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Von Koch Investigation

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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: …show more content…

⇒ The investigation of the area begins by supposing that C0, the initial curve (an equilateral triangle) has a total area of 1 unit2. [pic] The area of the initial curve is, therefore, [pic] [pic] [pic] = [pic] [pic] = 1 -First Iteration: *Since s2 = [pic], Area of C1 is [pic]units2. -Second Iteration: *Since s2 = [pic], Area of C2 is [pic]units2. -Third Iteration: *As s2 = [pic], Area of C3 is [pic]units2. Based on my findings in the area of the Koch curve, I created a table that will allow me to see general patterns much easily: |n (number of iterations) |Area (in unit2) | |0 |1 |1 | |1 |1+[pic]

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