FAME - Fractal Art Museum Enterprise - presenting:
     From Jo's Notebook
      Dr. Joseph Trotsky
     June, 2003
The T-Set Anti Fractals
V. The Duality of the T-Set Midgets.

 
The pictures on this page demonstrate an interesting feature of the T-Set: the duality of midgets. The pictures were taken in the equivalents of the sea horses valleys of the various members. In the above picture of the T-Set Rhombus you can see the a few minirhombs (rhombus midgets).
In the zoomed picture you will see the whole neighborhood of this section. The minirhombs are on the Julia formation at the right side. The cubic-like minibrot is located to the left of the junction point between the Julia formation and the big spiral in the border zone of the main T-Set body.
On the top-right side picture you can see this stretched familiar shape of the M-Set Cubic minibrot.
  The cubic minibrot has 2 main bulbs and the minirhomb has 4 main bulbs. The T-Set Rhombus is the Anti Fractal of the M-Set Cubic (3rd power). Thus the Midget Duality Principle of the T-Set says:
A n power T-Set fractal produces two kinds of midgets: those with n+1 vertexes (main bulbs) and those with n-1 vertexes.
The n+1 shape is the default shape of the n power T-Set fractal. The n-1 shape has a skewed/stretched shape of the n power fractal of the M-Set. Please note that it takes a difference of 2 between the number of vertexes of any n power of M-Set and T-Set. So we can phrase the T-Set Midget Duality Principle this way:
A T-Set fractal produces both T-Set and M-set shaped midgets.


 
This picture was generated by the T-Set Pentagon, which is the Anti Fractal of the M-Set power 4. According to the Midget Duality Principle there should be the T-Set 5 vertexes pentagonal shaped midgets and The triangular shaped minibrots of the quartic M-Set 4th power fractal. You can see both these midgets in the picture above. The quaternion is located to the left in the middle of the parallelogram structure.
On the zoomed picture you can see the entire neighborhood of this section.
  This is a closeup of the parallelogram shown in the left side picture. The standard somewhat distorted shape of the quartic M-Set minibrot is well defined and obvious in the center.
In the zoomed up picture you can see that all around this quartic midget there are again numerous pentagonal midgets.


 
This picture was generated by the T-Set Hexagon, which is the Anti Fractal of the M-Set power 5. According to the Midget Duality Principle there should be the T-Set 6 vertexes hexagonal shaped midgets and The rhombic shaped minibrots of the M-Set 5th power fractal. You can see very clearly both these midgets in the picture above.

   This picture was generated by the T-Set Octagon, which is the Anti Fractal of the M-Set power 7. According to the Midget Duality Principle there should be the T-Set 8 vertexes octagonal shaped midgets and The star of David shaped minibrots of the M-Set 7th power fractal. You can see both these midgets very clearly in the picture above.

Pleas notice as well, that I have already introduced the Trigon dual midgets in page tab "Trigon 2". With the trigon being the first in line Anti Fractal of the classic M-Set Mandelbrot. You can see the T-Set tri-vertexes midget, and the classic minibrot though somewhat stretched.

This concludes the present investigation of the T-Set Anti Fractals.
Actually there are here here two Sets of Anti fractals: the Anti M-Set Fractals and the Anti Julia Fractals. These Sets should be further investigated for their mathematical, morphological and model concepts.

Except for the short scientific-philosophical discussion in the intro page I have also not expanded this aspect. Nor have I discussed the meaning of Anti Fractals in the physical order of Nature. The concept that "clouds are not spheres..." was better understood by using the M-Set as a model to analyze Nature's phenomena. Now this concept could be expanded to include Nature's duality: Tornados are not stormy cones nor quiet circles, volcanos are not streaming polygons nor standstill lava planes and Black Holes are not infinite space nor zero derived Null. I do hope that the combined M-Set - T-Set concept will act as a dual model to achieve a better understanding of some of our universe riddles.

Author:
Dr. Joseph Trotsky

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