Von Koch Investigation

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  • Topic: Curve, Koch snowflake, Fractal
  • Pages : 11 (1267 words )
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  • Published : September 9, 2012
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Mathematical Investigation:

VON KOCH’S SNOWFLAKE CURVE

Ha Yeon Lee 11B
Mathematics HL

• Introduction:

➢ History of Von Koch’s Snowflake Curve

The Koch snowflake is a mathematical curve, which is believed to be one of the earliest fractal curves with description. In 1904, a Swedish mathematician, Helge von Koch introduced the construction of the Koch curve on his paper called, “On a continuous curve without tangents, constructible from elementary geometry”.

➢ In this mathematical task, I am going to investigate how the area and perimeter of a shape/curve changes and find out whether they increase by the same number every time,as the following process is repeated:

i. Start with an equilateral triangle.

ii. Divide each side of the triangle into three equal segments.

iii. On the middle part of each side, draw an equilateral triangle by connecting lines.

iv. Now remove the line segment that makes the base of the smaller triangle that was formed in step 3.

The above process (steps i~iv) can be repeated indefinitely. The shape that emerges is called “Von Koch’s Snowflake” for obvious reasons. An equilateral triangle, which is the shape used to start with to draw the Koch Snowflake curve, turns its shape similar to a star or a snowflake as each side of the previous curve is pushed out.

• Process:

In this investigation, the process of drawing the Koch curve has to repeat in order to generalize rules for both perimeter and area.

← Perimeter: Under the assumption that the equilateral triangle (so-called C0) at the very start has a perimeter of 3 units, find the perimeter for the next curves (C1, C2, C3, and so on), and eventually, find the perimeter of Cn.

During the second iteration, when extra equilateral triangles are added on the middle part of each side of the new curve, C1, the perimeter increases to [pic]. The perimeter of [pic] of C2 is the combination between the previous perimeter of C1 and an extra length of line segment:

Through the third iteration, the Koch curve, C3, gains extra line segments, and therefore, the value of the perimeter continues to increase from that of the previous curve:

The perimeter of C3 is ended up being [pic] unit.

Based on my findings in the perimeter of the Koch curve, I created a table that will allow me to see general patterns much easily:

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

↑Table: Investigating the Change in the Perimeter of the Curve after each Iteration

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

- Number of sides

After each iteration, I found that one side of the curve from the preceding stage has become four sides in the following stage. The construction of the Koch curve begins with three sides and therefore the formula for the number of sides can be expressed as below:

Number of sides = 3 [pic]4n (for the nth iteration)...
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