Fundamental enigma: pythagoras tree, duckies and amazing surf

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Offline Alef

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« on: May 11, 2018, 10:40:16 AM »
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The Pythagoras tree is a plane fractal constructed from squares. Invented by the Dutch mathematics teacher Albert E. Bosman in 1942, it is named after the ancient Greek mathematician Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. The finer details of the tree resemble the LÚvy C curve.
https://en.wikipedia.org/wiki/Pythagoras_tree_(fractal)



Angled pythagoras tree


https://en.wikipedia.org/wiki/L%C3%A9vy_C_curve


Ducks type fractals:





Menger based tree:



Amazing surface formula:




What does this mean? Any ideas?

This must be something fundamental. There certainly is a property of universality shared by mandelbrot. Jet it extends beyond higher dimension boundaries. Throught pythagoras tree is simmilar but not quite the same a surf or duckies trees or this could be just my visual perception.

This single could be described as interconnected pythagoras tree. This is what Mandelbox + rotation and asurf generates:



Removed Levy curves so that it would look smaller.
« Last Edit: May 15, 2018, 12:11:16 PM by Alef »
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Offline Alef

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« Reply #1 on: May 11, 2018, 11:27:38 AM »
Search gived me some pdf in latin which is not connected to this
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Entangling Fractals
We use the Heat Kernel method to calculate the Entanglement Entropy for a given entangling
region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal
dimension as well as the walk dimension. The power of the UV cut-off parameter is (generally) a
fractional number which indeed is a certain combination of these two indices. This exponent is
known as the spectral dimension. We show that there is a novel log periodic oscillatory behavior
in the entropy which has root in the complex dimension of a fractal. We finally indicate that
the Holographic calculation in a certain Hyper-scaling violating bulk geometry yields the same
leading term for the entanglement entropy, if one identifies the effective dimension of the hyper-
scaling violating theory with the spectral dimension of the fractal. We provide more supports with
comparing the behavior of the thermal entropy in terms of the temperature in these two cases.
https://arxiv.org/pdf/1511.01330v1.pdf

No, I was wrong. Tree pattern is identic to pythagoras tree. It too have conected branches just it is not so noticable becouse elements are more "fat". Of corse tree growth pattern is simmilar.
« Last Edit: May 11, 2018, 02:33:31 PM by Alef »

Offline gerson

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« Reply #2 on: May 11, 2018, 04:37:54 PM »
Maybe this is not conected with this issue because it is not Phytagoras but is a program to do 2-D Fractal Tree:
http://www.josechu.com/moving_fractal/index.htm

Offline Alef

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« Reply #3 on: May 11, 2018, 06:39:20 PM »
It too have bifurcating branches. H - tree maybe.

Of natural trees of temperate zone only willlows shows this growth pattern.  Say birch or oaks have leading vertical growth. Willows allways lose top bud.

Salix Fragilis Bullata:


Somehow 3D iterated formulas converge with pure geometrical construction. Only other fractal shape with property of universality is mandelbrot, I think.

Offline Fractal Institute

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« Reply #4 on: May 11, 2018, 09:08:06 PM »
great images!
I don't really understand what the enigma is, what is the question you are searching an answer for?

Offline Alef

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« Reply #5 on: May 15, 2018, 12:07:20 PM »
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great images!
I don't really understand what the enigma is, what is the question you are searching an answer for?
Thanks.
Same figure appears all ower again. Be it rotation and folding based iterated 3D fractals, folding based 3D IFS, geometric construction L-systems or nature. Does this have some mathematical basis? Could it be some universal figure like triangle or mandelbrot figure in 2D iterated fractals? (Well, it probably is.) But why? Probably this is just the most efficient growth pattern, self sustainable and it don't needs hudge genetic regulatory machinery - small branches don't need to know where they are on the tree.

This is even a question of extraterrestrial life. If fundamental laws are the same then maybe trees on other planets should look pretty simmilar.

Salix Fragilis Bullata, the most perfect resemblance for these generated trees:


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