Difference between revisions of "Koch's Snowflake 2"

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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.
 
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.
  
These properties are extremely easy to see with the Snowflake, which is why it is commonly used as an introduction to fractals. Its simple construction also allows for us to clearly calculate and understand fractal dimension, as well. The Snowflake also resembles some natural fractals such as coastlines, and fjords, where a finite area is surrounded by a complex and self-similar boundary.  
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These properties are extremely easy to see with the Snowflake, which is why it is commonly used as an introduction to fractals. Its simple construction also allows for us to clearly calculate and understand fractal dimension as well. The Snowflake also resembles some natural fractals such as coastlines, and fjords, where a finite area is surrounded by a complex and self-similar boundary.  
  
  

Revision as of 12:38, 23 July 2009


Sol-Koch
Sol-Koch.jpg
Fields: Fractals and Dynamic Systems
Image Created By: SolKoll
Website: common.wikimedia.org

Sol-Koch

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.


Basic Description

Curve Construction

Koch Construction.jpg


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.

First 7 iterations


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.

These properties are extremely easy to see with the Snowflake, which is why it is commonly used as an introduction to fractals. Its simple construction also allows for us to clearly calculate and understand fractal dimension as well. The Snowflake also resembles some natural fractals such as coastlines, and fjords, where a finite area is surrounded by a complex and self-similar boundary.


Click here, for more information about Iterated Functions.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Calculus in some sections
Infinite Perimeter of the Koch Snowflake

Fractal Properties

Self-similarity

The Koch Snowflake displays a property known as self-similarity, emphasized in the animation. This means that as we continue to magnify the Koch Snowflake, each magnified section continues to look similar to the larger perspective.



Fractal Dimension

2nd iteration of Koch Snowflake

The fractal dimension of a Koch Snowflake can be calculated using the formula for fractal dimension: \frac{logN}{loge}.

Taking the image shown to the left, the top diagram shows that the new new Koch Curve lengths are a third of the previous iteration's length after the second iteration, and so e = 3. The bottom diagram shows that there are now a total of 4 Koch Curves, so that N = 4.


Using the formula for fractal dimension: \frac{logN}{loge} = \frac{log4}{log3} \approx 1.26\,.

Demonstration

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Other Properties

Number of Sides (N_k\,)

Number of Sides

Let us observe the number of sides of a Koch Snowflake at its first iterations:

The starting triangle has a total number of sides: 3 = 3*1 = 3*4^0\,
The 1st iteration has a total number of sides: 12 = 3*4 = 3*4^1\,
The 2nd iteration has a total number of sides: 48 = 3*16 = 3*4^2\,

With the first three iterations, we can see a pattern emerging. The "3" coming from the sides of the starting triangle, and the "4" reflecting the number of new sides that emerge from each side of the previous iteration.

Thus, the total number of sides on a Koch Snowflake with a degree of iteration (k) is given by N_k = (3)4^k\,.


Side Length (L_k\,)

Side Lengths

The length of the sides of the Koch Snowflake at each iteration also follows a pattern. Each time the snowflake undergoes an iteration, every side of the fractal creates a protruding equilateral triangle so that the side lengths shorten by a factor of 3.

Each side of the starting triangle has a length: x \,
Each side of the 1st iteration has a length: \frac{x}{3} = \frac{x}{3^1} \,
Each side of the 2nd iteration has a length: \frac{x}{9} = \frac{x}{3^2} \,
The length of a side at any given degree of iteration (k) can be determined by L_k = (x)3^{-k}\,, where x is the side length of each of the three sides of the original triangle.


Perimeter (P_k\,)

Perimeter

As stated before, the perimeter of a Koch Snowflake lengthens infinitely. Let us observe the first few iterations of the snowflake to determine by what ratio the perimeter increases at each iteration. To calculated the perimeter of the fractal at any given degree of iteration, we multiply the number of sides by the length of each side:

P_k = N_k * L_k\,

P_k =(3)4^k * (x)3^{-k} = \left( \frac{4}{3} \right)^k(3x) = \left (\frac{4}{3} \right)^kP_0\,, where P_0 is the perimeter of the original triangle.

Thus, the perimeter increases infinitely by a ratio of 4/3 of the perimeter for the previous iteration.

As the number of iterations approaches infinity, we can see that the perimeter also approaches infinity: \lim_{k\rightarrow \infty}P_k= \left (\frac{4}{3} \right)^kP_0 = \infty.


Area (A_k\,)

First iteration of the Koch Snowflake
Second iteration of the Koch Snowflake
Third iteration of the Koch Snowflake

Finally, let us examine the area of the Koch Snowflake and prove that it is indeed finite. We assume that the area of the original triangle is A_0\,, and determine the area that each of the first few iteration adds to the total previous area (see the images to the right for visual illustrations).

The original area is: A_0\,
The 1st iteration adds: (3) \frac{1}{9}A_0 = \frac{(3)}{9}A_0\,
The 2nd iteration adds: (12)\frac{1}{9^2}A_0 = \frac{(3)(4)}{9^2}A_0\,
The 3rd iteration adds: (48)\frac{1}{9^3}A_0 = \frac{(3)(4^2)}{9^3}A_0\,

Thus the total area of the Koch Snowflake at the third iteration will be the summation of the expressions above: A_3 = A_0 + \frac{(3)}{9}A_0 + \frac{(3)(4)}{9^2}A_0 + \frac{(3)(4^2)}{9^3}A_0 \,.


Or more generally, to find the area (A_k) at a degree of iteration k:

A_k = A_0 + \frac{(3)(4^0)}{9^1}A_0 + \frac{(3)(4^1)}{9^2}A_0 + \frac{(3)(4^2)}{9^3}A_0 +...+ \frac{(3)(4^{k-1})}{9^k}A_0 .

A_k = A_0 \left(1 + \frac{(3)(4^0)}{9^1} + \frac{(3)(4^1)}{9^2} + \frac{(3)(4^2)}{9^3} +...+ \frac{(3)(4^{k-1})}{9^k} \right) \,.

A_k = A_0 \left (1 + \frac{3}{9}\left[1 + \left(\frac{4}{9}\right)^1 + \left(\frac{4}{9}\right)^2 + \left(\frac{4}{9}\right)^3 + ... + \left(\frac{4}{9}\right)^k \right] \right)\,

A_k = A_0 \left [1 + \frac{3}{9}\sum_{k=0}^\infty \left(\frac{4}{9}\right)^k \right], which contains a geometric series that will converge


\lim_{k\rightarrow \infty}A_k= A_0\left[1+\frac{3}{9}\left(\frac{1}{5/9}\right)\right] = \frac{8}{5}A_0\,

Thus, the Koch Snowflake is approximately 8/5's of the area of the original triangle.




Teaching Materials

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References

Cynthia Lanius, Cynthia Lanius' Fractal Unit:Koch Snowflake Larry Riddle, Koch Curve





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