The Infinite Coastline

In mid 20th century, mathematician Lewis Fry Richards discovers a strange paradox. For any coastline, the length of the coastline depends greatly on the ruler being used. The finer the ruler with more likely one will be able to measure bays can jagged edges that would be ignored by the larger one, thus the smaller the ruler the longer the measurement of the coastline. This brings up two paradoxical questions. With a fine enough ruler, is the coastline infinite and if so how can there exist only a finite amount of land?

Koch's Snowflake
The koch’s snowflake being generated. 

It turns out, according to an article by Benoit Mandelbrot, the coastline of landmasses such as England may actually be infinite however the area is finite. It also turns out that there are shapes such as some fractals that also exhibit the same property. The  Koch’s snowflake, is a fractal known an infinite perimeter but only a finite area. This is because on each iteration of the snowflake, the sides gets smaller but the number of sides increases at a faster rate. This means that once the sides are sufficiently small the number of sides is infinite.

Animated_fractal_mountainThe self similar nature and infinite nature of fractals can also be used to generate realistic coastlines with bays and jagged rocky outcrops. Fractals can also be used to generate entire islands and planets, as well as plants and trees. This technique is employed in many films and video games to produce realistic and awesome scenery.

A example of a fractal landscape
A example of a fractal landscape

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