Sunday, 26 February 2012

Mathematical Curiosities



To fractals and beyond


You might be wondering why I have placed a picture of a fractal pattern and a busy city together. One is very abstract, quite simple and represents an application of a simple mathematical rule. A city is very real: it contains many physical events (power stations, electric grid, pipes etc.), a city is covered in traffic which is itself a very complex system and of course we humans live and do our complicated stuff in these cities. 

So you are still wondering... what have they got in common? Well they are both systems. One is an abstract system and the other an extremely complicated physical system. The link between the two is mathematics or more generally 'rules'. The fractal pattern above is a mathematical function denoted by U(z) where z is the variable. The function U(z) literally lays down the rules for which z obeys, if one plots U(z) geometrically then you witness the fractal pattern. (I do not know the said function for this particular one).

Here is what we call a Julia set: 

The function for this set is a complex rational function : f(z) = p(z)/q(z)  where  p and q are complex polynomials. Then there are certain open sets which are left invariant under f(z). Further analysis will enlighten the mathematician. This is, however, the general formal definition of the Julia set.

Consider the complex equation:  f(z) = z^2 + C  where z is a complex variable and C is a complex parameter.

Different parameters which define C lead to different outcomes for f(z), therefore we should witness different fractal patterns. 


Here is a Julia set variation for f(z) where C = 1 - &  where & is the golden ratio. 


Here is a Julia set variation for f(z) where C = 0.70176-0.3842i
where i = sqrt -1 (imaginary number) hence C is a complex parameter. 

Just out of curiosity the last Julia set resembles a snowflake like structure and it also reminds me of shapes of certain galaxies and nebulae:



They obviously differ from one another, but are the functions which define them similar or related? Maybe and if yes, how interesting would that be. To know that there is a defined and explicit relation between a fractal pattern, snowflake and a galaxy? To me that reveals a very beautiful structure to reality.

'Beauty is the first test: there is no permanent place in the world for ugly mathematics'
G H Hardy

The miniscule glimpse we have just had at the beauty of fractals verifies Hardy's well poised comment. There is a unique beauty in our world also. If we go back to the busy city picture we know that this is one huge system that is probably defined by an (or many) extremely complicated differential equations which might take a computer too much time to solve. We can take this city and split it into different, more simpler systems. For instance the city will have a traffic grid and system, the way the roads connect (into nodes, average length of road and there general location) may obey rules, rules encouraged by geography and the size of the city, how dense the city is  etc. This part of the system is still governed by some sort of mathematical rule which does not have to be mechanical, it may of course be very non-linear.

How vehicles also obey rules of traffic e.g. must be on one side for one direction, be in correct lanes etc. is quite interesting. The rules may seem simple... the system may seem simple but over time we witness anomalies and breakdowns of the rules (this is most definitely due to humans mucking things up which again is a small eddy in another rule). 

The routines of many people are constrained by rules or conditions which could in themselves be expressed as variables and functions. We could picture the ideal or average routine of a human in our busy city. This would be like the standard Julia set we saw early on. With small deviations of parameters (in the case of the Julia set it was C) we see large deviations in the routines...  for instance there may be a constraint in the equation + or - R which represents the amount of children in the household. The more children the less time ( if the function represents the amount of time free for the mother then the constraint would be -R) a parent would have on themselves. Their routine would dramatically change over time as their children grow and establish other worldly connections, this routine would differ greatly with a person who has R = 0, that is... no children. So we see that, small systems in the busy city can 'act' like fractals in the sense that we can idealize one definition (equation) and then play out different variations... over time we shall see complicated systems. 

Some systems in the city may appear quite stable... take for instance the actual structure of the earth beneath the city. Over a period of time t let's say 40 years, some systems in our city will have changed dramatically but the earth's surface may have only change slightly. Over millions of years, the surface will have changed dramatically.

Part of a great game

Think of any system be it a persons life, the animal kingdom (evolution), culture, the universe and even a mm by mm group of algae. These systems are bound by mathematical rules which at the beginning (initial conditions) they seemed reasonably predictable, but over time these systems become complicated, chaotic and create beautiful patterns. They are all part of a game defined by rules. Is each system exploring every possibility of its rules? If this is not possible then are they taking unique routes through an infinitely long mathematical pattern?

On the surface things may appear to behave non-mathematically but through closer inspection we see that they resemble mathematical patterns and functions. 

Even if i haven't produced a paper or a theorem governing all these systems (which of course no one man or (even humans?) could do ). I have presented and elucidated a very simple idea, the idea that the universe around us is made up of intertwined complex and simple systems which each can be comprehended. 

These systems are beautiful and are governed by very beautiful mathematics. 

Though we must always take not of Einstein's thought in our brains: 

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.
Albert  Einstein

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