Tuesday, 28 February 2012

The Great Lessons of Pablo Picasso

It took me four years to paint like Raphael, but a lifetime to paint like a child.

Pablo Picasso

The lifetime work of this great man may seem very puzzling to any person studying his life or any artist looking at his style. Picasso was a prodigy in this creative field, who could draw and paint as good as the masters (Raphael, Da Vinci, Rembrandt etc.) at a very young age. To some people this meant Picasso had reached the peak of his artistic talent and powers. Some do not quite understand why he went against this academic-ism.

It is true by fact that no man or machine can yet perfectly imitate reality (photos are 1 dimensional so they do not perfectly imitate reality), so then why is there this tendency to attempt the impossible. Sometimes it is a show of great skill to produce works of fine detail but a great work of art contains ingenuity and character. Leave modelling and mimicking reality to the scientists.  

Picasso in his later works used prevailing ideas in art and mixed them around and played with them, he produced new ideas in art such as Cubism (with Georges Braque). Pablo broke barriers and surprised the world. 

The quote above contains one great lesson which I have learnt from it: Do not copy what your elders did put    become a child, that is, become extremely playful and curious. This will lead you to wonderful places.

As a student of physics this is a great lesson...  always stay curious about the world around you and as Isaac Newton said 'stand on the shoulders of giants' but don't just sit there... go to new heights.

Pablo was full of confidence and would love to publicly show his zeal and talents. I think confidence in any intellectual is very important but not necessary for a great intellectual. To be open with the public and to discuss ideas, views and even some misguided opinions helps one to improve their view of the world and creates debate within the public.

This man was very original and productive and is an inspiration to us all. I personally think that art is a therapy for the creative functions for ones brain and not a gateway to truth of any sort, for truth takes the form of logical certainty or empirical approximation which of course can only be expressed in language (mathematical and lingual). 

Here is a video showing Picasso as a productive, happy and confident young artist who pushed the conceptual and practical limits of his field. The music is also very catchy!

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

Friday, 10 February 2012

We must Improve mathematics education in England

By Luke Kristopher Davis

A recent study by the Royal Society of Arts concluded that only 15% of young people continue to study mathematics past GCSE level. In Japan this figure is 85% and in other countries such as New Zealand it is 47%  and in South Korea it is 57%. It seems as though the UK (not just England as the UK averaged 13-14%) is slacking in one of the most important areas of education.

It is almost certain that a significant percentage of young people will not want to continue a full academic education of mathematics up to university standard, but it is necessary that all young people study a degree of maths until aged 18. This proposed spectrum of mathematical education may disturb those of you who love and cherish pure mathematics. Those young people who want to venture out into vocational or non-academic careers must leave with some functional ability with computation and understanding of maths.
Susan Anderson, CBI Director for Education & Skills policy, said: 
"Maths is a subject of critical importance, and this report (By Taskforce and Carol Vorderman) rightly highlights that there needs to be more focus on teaching 'useful maths' that is relevant for future employment and day-to-day life." 
"Businesses are most concerned about basic levels of numeracy and it's alarming that more than one in five 16-19 year olds are considered functionally innumerate.
Susan also encourages the idea that students should prolong their education in mathematics:

"To help address this problem, all young people should continue to study some form of maths until the age of 18. Pupils with good maths ability should continue to study the full curriculum, but all pupils, regardless of ability, should go on to study a functional numeracy qualification."
I think this is a 'win-win' idea as it would save costs for businesses (private and public) on training new employees who do not have sufficient numerical ability. Also, depending on the standard and content of this new curriculum, the general financial awareness of the population will increase if we assume a correlation between financial competence and mathematical ability.

The outcomes of the CBI report which was out last August and the recent RSA study have slightly hit me.
Mathematics to me is a beautiful tool to help us humans describe and explain the world, sometimes mathematics describes the world so well you think that mathematics is actually the basis of the world.

The fact that a great number of young people (and of course older generations) may not realize how powerful and beautiful mathematics is, reflects the lack of depth, wonder and rigor that exists in our mathematics education.

In the economist 'The World in 2012' there are a few pages which contain statistical economic information about different countries. Our GDP growth  2011-2012 is 0.7%  which is 3.142857 smaller than Japans GDP growth (2.2%). China, which is also a nation keen on mathematics, has a huge GDP growth of 8.2%.

These countries are also expanding and growing scientifically, which we know necessitates a growth in mathematical education (in order to perform science at its best).

So mathematics will help us grow economically by saving businesses money spent on mathematical training, improving scientific research and technological growth, decreasing unemployment as major fields require people with mathematical competency. I also think that mathematics if its education becomes much more engaging, rigorous and playful will enrich the intellectual capacities in our population, which of course is a good thing.

Here is a video highlighting why mathematics is addictive, playful, beautiful and of course extremely useful: