Fractals: Where Art and Science Meet. Part 1.

Winter holidays have never been a particularly merry season in Oxford, but rather a quiet one. Most students have left, both undergraduate and graduate, and even last applicants coming here for interviews are packing their suitcases and going home, with hopes to return here next year. Many of my classmates went to their home countries, to return only in January, when our exams come. 

Nevertheless, no matter how strong homesickness may be, when our internationally diverse class finally graduate, some of us will eventually settle in the UK and, perhaps after 5-7 years of hard work, will decide to become British citizens. Or, to be a bit more precise, will try to, as becoming a subject to Her Majesty has never been an easy feat. Even if all the formal requirements are fulfilled, an applicant seeking British citizenship may still fail to score the last point - the Life in the United Kingdom test.

The test itself is a serious and funny thing at the same time. Serious because an applicant actually has to pass it to fulfil the formal requirement to become a citizen. Funny because of questions which may show up and people who may fail it. In the past few years, the average pass rate was not too high - just about 70%. People who score high typically have no problems adapting to the life in the United Kingdom. The pass rate for Americans, Australians, Canadians and other nationals of countries with imperial past is close to 95%. Life is not that easy for natives of countries like Iraq or Afghanistan - the pass rate for people with pretty different cultural background is below 50%. Nevertheless, the group scoring among the lowest usually comes at big surprise since the people who are the least adapted to the life in the UK are ... the Brits themselves! In the past few years, more than 11 000 citizens have taken the sample test with only about 14% passing it. Nothing to say, even David Cameron, the Prime Minister and Oxford graduate with a 1st in PPE, got in trouble when he was challenged with some UK history questions...

However, there is at least one question that they will almost certainly will never ask. (No, that's not "How long did the Hundred Years War last?", and don't expect Captain Obvious to help you). This question is "How long is the coastline of Great Britain?" (not to be confused with England, Scotland, United Kingdom etc.).

The problem behind this seemingly innocuous question is not just the difficulty of quantifying anything with the word 'British' in it (think 'British climate' or 'British cuisine'). The real one is the extremely complex structure and high curvature of this coastline: there are lots of bays, small peninsulas, and other structures which are pretty difficult to measure in straight lines. Take a look at these two: if you measure it with units of 200 km...

... the length is about 2400 km, but with a smaller unit of just 50 km...

... it is close to 3400.

So, as conventional wisdom suggests, the smaller the unit you use, the better estimate you get. But how small should it be? 50, 20, 10, or even 0,5 m? If you choose a very small one, you will be able to find even more curves to fit in...

After reading all this a pretty reasonable question may arise: "Does anybody care?". Well, perhaps it would have been a problem only for the mapping authority, American spies (the latter tend to underestimate the length of the UK coastline for some 5 000 km), and, of course, Geography majors, but all of a sudden, the issue of British coastline's convolution caught an eye of a Polish-born Franco-American Jewish mathematician, who developed the whole theory from it which was later applied to dozens of other spheres, most of which have nothing to do with British coastline. And that's how it all started...

Benoît Mandelbrot was not first to notice the coastline paradox, but it was he who gave this property a name. Even though he did not use the word 'fractal' in his original 1967 paper, he coined it few years later and was a pioneer of fractal geometry.

So, what's a fractal, after all? Let's take another aspect of British nature. It does not snow often in Britain (but when in does, it may be a disaster), but perhaps it was a snowflake which inspired another mathematician (this time a Swedish one) to describe one of the most simple fractals: a figure with infinite length and finite area.

How could that be? Let's take an equilateral triangle and make another equilateral triangle on each side. Then do it again and again:

If you are patient enough to repeat till infinity, you will get a Koch's snowflake. The nice property of this snowflake (and perhaps of a real snowflake too) is that it repeats itself:

That's why a flake (and perhaps Britain) have infinite length of a curve. And that infinite self-repeating property is exactly what makes a fractal. (If you really like Koch's snowflake and want to waste time understand it better, you can have fun here)

OK, I like beautiful British coastline and snow, but where else can I see fractals? The answer is: almost anywhere.

Take a look at:

clouds,

leaves,

lightning,

and even financial markets.

However, one of the most exciting part of all this fractal fuss is fractal art.

There are at least two interesting things about it:

• a pretty complex image can be generated using only one formula (however, nobody promises that it will be simple enough);
• you don't really need to understand the 'hard math' behind it: just get the program, plug in some numbers, and get a nice picture to pin on your dorm room wall to inspire fear in your friends and make your mom proud.

But that will be covered in the next part.

Stay tuned!