Fractal visionaries and enthusiasts:

There are very curious rectangles in one of the Julia sets of the Z^(2.003)+C Mandeloid.   To see one of them, calculate the Julia set with a C-value of -1.7435 and check the Z-coordinates at 0.00019+0.07388i at a magnitude of 75.

The orderly simplicity of the rectangle makes it an impressive fractal thing, especially when the surrounding railroad-track chaos is considered.   And its fragility makes it all the more interesting.   If imag(c) is changed even the slightest from zero, the rectangle distorts, fills with debris and disintegrates like a soap bubble.   And if the orientation within the Julibrot is changed by as much as one degree in any direction, the rectangle also distorts and disintegrates.

Such rectangles are pure Julia objects . . . or this is what I thought until I found today's image, which shows a minibrot in the area of the 2.003 Mandeloid that corresponds to the Julia rectangle.   To my surprise, I found that the area surrounding the minibrot is filled with rectangles exactly like those in the Julia set.   In today's image, these rectangles appear as tiny open areas where the bulky arms appear to branch out into smaller arms.   A single enlargement reveals them to be perfect rectangles however.

I have not yet done much investigation of these Mandelbrot rectangles.   Maybe some of them are not so perfect.   Maybe some squares, trapezoids, parallelograms, or even more exotic figures with more than four sides, such as octagons, lie hidden in the scene.   Too bad we're already near the limit of resolution.   We may never find out.

But I don't give up so easily, especially when it comes to fractals, and I have a funny feeling that many more surprises lie nearby.   So I'll be doing a lot more exploration in the area.   Stay tuned.   Who knows what might turn up.

I named the image "Rectangle Holiday" for the fun of it.   I rated it at an 8 because I think it's pretty good.   The calculation time of 3-3/4 minutes will pass in a few flashes, or the flashes may be escaped by going to the FOTD web site at:and seeing the finished image posted there.

Tuesday produced enough sun to keep the fractal cats happy.   The temperature of 68F 20C was a little chilly for the date, but well within reason.   My day was just eventful enough to prevent boredom.   The next FOTD will be posted in 24 hours.   Until then, take care, and when you find the ultimate truth, what will you do with it?


Jim Muth
jamth@mindspring.com
jimmuth@aol.com

START PARAMETER FILE=======================================

Rectangle_Holiday   { ; time=0:03:45.28-SF5 on P4-2000
  reset=2004 type=formula formulafile=basic.frm
  formulaname=SliceJulibrot4 passes=1 float=y inside=0
  center-mag=+0.00002587319771205/+0.0000219042802989\
  2/3.8e+010/1/67.5/0 periodicity=10 bailout=10000
  params=0/0/0/0/-1.7435/0/0/0/2.003/0 maxiter=3600
  logmap=516
  colors=00089K9AM9BO9CQADTAFVAGXBJZBL`BObCRdCUfCWhD\
  ZjD`lD`nCboCeqCfsCguChwCixCjyCkwBlwBmwBnwBowBowBow\
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  zzdzzezzezznzzlzzkzzjzzizzhWzgTzfdz7_zJVzVPzOQzURz\
  _EzlLzhyzwsztmzqgznHzj9zi }

frm:SliceJulibrot4   {; draws most slices of Julibrot
  pix=pixel, u=real(pix), v=imag(pix),
  a=pi*real(p1*0.0055555555555556),
  b=pi*imag(p1*0.0055555555555556),
  g=pi*real(p2*0.0055555555555556),
  d=pi*imag(p2*0.0055555555555556),
  ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
  sg=sin(g), cd=cos(d), sd=sin(d),
  p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
  q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
  r=u*sg+v*ca*sb*cg, s=v*sin(a),
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=z^(p5)+c
  |z| <= 9 }

END PARAMETER FILE=========================================