Koch's Snowflake 2
- The image is an example of a Koch Snowflake, a fractal that first appeared in a paper by Swede Niels Fabian Helge von Koch in 1904. It is made by the infinite iteration of the Koch curve.
The Koch Snowflake was first described in 1904, but it waited until 1967 to gain what is today considered its main significance. That year, Benoit Mandlebrot published a paper explaining an earlier finding that as the length of a country's border is measured with increasingly fine measurements, the measured length does not get closer and closer to a specific, "true" measurement of the border. Surprisingly, the measured length instead grows exponentially, suggesting that the true length of a border is infinite.
Mandlebrot's paper, entitled "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", showed that this strange finding could be explained if the border of a country were a fractal. Fractals, which were later found to occur not just in borders, but also throughout nature in such places as the shape of coastlines and the branching of blood-vesslels, are marked by self-similarity, meaning that no matter how much you zoom in on one small section of a fractal, parts of it continue to look like the whole original border, with just as much detail.
Mandlebrot also showed in "How Long is the Coast of Britain" that this self-similarity could give rise to other strange properties in fractals. For instance, it turns out that it doesn't make sense to describe most fractals as being 1 or 2 dimensional, but rather as something in between.
To illustrate these properties, Mandlebrot didn't actually use coastlines, but rather used relatives of the Koch Snowlake, which is one of the simplest of all fractals. By using this shape rather than a coastline, Mandlebrot was able to show how these surprising properties could exist in any fractal and provide some insights into how they might be measured.
The curve begins as a line segment and is divided into three equal parts. A equilateral triangle is than created, using the middle section of the line as its base, and the middle section is removed.
The Koch Snowflake is an iterated process. It is created by repeating the process of the Koch Curve on the three sides of an equilateral triangle an infinite amount of times in a process referred to as iteration (however, as seen with the animation, a complex snowflake can be created with only seven iterations - this is due to the butterfly effect of iterative processes). Thus, each iteration produces additional sides that in turn produce additional sides in subsequent iterations.
An interesting observation to note about this fractal is that although the snowflake has an ever-increasing number of sides, its perimeter lengthens infinitely while its area is finite. The Koch Snowflake has perimeter that increases by 4/3 of the previous perimeter for each iteration and an area that is 8/5 of the original triangle.
Click here, for more information about Iterated Functions.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Calculus in some sections
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