[Musicforawhile (OP)]

Branches of trees, blood vessels, lightning, lung alveoli etc all have the same branching (fractal) shape. Also, I paint and when I've experiment with printing  have found when you press two surfaces together, with one or both thickly painted, when you pull them apart you will get the same branching pattern. There is a word for this art technique, but I can't remember it right now...so I've heard that this shape is created because that's the way energy travels, but can someone explain that to me? What is it about the way energy travels that creates these branching shapes?

[yor_on]

Maybe it is about constants, defining the mathematics hiding behind it, and energy costs? You have entanglements finding the 'cheapest' ways transporting energy in plants, so maybe that could be one reason. A lot of the universe seems to be about what will cost the least.

[RD]

lightning ... branching (fractal) shape ...

http://en.wikipedia.org/wiki/Lichtenberg_figure

The lightning pattern is tree-like , but not exactly the same [fractal] geometry as a living tree,
the lightning pattern is an example of ... http://en.wikipedia.org/wiki/Diffusion-limited_aggregation

... I paint and when I've experiment with printing  have found when you press two surfaces together, with one or both thickly painted, when you pull them apart you will get the same branching pattern ...

http://en.wikipedia.org/wiki/Hele-Shaw_flow  [ type of Diffusion-limited_aggregation ]

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https://www.google.com/search?=Hele-Shaw_flow+

[Musicforawhile (OP)]

You have entanglements finding the 'cheapest' ways transporting energy in plants, so maybe that could be one reason. A lot of the universe seems to be about what will cost the least.

How do the energy particles/waves know what is the cheapest route?

[RD]

How do the energy particles/waves know what is the cheapest route?

By trying routes in all directions and following the easiest one they encounter ...

The random Brownian motion of the particles mean they try to move in all directions,  and will follow the path of least resistance they encounter, e.g. a patch with lower pressure, or region with less electrical repulsion ,  ( the filaments in Lichtenberg figures are evenly spaced because they repel one another ).


http://en.wikipedia.org/wiki/Patterns_in_nature#Trees.2C_fractals

[evan_au]

The challenge for the blood supply of a living organism is to:

• Get a supply of fresh blood adjacent to every cell
  
• While wasting a minimum amount of energy
  
• ...which means having close to the shortest possible path from lungs to the cell
  
• (and from the cell back to lung and kidneys to get rid of waste)
  

One approach would be to have a fine capillary vessel which winds its way past every cell in the body, but:

• The viscous energy losses would be enormous
  
• The first few cells passed by this capillary would get a good supply of oxygen, and will be able to dispose of CO₂.
  
• However, the later cells would suffocate as there would be no O₂ left, and would find the blood already saturated with CO₂
  
• If you got a small cut or blockage to this capillary, your blood supply would be cut off.
  
• So this wouldn't work very well!

Another approach would be to have a fat artery (like the aorta) which winds its way past every limb in the body, but:

• This fat blood vessel has a small surface area
  
• This would deliver oxygen to just a small number of cells
  
• So this wouldn't work very well, either!

A compromise is to take a wide artery from the heart,

• divide it into every part of the body (so it carries blood a long way with low losses)
  
• divide it again to get into the head and every limb
  
• divide it again... and again
  
• until it passes close to every cell for a short distance, giving up its O₂, but having fairly low viscous losses
  
• then join together into fat veins that travel back to the heart
  

Blood vessels are a "space-filling curve", with a fractal dimension just under 3, since they must fill up the 3-dimensional volume of your body.

Similar logic applies to the 3-dimensional structure of your lungs.

A creature which has a more efficient circulation system will tend to have more descendents that one with a less efficient circulation system. So this will tend to converge on the most efficient structure for the environment in which the creature lives.

[Musicforawhile (OP)]

How do the energy particles/waves know what is the cheapest route?

By trying routes in all directions and following the easiest one they encounter ...

The random Brownian motion of the particles mean they try to move in all directions,  and will follow the path of least resistance they encounter, e.g. a patch with lower pressure, or region with less electrical repulsion ,  ( the filaments in Lichtenberg figures are evenly spaced because they repel one another ).


http://en.wikipedia.org/wiki/Patterns_in_nature#Trees.2C_fractals

And is the reason why they can encounter routes in all possible directions due to wave-particle duality? Because the wave property is able to simultaneously try every possibility and therefore will "select" the path of least resistance?

[Atomic-S]

I'm not sure we need to invoke quantum mechanics to explain any of this. Classical physics would seem adequate, if we take into account the concept of unstable equilibrium.  That is a situation in which a system in balance, if slightly perturbed, develops a force that tends to further unbalance it.  These fractal situations could start out as a continuum (considered classically) that is uniform everywhere to a very good approximation, such as the paint between two surfaces. Simple-mindedly, we might expect that on disturbing the system, as for example separating the surfaces, the uniformity would cause a uniform phenomenon, such as shrinking the paint down into a single circular glob. However, if the equations of fluid flow are such that an increase in flow in one area would attract further fluid into that area, then any slight nonuniformity would be quickly amplified as the process proceeded, resulting in a relatively chaotic outcome.  As for why the same pattern would repeat over and over at different scales: that may simply be an outgrowth of the fact that the same physics start operating all over again at every opportunity, which would tend to cause a replication of pattern. As pertains to thing such as biological organisms adapting to their environment, the physics might be very different than those of paint but the basic mathematical principles are likely quite similar. That would be the principle that a slight deviation from uniformity creates a situation favorable for its own increase.  That is satisfied in a situation where a branching circulatory structure results in a more efficient organism than one having only one central blood vessel.

[Craig W. Thomson]

How do the energy particles/waves know what is the cheapest route?

By trying routes in all directions and following the easiest one they encounter ...

Yes. A photon is energy and has wave properties. A wave spreads out in space rather than existing as a point. As such, it can "explore" different routes at the same time in order to find a place for the wave function to collapse.