Mathematical Investigation:

VON KOCH'S SNOWFLAKE COMPETITION

Ha Yeon Lee 11B

Mathematics HL

• Advantages:

➢ Great Von Koch's Snowflake Competition

The Koch snowflake is a mathematical curve, which is believed to be among the earliest fractal curves with description. In 1904, a Swedish mathematician, Helge vonseiten Koch presented the construction from the Koch contour on his newspaper called, " On a ongoing curve without tangents, constructible from fundamental geometry”.

➢ In this mathematical process, I am going to research how the area and edge of a shape/curve changes to see whether they boost by the same number whenever, as this process is usually repeated:

i. Start with a great equilateral triangle.

ii. Split each side from the triangle in to three equal segments.

3. On the midsection part of either side, draw a great equilateral triangle by connecting lines.

4. Now remove the line section that makes the base of the smaller sized triangle that was formed in coordination 3.

The above procedure (steps i~iv) can be repeated indefinitely. The design that comes forth is called " Von Koch's Snowflake” to get obvious reasons. An equilateral triangle, which can be the shape accustomed to start with to draw the Koch Snowflake curve, transforms its form similar to a legend or a snowflake as both sides of the earlier curve will be shifted out.

• Process:

In this investigation, the drawing the Koch curve has to replicate in order to extend rules intended for both perimeter and region.

← Perimeter: Within the assumption which the equilateral triangle (so-called C0) at the very start has a perimeter of three units, discover the perimeter for the next curves (C1, C2, C3, so on), and ultimately, find the perimeter of Cn.

During the second iteration, once extra equilateral triangles are added around the middle element of each side of the new curve, C1, the perimeter boosts to [pic]. The perimeter of [pic] of C2 is definitely the combination between the previous perimeter of C1 and extra length of collection segment:

Throughout the third version, the Koch curve, C3, gains extra line sections, and therefore, the significance of the edge continues to increase from that with the previous competition:

The perimeter of C3 is became [pic] product.

Based on my personal findings inside the perimeter in the Koch shape, I a new table that will enable me to determine general habits much quickly:

|n (number of iterations) |Perimeter (in unit) | |0 | three or more | |1 |4 | |2 |[pic] | |3 |[pic]

↑Table: Examining the Difference in the Perimeter of the Competition after each Iteration

If I list all values in the perimeter in a row, I can see that following each iteration, [pic] (common ratio) is constantly multiplied.

- Number of edges

After each version, I found that one side from the curve from your preceding stage has become four sides in the following level. The construction with the Koch contour begins with three attributes and therefore the solution for the number of sides may be expressed because below:

Range of sides sama dengan 3 [pic]4n (for the nth iteration)...

Cited: Brownish, Diana. Diagrams of Vonseiten Koch Snowflake Curves. March, 12th, 2009

" Fractal”

" Koch Snowflake”. Wikipedia. 2009. Wikimedia Foundation. August, 18th, 2009.