Fractal visionaries and enthusiasts: 

Today's image is part of the Julia set of a rectangle in the Z^(2.003)+C Mandelbrot set.  Surprisingly, there is no rectangle at the center of the image, but rather an octagon with in-curved sides.  The Julia set of a rectangle would normally be expected to be another rectangle, but fractals, like so many theories of modern physics, do not always do the intuitive things.  For some reason, objects that appear as rectangles in their Mandelbrot aspect appear as octagons when viewed in their broader Julia aspects.

This Julia doubling is not actually so surprising, considering that the Julia sets of the Mandelbrot radical 'stars' have double the number of arms as the Mandelbrot stars themselves.  Thus, the Julia set of the two-branched star located around -1.543 on the main spike of the M-set has four main branches.

The name "Truncated Rectangle" is another way of describing an "Octagon", which to be more precise, is a truncated square.

I rated the image at an 8, due mostly to the coloring, which is a combination of my efforts and a random shot of the Fractint program.

The calculation time of 4 minutes borders on slowness, but at least IMO, the image is worth the effort.  And the web sites can remove even most of that effort.

The pleasant late summer weather took a break here at Fractal Central today, when heavy clouds moved in, followed by lively thunder-showers.  The temperature of 84F 29C felt warmer than usual due to the high humidity.  The fractal cats got into a bit of a spat when they both wanted to lie on the table beside my favorite chair, but reason finally prevailed and they agreed to share the space with tails thrashing in unison.  The next FOTD is soon to be posted.  Until whenever that might be, take care, and if we only go around once in life, as conventional wisdom says, then when do we get to use the lessons we have struggled so hard to learn during our time on earth?

Jim Muth
jimmuth@earthlink.net


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

TruncatedRectangle { ; time=0:04:00.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=SliceJulibrot4 center-mag=-0.000021470\
  16991775/+0.00000106606499224/269396.1/1/-12.5/0
  params=90/0/90/0/-1.743492647282116/2.074078440319\
  529e-008/0/0/2.003/0 float=y maxiter=3200 inside=0
  logmap=994 periodicity=6 mathtolerance=0.05/1
  colors=000BBCFFHIINLMSOR`RWfUZmW`wYdz`gzcjzfmzirzm\
  vzrzzmszhmzccpZUfVKUSJKPJPMIVJI_GHeDHjccEYgOTjXOmf\
  JpoLiiNbdOW_QQURJPTCKU6FfDScLO`TLY`HVhESpAQw7cVMp3\
  `bVfQvkPqlOmlNimNemQhlSkkUnjWhfXcbYZZ_UV`PRaKNZLKW\
  LHTMERMBON8LN5IO2GO0HL9IIIJFRKC_LAhXDihGjsJkmPhhVf\
  c`dZfbnLPkQUhVYf_acce`hjZmnWrrUvvdbmoJdkIXgIPcIH_I\
  AWNHSRNPWTL_ZIddEhjBlpLngVoZdpQmqInkGofFpaEqXDrRBs\
  MAtH9uC8zSgzVbzYYz`TzcPzfKziFzkBzfAzaAzXAzSAzNAzJA\
  zSHz`NgiUfr__iSTaLMTEGL7KJEOHKRGRVEXYDb`9fc6if3mh0\
  pnXHmbPlgXlldkqlkvthuuftvcswarxZqyXqyUrwSsuQtsOuqM\
  voKwnNsoQppTmqVirYfs`ctb`ucdqdgmejifnegqahtYiwUakP\
  U_LUOKECC618G4AP7BZ9DgCEpEFfMUXUhN`wO`rO`nO`jO`fO`\
  bPaZQbWRcSScPUvN`xcfzs`uzWqpRmoLinGemBal6YkFxrLpqQ\
  ipVao_VndNmiGln8ks1kt3kt4ku6ku7ktDhsJesPcrV`r`ZqfW\
  qlUhiY`gaSeeKciBam3_p7XoAUnDRmGOmJLlMIkPFjSDjTAfU7\
  bV5ZW2VW0SX4VGJV5PK8JEBE8 }

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=========================================