Fractal of the Day
by Jim Muth

Perfect Proportions ©
Jim Muth's fractal image in GIF format (640x480).

FOTD -- April 09, 2008   (Rating 6.5)

Fractal visionaries and enthusiasts:

Although the minibrot at the center of today's image looks like a quadratic one, it is not.   The parent fractal was generated by the formula Z2.0000001+C rather than the classic Z2+C. On the surface, the parent cannot be distinguished from a perfect classic Mandelbrot set, but down deep, especially along the negative X-axis, things are quite different.

The little minibrot in today's image lies at the edge of the infinitely divided X-axis in the vicinity of East Valley of the large minibrot at -2.475 on the X-axis.

Yes, I know that the large minibrot is centered at -1.75 of the negative X-axis, and not -2.475, but today's parent is not the standard Mandelbrot set.   It is the fractal I call the shadow Mandelbrot set, which lies in the same Julibrot figure as the classic M-set but is double rotated 45 and 45 degrees toward the Julia orientation.

This less-familiar shadow set has twice the area of the classic set, and the linear dimensions are therefore 1.414 times those of the classic set.   As for double rotation, don't waste mental energy trying to picture it.   It's a four-dimensional thing impossible to visualize in three dimensions except in a projection.

With its near-perfect square pattern, the minibrot in today's image is quite striking in its own right.   Enjoy it while you have it, for we will see another view of it in tomorrow's FOTD, when we see what a difference a little rotation can make.

The image rates a not-too-bad 6-1/2.   The name "Perfect Proportions" refers to the strikingly perfect square pattern, which will not remain perfect for long however.

The calculation time of only 35 seconds makes running the included parameter file a pleasure.   Equally pleasurable is the trip to the FOTD web site at:where the finished image is posted for instant viewing.

A dull, dreary morning gave way to a brilliant blue sky at noon, followed by a balmy sunny afternoon with a temperature of 63F 17C.   The cats enjoyed the balminess from their shelf in the front window.

My day was busy, but still well under control.   The next FOTD, a shocking scene indeed, is already in the bag and will be posted in 24 hours.   Until then, take care, and be ready for the big change.

Jim Muth

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

PerfectProportions { ; time=0:00:34.89-SF5 on P4-2000
  reset=2004 type=formula formulafile=allinone.frm
  formulaname=SliceJulibrot4 inside=0 periodicity=10
  center-mag=0/0/2.154207e+008/1/-179.25/0 logmap=181
  float=y maxiter=1800
  hjzZezPDzrHzkLzdPzZTzSXzL }

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

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

Want to view, create, or know more about fractals?

Go to my Fractal Links webpage,
to the renowned Fractal Census,
or to the revised Spanky Fractal Database.

Plus, to my hostings of:
O's Fractal Art Gallery.

Go to my Fractal Pages on this site, or
to his latest web pages for Fractals.

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