December 23, 2021
Since the dawn of humanity, maths and geometry have been proven to be very useful and critical tools in the process of deciphering the world around us. Many claim that these sciences are “the language of the universe,” and for a good reason: nature is indissolubly linked with mathematical notions, from the simplest examples of organisms handling numbers and simple mathematical procedures, to complex geometric patterns and functions found in animals and plants.
What’s impressive about this connection between maths as a science and nature, is that relatively simple procedures, like measuring distance or demarcating space, could reach an impasse or a paradox. Starting from two-dimensional mathematical operations on paper, and touching upon the three-dimensional world around us, we notice many deviations and ambiguities. However, even in the most extreme of exceptions, there’s an equally extreme and well-thought mathematical model that will help us comprehend and decipher the phenomenon. The Coastline Paradox is a distinctive example.
This paradox was observed as a phenomenon for the first time by the English mathematician Lewis Fry Richardson, in his attempt to explain the deviation in measuring the shared borders between Spain and Portugal. But this is the problem: Let’s suppose that geographers and cartographers attempt to measure Great Britain’s coastline for scientific and informative purposes. Obviously, as is evident, coastlines aren’t straight lines. As scientists magnify the image of a line that appears straight, they notice how it’s made up of countless curves and turns. Actually, as much as someone tries to measure a coast with great precision, they never meet any straight lines for the process to get easier. This led scientists to use the smallest unit of distance measurement apt for each coastline, in order to have an approximative image. For example, if Great Britain's coastline is measured in units of 100 kilometres, its total number will be 2800 kilometres, while if units of 50 kilometres are used, the total is 3400 kilometres. In general, we come to the conclusion that the smaller the unit used in the measurement process, the bigger the total result. Therefore we cannot measure the real values of each country’s coastlines worldwide, but we can only approximate them. In theory, one might delve deeply into the molecular level and measure the straight lines that unite the coasts’ molecules, with the purpose of measuring the most precise value to exist. What’s impressive with this mental experiment is that, if someone tries to measure the molecular distance, the result that will come up will approach infinity. It’s riveting how infinity, a concept so out of this world and distant, is observed in something so humane and terrestrial, like the curves of a beach in Great Britain.
As mentioned above, though, there will always exist the appropriate mathematical schemes to partially explain this paradox. In geometry we have a self-repeated geometrical shape, the fractal. Its characteristic is that, as you zoom in a fractal’s image, you notice repeated, self-identical patterns of shapes, in different scales, that emerge through the repetition of any mathematical procedure. They’re shapes that enclose a measurable square footage but have an approximately infinite circumference. Anyone can think of a shape like this if they just imagine a snowflake, with small branches of ice that stretch equally towards different directions. Each snowflake is different, but it also contains a repeated pattern. Similar examples like this exist everywhere on earth, like tree and river branches, cloud shapes, various flowers, mountain shaping, and of course, coastlines.
So, the phenomenon of measuring a coastline, that humanity is unable to interpret using Euclidean geometry, was processed based on fractals and their properties. It’s a theoretically and approximately solved problem that proves the daily proximity of maths and geometry. What personally fascinates me the most in this connection is how complex and infinitesimal mathematical systems that arise after centuries of research and theoretical labour on different science branches, were inhabiting plants, the earth, and mountains, practically under our noses, in the first place.
Translated by Kiriaki Arnaouti, Reviewed by Panagiota Katsaveli
Recently turned 18, born and raised in Thessaloniki and he loves that city. He’s a first-year student in the department of Physics, the mother of all sciences, which has been his dream since he was a child. He’s someone with greatly high energy levels without always knowing where to channel it, something that makes him hyperactive. He loves electronic music and is an amateur dj (self-taught, don’t judge) while he’s also a devoted supporter of the Marvel cinematic universe (Team Captain America). The most important force for him is inspiration and it’s the only thing he wants to transmit to the people around him.
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