Fractal visionaries and enthusiasts:

The math expression that was iterated to create today's image is Z^2+C, the same expression that gives us the well-known classic Mandelbrot set with all its perturbed sets, as well as all the Julia sets.  Yet the image is clearly nothing at all like a Mandelbrot set, pure or perturbed, nor does it resemble any Julia set.  This is because it is a slice through the Z^2+C Julibrot in what I call the Rectangular direction, which is defined by imag-C,imag-Z.  The rectangular shape of the slice shows why I call this particular orientation the Rectangular direction, and also explains the name I gave to the image -- "Rectangular Reason".

The large open area that fills a good part of the frame is a different view of the main bay of the M-set.  The narrow bluish stripes running horizontally at the top and bottom of the open area are what I call 'bridges'.  In this case they are sideways views of the west branches of the valleys leading to the large north and south period-3 buds of the M-set.  The horizontal open areas above and below these 'bridges' are sideways views of the buds themselves.

I gave no rating to the image, since almost all its interest is in its mathematics.  But don't downplay the math interest.  To make things much clearer than I can explain in words, gradually decrease the value of the real(p1) parameter from 90 to zero in increments.  (Keep the imag(p2) parameter unchanged.)  The images are very fast and the process will take little time.  By the time the value is reduced to 50 the destination will be apparent.  Doing a 360-degree rotation would make an interesting animation, but I doubt that I'll get the chance.  (Decreasing the imag(p2) parameter of today's image to zero while leaving the real(p1) parameter unchanged brings us to the Elliptic orientation, but that's another story.)

To create today's image, I oriented the 4-D Julibrot figure so that the imaginary Z-axis is perpendicular to the Mandelbrot set on the screen and ignored the parts of the Julibrot that did not fit into our 3-space, leaving an imaginary invisible 3-D slice of the Julibrot intersecting the screen.  Then I rotated the screen image around the line perpendicular to the X-axis at the point -0.2.  The resulting image is what would be seen by a viewer located in the plane of the screen just to the left or right of the screen when the invisible 3-D slice was sliced at the point of -0.2 on the real-C axis.

All this talk does not mean the image is a slow one.  In fact, it finishes in a lightning fast 5 seconds.

Heavy clouds and a temperature of 50F 10C made today a day to be forgotten here at Fractal Central.  The occasional showers helped none at all.  And the experts assure us that more rain is on the way tomorrow.  The fractal cats always live in the present, so they usually forget a day's events at the first stroke of midnight.  Many humans are not so forgetful.  They hold long term grudges and resentments that fester until war breaks out.  FL and I fall between the cats and the long-term resenters.

The next FOTD, a continuing adventure in 4-D hyperspace, will be posted in 24 hours.  Until then, take care, and I just read that there is some evidence that the sub-atomic particles known as neutrinos might travel faster than the speed of light.  The claim is not yet confirmed, (In fact, I doubt it), but if this proves to be true, then neutrinos can move backward in time.  This would surely play havoc with our scientifically accepted ideas of an effect following its cause, but after a century of relativity and quantum mechanics, we should not be so surprised when we find one more thing that violates our common-sense view of reality.

Jim Muth
jimmuth@earthlink.net


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

Rectangular_Reason { ; time=0:00:05.66-SF5 on P4-2000
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=SliceJulibrot4 center-mag=0/0/0.73/1/\
  90/0 params=90/0/0/90/-0.2/0/0/0/2/0 float=y
  maxiter=5000 inside=0 periodicity=6
  colors=000D08F09H0AK0BQ0CW0E_0Gb0Ih0Kn0Mp0Ot0Qm5Pc\
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  sc7sc6sc5tb6sc5sc5sc5sc5sc5sc5sc5sc5sc5sc5sc5sc5sc\
  5sc5sc5sd5sd5sd5sd5sd5sd5 }

frm:SliceJulibrot4   {; draws all 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), esc=imag(p5)+9
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=z^(real(p5))+c
  |z|< esc }

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