Fractal visionaries and enthusiasts: 

The day we spent yesterday back at Old Fractal Central went quite well.  But the fractal cats disapproved of our all day absence and our 10pm return.  A quick treat of tuna and cheddar restored their good spirits.

Today's image shows both the Mandelbrot and Julia aspects of the same Julibrot scene together at the same time.  Since I will attempt to explain how this is possible, I have named the image "Fractals-101".  The reddish outer parts outline the Julia aspect of the scene, while the bluish inner parts show the Mandelbrot bud aspect.

The bluish Mandelbrot bud near the center is about 60 times its apparent size when it is viewed as a straight Mandelbrot shape.  To see the pure Mandelbrot view, reset the real(p2) and real(p3) parameters to zero and reset the magnitude to 60.  (The value of 60 is necessary because it is close to the value of the tangent of 89.)  But how can a Mandelbrot image be enlarged by merely slicing it at a very sharp angle?

The answer lies in the 4-dimensional shape of the high-iteration Mandelbrot features of the Julibrot, which are extended to infinity in two dimensions, and display the typical compact Mandelbrot shapes in the other two dimensions.

Imagine an infinitely long cylinder in three dimensions with a circular cross-section.  If it is sliced straight across, the slice is a circle.  But if it is sliced at an ever increasing angle, the slice is an increasingly stretched oval.  And slicing the cylinder along its length produces an infinitely long straight rod.  The angle of the slice causes the expansion of one dimension of the slice.

In four dimensions we may have an object that is circular in its two compact dimensions and extends to infinity in its two extended dimensions.  Such an object may be sliced at two independent sharp angles.  Doing this will give a 2-D slice that is stretched in two directions, and if the angles are equal, the cross section will be enlarged proportionally without being stretched.  We have a Mandelbrot microscope.  The Mandelbrot part of today's image was enlarged just this way.  (Unfortunately, we can get no extra magnification of Mandelbrot objects by doing this.)

The image itself is pretty routine.  The most interesting part is the curiously shaped bluish Mandelbrot bud at the center.

The calculation time of 2 minutes is reasonably fast, but the web sites are always there to make things even easier.

A mix of sun and clouds, with a temperature of 64F 18C made today quite typical of the end of September here at Fractal Central in Central Pennsylvania.  The fractal cats enjoyed the sunny periods on their shelf, and endured the cloudy times in sulk mode.  The humans spent most of the day recovering from yesterday's trip to Old Fractal Central, where my sister is putting out food for the neighborhood cats that are starting to gather in increasing numbers for a free meal.  The next FOTD will be posted in a reasonable amount of time.  Until whenever, take care, and if we suddenly lost all our advanced technology, how many, I wonder, would survive and how many would curl up and expire.

Jim Muth
jimmuth@earthlink.net


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

Fractals-101       { ; time=0:02:00.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=JulibrotMulti2 function=recip passes=1
  center-mag=0/0/1.572327 params=6.5/-6.5/89/0/89/0/\
  -1.626/0/0/0 float=y maxiter=24000 inside=0
  logmap=12 symmetry=xaxis periodicity=6
  colors=000S05U17W28Y39_4Aa4Bc5Ce6Dg7Ei8Fk8Gm9HoAIq\
  BJtCKwCLuEMsGMqINoKNmMOkOOiQPgSPeUQcVQbUPbUPbUOaUO\
  aUNaTN`TM`TM`TM_TL_SL_SKZSKZSJZSJZSJVTKSULPVMLWNIX\
  OFYPBZQ8_R5`S2`T9cQGeNNhKUjH`lEgoBnq8us5so9rlDpiGo\
  eKmbNl_RkWViTYhQafMdeJhdGkdGkcHkcHlcHlcHlbIlbImbIm\
  bImaJmaJmaJnaJn`Kn`Kn`Ko`Kz_Lz_Lz_Lz_LzZMzZMzZMzZM\
  zYNzYNzYNzYNzXOzXOzXOzXOzWNzWNzWNzWNzWNzWNzWNzWNzW\
  MzWMzWMzWMzWMzWMzWMzWMzWLzWLzWLzWLzWLzWLzWLzWLzVKz\
  VKzVKzVKzVKzVKzVKzVKzVJzVJzVJzVJzVJzVEzVJzVJzVJzVJ\
  zVIzVIzVIzVIzVIzVIzVIzVIzUHzUHzUHzUHzUHzUHzUHzUHzU\
  GzUGzUGzUGzUGzUGzUGzUGzUFzUFzUFzUFzUFzUFzUFzUFzYKz\
  XKzXKzXKzXKzXKzXKzWKzWKzWKzWKzWKzWKzVKzVKzVKzVKzVK\
  zVKzUKzUKzUKzUKzUKzUKzaKz`Kz`Kz`Kz`Kz`Kz_Kz_Kz_Jz_\
  Jz_Jz_JzZJzZJzZJzZIzZIzYIzYIzYIzYIzYIzYIzXHzXHzXHz\
  XHzXHzWHzWHzWGzWGzWGzWGzVGzVGzVGzVFzVFzUFzUFzUFzUF\
  zUKzKKzKKzMKzLKzLKzKKzKKz }

frm:JulibrotMulti2  {; draws all slices of Julibrot
  pix=pixel, u=real(pix), v=imag(pix),
  a=pi*real(p2*0.0055555555555556),
  b=pi*imag(p2*0.0055555555555556),
  g=pi*real(p3*0.0055555555555556),
  d=pi*imag(p3*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),
  aa=-(real(p1)-2), bb=imag(p1),
  c=p+flip(q)+p4, z=r+flip(s)+p5:
  z=(bb)*(z*z*fn1(z^(aa)+bb))+c
  |z|< 6 }

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