FOTD -- December 20, 2001 (Rating 5) Fractal visionaries and enthusiasts: On Tuesday, Jon K. posted a letter to the Fractint list, commenting on the FOTD of December 17, which is a scene in the hypercomplex Mandelbrot set. In the letter he asked whether an easy way exists to find the locations of the top layer and bottom layer in the hypercomplex set. This question is more complex than it seems. To start, the appearance of different layers in the FOTD of December 17 is an illusion. The hypercomplex M-set is 4-dimensional indeed, but only a single 2-dimensional slice of it can ever be displayed by the hypercomplex formula. (The julibrot formula can display 3-dimensional slices.) What is happening in the 12-17 image is that the hypercomplex figure has been sliced in an area where different parts of it are merging, actually trying to occupy the same points of hyperspace, with the higher-iteration features being obliterated by the lower-iteration features. In the 12-17 image, we are not looking through an eroded Mandelbrot shaped hole at features that lie on the other side, we are seeing all that remains of the higher-iteration features, the rest having been obliterated by the surrounding low-iteration stuff. Jon asked whether I know of an easy way to find scenes such as 12-17's. The answer is no. Both 'layers' move in an unpredictable manner as the parameters are changed. The only way I know to find such scenes is to track them down by trial and error bracketing and to fine-tune the image with the Fractint evolver. As I noted in the 12-17 discussion, the hypercomplex formula of Fractint does not draw all possible slices of the hypercomplex Mandelbrot set. The formula I wrote for today's image draws more, though not all slices. Today's formula has a drawback however in that the periodicity must be shut off in order for it to work correctly. The p3 parameter of today's HyperMandelbrot formula works the same as the cj and ck parameters of the hypercomplex formula. (I have no idea what the unused parameters of the hypercomplex formula are eventually supposed to be used for.) The p1 parameter of today's HyperMandelbrot formula works roughly the same as varying the initial Z parameters of the Mandelbrot set. The p2 parameter does strange things indeed, things that are almost unbelievable. Since it alters the effect of the other parameters, it is changed only at the risk of the user. Enjoying risk, I set it to 0.5 for today's image, and when I set p3 to 0.008, just look at what it did to our Mandelbrot set. I named today's image "HyperMandel Madness" as a poetic description, and rated it at an average 5. With a running time of 12 seconds, today's parameter file is virtually instantaneous. As always, the completed GIF image is also available at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> With a little more sun and a little less wind, the fractal weather Wednesday was somewhat improved over the day before, but the dynamic duo still would have preferred a temperature of 77F 25C rather than the 54F 12C that actually happened. They were in a surly mood until the sound of some stray cats fighting somewhere in the brush sent them dashing to their lookout shelf by the window, where they spent more than an hour. Then, when their evening mealtime arrived, they completely forgot their troubles. It's now time to accomplish something a bit more practical. Until next time, take care, and don't get hyper. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ HyperMandelMadness { ; time=0:00:12.14--SF5 on a p200 reset=2002 type=formula formulafile=allinone.frm formulaname=HyperMandelbrot center-mag=-0.546672/1\ .33227e-015/0.9318636 params=0/0/0.5/0/0.008/0 float=y maxiter=480 inside=0 logmap=yes periodicity=0 colors=000KAcKAbKAaKA`KA_KAZJAYIAXHAWGAVFAUEATDASC\ ARBAQAAP9AO8AN7BM6AL6AK000AQlBSlCTlDVlEWlFYlGZlG`l\ HblIclJelKflLhlMilNklNllR`lVPmZDmb1ma4na6n`8n`Ao`C\ o_Eo_Go_IpZKpZMpZOpYRqYTqXVqXXrXZrW`rWbrWdsVfsVhsV\ jsShpPgmMfjJegGdeDcbAb_7aX4`U2_S3aQ4cO4eM5gK6iI6kG\ 7mF8oD8qB9s9Au7Aw5By3Bz2Ew5Gu7Is9KpBMnDOlFRiITgKVe\ MXcOZ`Q`ZScXVeUXgSZiQ`kObmLdpJgrHitEkvCmxAom8qcSoP\ RpUQpPPpKOpFNpKMpGLpIKpKJpMIpOHpQGpSFpUEpWDpYCp_Bp\ aBpcEneGlgIjiKhkMfmOdoQbqT`sVZuXXwZVy`TzbRzdPzfNzg\ PzgQzhSzhTziVziWzjYzjZzk`zkazlczldzgbzcaz__zWZzRYz\ NWzJVzFUzKRzPOzTLzYJzaGzfDzjBzfDzcEz`FzYGzVHzSJzPK\ zMLzJMzGNzFRzFUzEYzE`zDdzDgzDjzFhzHgzJfzLezNdzOczQ\ bzSazU`zW_zXZzZ_z`_zb_zd_zf_zh_zj_zk_zm`zo`zq`zs`z\ u`zw`zy`zz`zz_zz_zzZzzZzzWzzTzzQzzNzzKzzHzzEzzBzz8\ zz5zz2zz0zzBzzLzzVzzdzzbzz`zzZzzXzzVzzUzzSzzQzzOzz\ MzzLzzFzzPzzQzzRzzSzzTzzU } frm:HyperMandelbrot {; periodicity must be turned off a=(p1),b=(0,0): q=sqr(a)-sqr(b)+pixel, b=(p2+2)*a*b+p3, a=q, |a|+|b| <= 100 } END 20.0 PAR-FORMULA FILE==================================