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Cross-referential fractal re-expression of approaching infinity/nothing as seen in rivers/trees.

The tree
Follow one branch;
As an acorn sprouts into a tree it grows a trunk and produces a branch which produces a branch...reaching for an infinite number of branches just from one main branch (the trunk).  It progresses in the same way as infinity;   and 0, and 1, and 2...and Xinf.  In this way a tree is a natural fractal re-expression of infinity.  Of course, it never reaches infinity because the information required to describe it hits limits of resolution just as a wave function hits the Planck resolution limit making the universe's boundary conditions finite.

The river
does not grow from a source that becomes a trunk which adds branches.  It goes the other direction.  Where I had compared the history of cosmology to a river system with branches leading to "dead ends", a real river begins at those dead ends.  Each dead end is similar to the "and not" description of nothing as water starts to flow and that flow joins others to work down to a single flow.  Of course it never reaches nothing, it empties into a lake, sea, or ocean...but it does interface with "not river".  Just as "nothing" interfaces with "not universe".

Why do trees and rivers appear to be fractally similar to each other?
They are cross-referential fractal re-expressions of the same universal scale, the direction toward infinity vs direction toward nothing.  Both are easily represented in two dimensions.

It might be a useful thing to develop a classification system of fractal expression that is cross-referential to fundamental properties of the universe.  Such as spin/revolution, and volumetric efficiency, and properties of fields...and especially the property of causality.

Just to see where it might lead.

Yes, that's the one. It is a classification based on how different scale-symmetric shapes self-connect, regardless of their exact geometry... so it is more topological and so natural objects fit in the categories. A river being a 2D tree and a real world tree being a 3D tree.

The upper right triangle of the table is very speculative, so I invite everyone here to consider what these table entries actually should be, in order to be consistent with the rest of the table. Communal discoveries are always better :-)



I like your idea of a classification based on what the object is trying to optimise...
"Why do trees and rivers appear to be fractally similar to each other?"
I think both are maximising 'coverage per total length', in the case of the river, the large coverage means water doesn't flow far overland before reaching a (more efficient) groove/stream to flow down and smaller total river length reduces the distance to the mouth, so the river maximises the flow of water to the mouth. For a tree the coverage maximises the leaf access to the sun from all angles and the total branch length minimises the distance sugars etc need to flow in order to maintain those set of leaves. Something like that.

Another question is why are coastlines fractal? a suggested answer is that they maximise the dispersion of wave energy. i.e. a flat coastline will get battered by waves until it takes on a rougher shape, the wave energy will decrease as the coastline gets rougher, until it reaches some equilibrium.

so perhaps most fractals in nature are there to optimise something... bird bones are a fractal sponge to maximise the strength for a given weight... etc. 

What is the true reason for what?


Wow, perhaps it really makes sense to start asking WHY something is fractal. I love your conclusions. Never thought about the coastline-fractal in that way.
Same things as in trees for our lung. and our circulatory system as in rivers.

I wonder if  spirals like galaxies or hurricanes work in that way too..