0A0EADEADD9CC9BC8BB8AA79A79C6BB6DK6FN6HP6JR\
6LT6NV6PW5QX5SY5UZ5W_5Y`5_a5ab5bc5cb7b`8`ZAZXBXVCV\
SETQFROGPMINKJLHLJFMHDNFBOD9PB8QB7RA6QA6PA5OA5NA4M\
A4LA3KA3JA5MC6OD7RF8TG9WIAYJB`LDbMEeOFgPGjRHlSIoUJ\
qVHmTFiREfPCbNB_L9WJ8TH6PF5MD3IB1D72F93GA4IC5JD6LE\
6MG7OH8PI9RKASLAUNBVOCXPDYRE_SG`TE`TDN8CMDBLIAKN9J\
S8IX7Ha6Gf5Fk4Ep3Du2Dy5Bv79tA8rC6pF4nH3lLCjPKiTShW\
_g_gfcoemzbfwd_tfTqhMnjFkl8hn2fo2fo2fo4gm6gl8gkAgj\
CghEggGgfIgeKgcMgbOgaQg`Sg_UgYWgXYgW_cVagTcgSegRgg\
QhgPgeRgcTgcVfbXfaYf__eZaeYceXeeWfdUgdTgcShcRhbQhb\
PiaOiaNi`Mj`Lj`Kj_Jk_IkZHkZGlYFlYElXDmXCmXBmWAnW9n\
V8nV7oU6oU5oT4pT3pU0oT2pT3pT5pS6pS8qS9qRAqRCqRDrRF\
rQGrQHrQJsPKsPMsPNsOOtOQtORtOTtNUuNVuNXuMYuM_vM`vL\
avLcvLdwLfwKgwKhwKjxJkxJmxJnxJoxabatQFsRGsRGrSGrSH\
rSHqTHqTIqUIpUIpUIoVJoVJoWJnWKnWKnXKmXKmYLlYLlYLlZ\
MkZMk_Mk_Mj_Nj`NkaFlbFzzz }
frm:triternion { ; adapted from Morgan Owens' formula
c1=real(pixel)+(p1),
c2=imag(pixel)+(p2),
c3=(p3),z1=z2=z3=0:
nz1=z1*z1+z2*z3+z3*z2+c1
nz2=z1*z2+z2*z1+z3*z3+c2
nz3=z1*z3+z2*z2+z3*z1+c3
z1=nz1,z2=nz2,z3=nz3
z=nz1+flip(nz2)
|z-pixel|<100 }
END 20.0 PAR-FORMULA FILE==================================
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