Imagine you wanted to measure the coastline of Great Britain. You might remember from calculus that straight lines can make a pretty good approximation of curves, so you decide that you’re going to estimate the length of the coast using straight lines of the length of 100km (not a very good estimate, but it’s a start). You finish, and you come up with a total costal length of 2800km. And you’re pretty happy. Now, you have a friend who also for some reason wants to measure the length of the coast of Great Britain. And she goes out and measures, but this time using straight lines of the length 50km and comes up with a total costal length of 3400km. Hold up! How can she have gotten such a dramatically different number?
It turns out that due to the fractal-like nature of the coast of Great Britain, the smaller the measurement that is used, the larger the coastline length will be become. Empirically, if we started to make the measurements smaller and smaller, the coastal length will increase without limit. This is a problem! And this problem is known as the coastline paradox.
By how fractals are defined, straight lines actually do not provide as much information about them as they do with other “nicer” curves. What is interesting though is that while the length of the curve may be impossible to measure, the area it encloses does converge to some value, as demonstrated by the Sierpinski curve, pictured above. For this reason, while it is a difficult reason to talk about how long the coastline of a country may be, it is still possible to get a good estimate of the total land mass that the country occupies. This phenomena was studied in detail by Benoit Mandelbrot in his paper “How Long is the Coast of Britain" and motivated many of connections between nature and fractals in his later work.
late 50s, happily married later in life, generally optimistic (but with the occasional deeply dark mood), eager to explore all forms of artistic expression, published 3 fantasy novels as a young man (2 of which were horribly overwritten), introverted with a few close friends but much more out of my shell these days :)
tenuous: Clouds over Mars, photographed by Mars Express, 6th June 2008.
81°N 248°E to about 24°N 270°E, on the Vastitas Boralis just south of the Rupes Tenuis, a cliff forming part of the edge of the north polar plateau. The dark region in the 2nd image is a small dune sea, and the clouds seen may be high-altitude water-ice clouds.
Drawn from the martian Classical melange, tenuis is a Latin word meaning thin or fine, from whence the English tenuous.
Composite of 5 frames.
Image credit: ESA. Animation: AgeOfDestruction.
"Bam! This is the 3rd single from the #XXII art project! It features #SonsOfLiberty forming Voltron again for the first time since 2013! It features all recording artistes and is an anthem for all our supporters, including #TeamKB Enjoy!" - King Biggs
Produced by The Defenders (King Biggs x Heatwave) and J Chris Beats. Mixed by Heatwave. Artwork by Michael Farquharson.