Check out the deviation Celestial Shuriken 2 [link] uploaded by and follow the links under Artist’s comments. He have found exact the same ghostlike phenomena [link] described in this journal The difference is that in this case the phenomena is visible thanks to inside filter. Whiteout inside filter the central basin in Celestial Shuriken would be completely black Also check out the other part of his wonderful fractal gallery

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This journal refers to some Ghostlike phenomena I have found using the Compass formula z -> z^d - da^(d-1) z (see article 27 Compasses in my Chaotic series [link] ). It’s maybe the most perplexed phenomena I’ve found in the mysterious of fractals

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:

GhostStart [link]
GhostZoom1 [link]
GhostZoom2 [link]
GhostZoom3 [link]
GhostZoom4 [link]
GhostZoom5 [link]


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 [link] 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 [link] . At the border to the right, you see a green rectangle, denoting the zoom to the next image.

3) GhostZoom2 - GhostZoom4 [link] [link] [link] . 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 minibrots are more like ghostlike shadows of minibrots! This is quite obvious if you look at GhostZoom4 [link] . 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 (near the neck) and one big component in the front of the head, which 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 the shapes 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 [link]
, our very last image. Her it is very clearly seen, that every black component has 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. That device is another phenomena in this type of fractals!

In other words: IN THIS BLURRY REGION THERE ARE SETS OF JULIA-LIKE COMPONENTS BUILDING UP GHOSTLIKE SHADOWS OF THE MANDELBROT SET! 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?

The parameter files attached under Artist’s Comments refers to the formula "ExtendedCopasses" z -> z^d - da^(d-1) z + b, which means I have added a parameter “b”. In these images "b" is fixed to zero so the images could be drawn by the simpler Compass formula as well However by this extended formula you, in the same way as in cubic parameter space obtain a four dimensional (a, b) - space, where you can glide along the non plotted axis’. Moreover you can perform rotations in the same way as describe in articles 21 and 22 in my Chaotic series [link]

If you don't have the formula, please download the modules from [link] .

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 a very hard work for your computer if you run the parameter files from the blurry region. Have a nice play