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From differential solution set to Fractal appearance set.


I've given some thought to the recent research published by a team of Scientists that the large scale structure of the Universe looks a lot like the small scale structure of neuronal connections and like the planet scale structure of the connections between computers on the internet.

Here is my explanation for why this is so (yes, this is the type of conversation I prefer to have to the detriment of my dating life) it's simple. From exploration of the class of equations known as differential equations we know that defining the initial conditions of an equation gives rise to a set of solutions for those initial conditions that range often over continuous space. Even if the solutions are not continuous the variation in variables (known as the order of the equation) can be infinite, leading to an open set of possible solutions to the initial equation.

Thus for sufficiently chosen initial conditions any arbitrary solution can be identified for any given equation. It turns out that this is true for those particular equations that have range across continuous space. These tend to have non linear relationships and dynamic variable relationships in dimension (partial)...these type of differential equations are in fact very special...they are the type that are most fitting for solving some of our most complex dynamic systems in the physical world.

Some n'th(where "n" is high not necessarily infinite) order, non linear , partial , differential equations of note:

The Heat equation (how heat energy convects between materials):

The Navier Stokes equation (how fluids [which include gases] flow):

The Schrodinger equation (the governing equation of all quantum mechanics):

The Gravitational Equation (Einstein Field Equation, governs how energy and matter distort space time):

:Looking at those it should be clear why over sufficiently varied initial conditions they seem to give systems of solution that appear similar as the initial conditions vary across wide, near infinite ranges, all vary enough away from any given set of conditions as to appear nearly random....*chaotic* but based on the same kernel the same dna in the form of the initial differential equation and the defined initial conditions. This sounds like self similarity across the solution set and corresponding initial conditions. Fractal systems are a special subclass of Chaotic systems and so from the variations of parameters over infinite sets all those equations appear to create fractal solution sets...which in the aggregate indeed do look alike (not quite random noise...but no real repeatable structures outside of self similar ones). Fractals.

This shouldn't be surprising at all to us, it's not the first time such similarities have been noted but I find it interesting that the connection between the family of solutions that come out of the types of equations that model these systems and how they appear at different scales (essentially different initial condition values) and the similarity across those equations in this because of their identical mathematical form has not been (at least to my knowledge yet made) so I make it here.

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