Fractal visionaries and enthusiasts:

I have decided to make the fourth dimension the FOTD theme of the month for October.  4-D hyperspace is not that much of a departure from the 3-D space we are so familiar with, since the Z^2+C Julibrot, (the composite of all the Julia and Mandelbrot sets), is a 4-D figure anyway.

Today's image shows a small part of the shore line of the period-4 bud that lies on the northeast lobe of the main bay of the Mandelbrot set.  Actually, the scene shows a section of the shore-line as it extends into hyperspace

The entire scene has been stretched by a factor of over 57, (the tangent of 89 to be exact), in the direction of the Mandelbrot Y-axis.  This stretching was necessary to keep the Mandelbrot ghost buds in their normal near circular shape as the view was rotated from the Mandelbrot orientation to within one degree of the Oblate orientation.

The result of all this stretching and the additional skewing that resulted from rotating the image is a scene that vaguely resembles a scene on an alien planet with wispy cirrus-like clouds in the sky.

I could have given the image a name such as 'Alien Cirrus' but I decided to name it simply "The Fourth Dimension", which quite nicely leads off a month that will be devoted to Julibrot scenes in 4-D hyperspace and hyper-objects such as the six regular 4-D polytopes.

The extra half-point in the rating of 8-1/2 is due to the coloring.  My modesty prevents me from granting myself any additional bonus points, even though they were deserved.

The calculation time of 13-1/2 minutes is somewhat slow, leaving it to the individual fractalist to decide whether the 4-D scene is worth the wait.

Today's weather left much to be desired here at Fractal Central.  The fractal cats felt that the temperature of 52F 11C was too chilly for comfort, while FL and myself felt that the heavy clouds that blocked the sun and the occasional rain that fell through the day were too unpleasant for the first day of October, which is supposed to bring bright blue weather.

The next FOTD will be posted in 24 hours.  Until then, take care, and the formula for the hypervolume of a 4-dimensional hypersphere is 1/2(pi^2)*(r^4).  How long will it be until anyone other than a physicist actually uses this formula?

Jim Muth
jimmuth@earthlink.net


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

TheFourthDimension { ; time=0:13:38.22-SF5 on P4-2000
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=SliceJulibrot2 passes=t
  center-mag=-0.09254657031508136/+0.561631274103498\
  60/855.02/0.03822/-1.073/-87.9929489969829319
  params=0/0/89/0/0.25/0/0/0 float=y
  maxiter=125000 inside=255 logmap=140 periodicity=6
  colors=000UizUhzUgzUfzUezUdzUczUbzUazU`zU_zU`zUazU\
  bzUczVdzWezVfzWgzVhzUizThzShzRgzRfzRezQdzQczPczPcz\
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  zQNzRTzRSzSSzSRzSPzSOzSNzSMzSLzSLySLxSLwSLvSLuSLsS\
  LqSLqSLqSLqSLqSLqSLqSLqSLpSLoSLnSLmSLlTLkTLjTLiTLh\
  TLgTLfTLeTLdTLcTLcTLbTLbTLbTLaTLaTLaTLaTLaULbVLcWL\
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  rKzsKzsKztKztKzuKzuKzvKzvKzwKzwKzyKzyKzyKzyKzyKzyL\
  zyMzyMzzNzzNzzOzzOzzPzzQzzQzzRzzRzzSzzSzzTzzUzzVzz\
  WzzWzzXzzYzzYzzZzz_zzazzazzbzzbzzdzzdzzezzfzzfzzgz\
  zgzzhzzizzjzzjzzkzzlzzlzzmzzmzzmzzmzzmzzmzzmzzmzzm\
  zzmzzmzznzznzzozzpzzpz00H }

frm:SliceJulibrot2   {; 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=sqr(z)+c
  |z|<=9 }

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