Koch's Snowflake 2
From Math Images
Basic DescriptionThe Koch Snowflake was first described in 1904, but it was not until 1967 that some of its most significant properties became apparent. 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 measurement devices, the measured length does not approach a specific, "true" measurement of the border, as one would expect with increasingly precise measurement. Instead, the measured length grows exponentially, suggesting that given a fine enough measuring device, there would be no upper limit on how long the measured length of the border could be.
Mandlebrot's paper showed for the first time that this strange finding could be explained if the border of a country resembled a fractal, shapes which are largely defined by self-similarity , and have since been found throughout nature in such places as coastlines and the branching of blood vessels.
Mandlebrot's paper went on to show that this self-similarity could lead to other counter-intuitive properties in fractals as well. For instance, Mandlebrot showed that it doesn't make sense to think of most fractals as being 1 or 2 dimensional, but rather as something in between.
To derive and demonstrate how self-similar shapes can behave in these ways, Mandlebrot didn't actually use coastlines, but rather used variations of the Koch Snowlake, because of its relative geometric simplicity. With this example, he was able to show how these surprising properties could exist in any fractal and to provide insights into how they might be measured and interpreted. These insights continue to form the basis of much of our analysis of fractals today.
Koch Curve Construction
The curve begins as a line segment and is divided into three equal parts. A equilateral triangle is then 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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