The Coastline Paradox (or How to Fail a Student in Geography)
We are used to thinking that mathematics is a tool that allows other sciences to describe the world around us more accurately and sometimes find patterns where they were not visible at first glance. However, today we will talk about how the opposite happened and how a funny geographical incident led to the emergence of a completely new section of mathematics.
If you ask Google “What is the length of the shoreline of Lake Baikal?”, different websites will offer you different answers: from 1850 to 2100+ kilometers. Doesn't that bother you? Yes, the discrepancy in data is more than 250 km! *Those who know what the trick is, please don't spoil it*. But how did it happen, are our people really so careless about geographical measurements? However, Lewis Fry Richardson faced the same problem back in the early 20th century.
Richardson was a very interesting person in his own right: an English scientist, fascinated by everything from mathematics (maybe someone has heard of Richardson's jaw-dropping extrapolation) to meteorology (his work in this area is the basis of modern weather forecasting). In addition, he was one of the first inventors of the echolocation method of detecting icebergs – the man was so impressed by the tragedy of the Titanic. In general, he managed to shine everywhere. But we will not dwell on his personality, otherwise a separate article can be written here, let's just say that Richardson was, in addition to everything, a deeply convinced pacifist, so his research often touched on the theoretical causes of military conflicts.
Trying to understand whether the probability of war between two countries depends on the length of their common border, he noticed a significant discrepancy in the values from different sources. According to official data from the Spanish, the length of their border with Portugal was 987 kilometers, while the Portuguese measured it as 1214 kilometers. Again, a spread of 200+ km. So which value should be used for the study? It seems that the Europeans also had problems with counting…
There is no secret here, and as most have already guessed, the answer lies in the measuring instrument: the smaller your, so to speak, “ruler”, the more unevenness of the relief can be taken into account. So there is no wrong data (however, just as there is no right one), and the Portuguese, in all likelihood, used a shorter ruler than the Spanish. The same applies to the coastline of Baikal: in order to know its length, it is necessary to indicate the scale of the map on which the calculations were made, because the larger the scale, the more bays, coves and bends are displayed on it.
*Well, yes, well, yes, I understand that this is all obvious, but foreplay is a must*
Richardson, of course, also immediately understood what was going on, but, unlike his predecessors, who ignored the discrepancies in the data, he was deeply affected by this fact. In his article, published in 1961, he concluded that as the unit of measurement decreases, the length of a coastline or borderline tends to infinity and has no fixed value in principle *unless you start measuring everything by atoms*.
As often happens, the article didn't get much attention back then, but today this amazing discovery is called the Richardson effect or the coastline paradox. That's it. Thanks for your attention, like, subscribe… Oh yeah, I thought we were talking about mathematics. How did it get in here among the rulers, maps, and the Spanish and Portuguese?
The thing is, without realizing it, Richardson became the discoverer of a new discipline. Six years later, in an article in Science, mathematician Benoit Mandelbrot again described the coastline paradox in detail, and in 1975 he came up with a name for such infinitely complex objects as the jagged border of coastlines. Thus was born the beautiful word – fractal.
For an object to deserve the title of a fractal, it is enough to have a complex structure, regardless of the scale of consideration – this is the most understandable (but also the most amateurish) definition of this term. In the real world, of course, we cannot infinitely increase the scale or infinitely decrease the units of measurement, but still the Spanish-Portuguese border is a fairly close analogue of a fractal, and a very visual one at that.
Mandelbrot, who enthusiastically developed the unexplored field of mathematics, called it “beautiful, devilishly difficult and more valuable every day.” Other scientists followed him, discovering more and more new types of fractals, already purely mathematical.
An example of one such fractal is probably known to many, it even appeared somewhere on the cover of a textbook – this is a snowflake or Koch curve. It is funny that Helge von Koch himself described this figure back in 1904, although for him it was just entertaining mathematical gymnastics, nothing more. It is easy to construct: divide the segment into three parts, build an equilateral triangle on the central part, erase the base and repeat the actions from the beginning on all the new segments obtained.
If we take and connect the ends of this curve, we get a beautiful snowflake, which was called the Koch snowflake *how unexpected*. At the same time, the resulting figure still has an infinite length! – no matter how you zoom in, no matter how much you increase the scale, you will always see a complex jagged border. Just like with the shoreline of Lake Baikal. By the way, this fact is one of the extremely interesting properties of closed fractals – the curve that forms them has an infinite length, but at the same time limits the precisely calculated final area, because both the Koch snowflake and Lake Baikal can be completely placed in a circle.
So, while the surface area of Lake Baikal can be measured quite accurately, the length of its shores will vary in different sources. Think about it if you are a geography teacher and feel the urge to torment one of your students with tricky questions.
Author: Alexander Griboyedov
