This article refers to some Ghostlike phenomena
I have found using the Compass formula z ->
z^d - 2a^(d-1) z (see article 27) Comp***** in
my Chaotic series,
http://klippan.seths.se/fractals/articles
Interesting patterns will appear when the real
part of the exponent is a negative non-integer
number. So let's put the exponent "d" to
-3.65454654168845+0.4654564654654654i and
thereafter perform a little zoom. Thus:
http://klippan.seths.se/illustrations/illustrations/GhostStart.jpg
http://klippan.seths.se/illustrations/illustrations/GhostZoom1.jpg
http://klippan.seths.se/illustrations/illustrations/GhostZoom2.jpg
http://klippan.seths.se/illustrations/illustrations/GhostZoom3.jpg
http://klippan.seths.se/illustrations/illustrations/GhostZoom4.jpg
http://klippan.seths.se/illustrations/illustrations/GhostZoom5.jpg
>From GhostStart to GhostZoom3 the next zoom
is shown by a rectangle. In GhostZoom4, however
the next motive (the last) is appointed by
an arrow. So let's go:
1) GhostStart is the parent fractal Note that
small copies of the entire Mandelbrot set are
visible in the periphery. Also keep in mind
that the small holes that repeats in smaller
and smaller scales often have the shape of the
central basin.
2) GhostZoom1. At the border to the right, you see
a green rectangle, denoting the zoom to the next
image.
3) GhostZoom2 - GhostZoom4. We zoom towards a
minibrot in the blurry region near the border
of the central basin. But what! The minibrots
we see under ways in that blurry region seems
to be disconnected! First of all there are holes
with the same shape as the central basin of the
whole fractal. Second, the rest of the minibrots
seems to be constituted of smaller disc-like components.
That is the minibrot are more like ghostlike shadows
of minibrots! This is quite obvious if you look at
GhostZoom4. And, if you look at the left lower
part of that image, you will see the contours of
another minibrot, yet more disconnected!
But, what forms have those components building
up the shadows of the minibrots? If we look at the
minibrot to the right there is one big component
in the body (neat the neck) and one big component
in the front of the head, that makes me shout.
The first mentioned have the shape of a 1-periodic
Julia set, and the second the shape of a 2-periodic
Julia set. All this in the very way these Julia
sets would have if they were drawn using parameter
seeds from respective spot of the ordinary Mandelbrot
set! To verify this hypothesis, I made a zoom in
the back of the head at the spot appointed by the
green arrow, and received ..
4) ..GhostZoom5, our very last image. Her it is
very clearly seen, that every black component have
the shape of 2-periodic Julia sets! The speed
of the 2-armed spirals increases the more near
the neck (outside the image in the right upper
direction) the position is located. The right
part of the biggest component is cut off. This is
another phenomena in this type of fractals!
In other words: IN THIS BLURRY REGION THERE ARE
SETS OF JULIA-LIKE COMPONENTS BUILDING UP GHOST-
LIKE SHADOWS OF MINIBROTS! This in such a way that
components of the shape of 1-periodic Julia sets
produce the shape of the body, components of the
shape of 2-periodic Julia sets produce the shape
of the head, etc. This is a phenomena I never seen
or heard of anywhere! Wonder if the great mathematicians
have noticed, and maybe explained this phenomena?
If you use Ultra Fractal, the parameter files for
the above images(in fact referring to the formula
"ExtendedCop*****", "b" fixed to zero, can be
downloaded from:
http://klippan.seths.se/illustrations/illustrations/GhostBrots.zip
If you don't have the formula, please download the modules from
http://klippan.seths.se/fractals/articles/modules.zip
The above illustrations are rendered to 3200x2400
resolution (GhostBrot2 to 6400x4800) and thereafter
resized in a graphical application to 1067x768
resolution. I will prepare you for very hard work
for your computer if you run the parameter files
from the blurry region. Have a nice play!
---------------------
Regards,
Ingvar Kullberg
www.come.to/kullberg
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