Fractal visionaries and enthusiasts:

With today's FOTD we find ourselves roaming once again in four-dimensional hyperspace, as we will be all the month of October.  Every image this month will be created with the Mandelbrot expression, Z^2+C, but none of the images will resemble the usual Mandelbrot or Julia scenes.

Today's image is a view of Scepter Valley, which is the valley at -1.25 on the negative X-axis of the Mandelbrot set.  This is the inspiration for the name "Royal Scepter", though the image looks nothing at all like the Mandelbrot or Julia aspects of Scepter Valley.

The scene is sliced at an odd angle oriented 75 degrees from the Rectangular direction toward the Julia direction of the Julibrot figure.  The brilliant pink vertical stripe in the center of the image is part of the open space between the two branches of the valley, viewed from the side.

Though the mathematical interest rates higher than usual in today's image, I could find nothing of artistic value in it that might raise the rating above that of an average 7.  I did spend some extra time on the coloring however.

In 4-D space, such as the space in which the Julibrot figure lies, there are six perpendicular planes through every point.  Every plane is perpendicular in the normal 3-D manner (intersecting in a line) to four other planes, and absolutely perpendicular (intersecting in only a single point) to one other plane.  In the Julibrot the absolutely perpendicular pairs of planes are Mandelbrot-Julia, Rectangular-Parabolic, and Oblate-Elliptic.  All other pairs of planes lie in the same 3-space and are perpendicular in the normal 3-D manner.

Today's image is oriented between the Rectangular and Julia directions, which implies that the rotation is of the normal 3-D kind.  In this case, the axis of rotation is the imag(z) line that intersects the M-set at -1.25 of the X-axis.

In today's image, the rotation may be seen in progress by varying the real(p2) parameter between zero and 90, which takes us on a trip from the Rectangular to the Julia directions.  But be warned, the calculation time increases dramatically as the value of real(p2) gets close to 90.

The higher dimensions have always been a mystery, but over the years I have come to see 3-D space as a construct of the mind, and also as an indicator of our level of consciousness.  We cannot imagine spaces of four or more dimensions because we have not yet reached a level of consciousness where it is possible.

Will the human race eventually evolve to a level where higher spatial dimensions are within our grasp?  Will we then find ourselves in a world with four or more spatial dimensions?  Physicists tell us that the higher dimensions (11 at the latest count) are there, but we cannot percieve them.  If this is so, who knows what mysteries the universe yet holds?

These questions will not be answered in today's image, but the image does calculate in only one brief minute.  The image will make the questions just a bit more significant however.

Heavy clouds continued here at Fractal Central today, with a very autumnal temperature of 54F 12C.  The fractal cats, solidly in cold weather mode, spent the day doing warm things such as curling up on the blankets.  FL and I, the local humans, spent the day keeping things running, which is not always as simple as it first appears.

The next FOTD will be posted in 24 hours.  Who knows where in the Z^2+C Julibrot figure we might find ourselves or what we might find there.  Until then, take care, and is three-dimensional physical reality really real?

Jim Muth
jimmuth@earthlink.net


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

Royal_Scepter      { ; time=0:01:01.24-SF5 on P4-2000
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=SliceJulibrot2 center-mag=0/0.0169994/\
  2.456186/1.1123 params=90/0/75/90/-1.25/0/0/0
  float=y maxiter=50000 inside=255 logmap=3
  symmetry=yaxis periodicity=0
  colors=000agobgocgodgoegnfgmggmhfmgfmgemfdmdcmbaj`\
  _gZYcVW_RTWOPSLMOIJLFGHCDE9AA67734334635936C38F3AK\
  3CO3ES3GW3I_4Lc4Nf4Qi4Sl4Xo4Zr4au4bx4dz9dzDdzIczMb\
  zR_zQYzPWzOUxNRvMOtMOqNOoNOnNOmOOlOOkPOjPNiPNhQMgQ\
  LfRLeRLdSKcSKbSKaTJ`TJ_UJ_UIZUIZVIYVIYWJXWIXXIXXJW\
  XJWYJVYKVZKUZKUZLU_LT_LT_MT`MS`MS`MRaMRaLRbLQbLQcK\
  PcKPcKOdKOdKOeKNeKNeKMfKMfKLgKLgKKhKKhKKhKJiJJiJIj\
  IIjIHjHHkHHkGGlGGlFFlFFmFEmFEnGEnIDoKDoMCoOCpPBpQB\
  qRBqSAqTArU9rW9sY8s_8sa3sb4sZ4oU4kP4iK4iF4iA4i85i8\
  5i95i95iA5iA5iA5iB6iB6iC6nC6nD6nD6nD6nE7nE7nF7nF7n\
  G7nG7nG7nH8nH8nI8nI8nJ8nJAr9FsAKsAPsAUsAZsAcsAhsBm\
  sBrsBvsBzsBzsBzsCzsCzsCzsCzsCzsCzsDzsDzsDzsDzsDzsD\
  zsEzsEzsEzsEzsEzsEzsFzsFzsFzsFzsFzsFzsGzsGzsGzsLzs\
  LzsLzsMzsMzsMzsMzsMzsMzsNzsNztUzuUzvUzwUzxUzyUzzUz\
  zUzzZzzZzzZzzZzzZzzZzzZzzZzzczzczzczzczzczzczzczzc\
  zzczzczzczzczzczzczzczxbx }

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