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 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.
While the Koch Snowflake itself rarely appears in nature, it acts as a sort of template for shapes, such as coastlines and fjords, that do appear in many real places. These objects, called fractals, have many properties that defy common intuition, despite their ubiquity. For example, as was described above, such shapes can enclose a finite area and yet have infinite perimeter. See the fractals page for more examples of the strange behavior of fractals.
Understanding how these properties can arise in a relatively simple fractal like the Koch Snowflake can help mathematicians and lay-people alike get a glimpse of how such strange behavior could exist in more complicated, but fundamentally similar, objects that appear in so many natural phenomena.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Calculus in some sections
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