Von Koch’s Snowflake Curve Investigation
Von Koch’s Snowflake is named after the Swedish mathematician, Helge von Koch. He was the one who described the Koch curve in the early 1900s. The Koch curve is a mathematical curve that is continuous, without tangents. In this investigation, we will be looking at the particularities of Von Koch’s snowflake and curve. Including looking at the perimeter and the area of the curve. This investigation is continued by looking at the square curve as well as the triangle’s curve. The Von Koch’s snowflake is constructed by starting with an equilateral triangle. Then remove inner third of each side and build another equilateral triangle at the same place the side was removed. This process is followed over and over again, indefinitely until it creates a snowflake shape.
The process described is shown in this picture:
The Koch curve starts with a line segment with zero separated lines. Then the single line segment …show more content…
The main difference is that instead of an equilateral triangle, it is an equilateral square. To create the square curve, the sides of the square are removed and are replaced by another equilateral square. This process is followed indefinitely. After one segment of an equilateral square is formed into three segments, the equilateral square is formed into five segments. This is different than the segments from the snowflake because three segments became four, not five. Both the length of the perimeter and the total area are determined by geometric progressions. The progression of the area converges to two while the progression of the perimeter goes to infinity, just like the Koch snowflake. Another similarity to the Koch snowflake is that there is a finite area and an infinite fractal curve. The total area to the nth iteration is shown by An= (⅕)+ (⅘) (the sum of k=0 to the n terms) (5/9)^k. The perimeter is
Von Koch’s Snowflake is named after the Swedish mathematician, Helge von Koch. He was the one who described the Koch curve in the early 1900s. The Koch curve is a mathematical curve that is continuous, without tangents. In this investigation, we will be looking at the particularities of Von Koch’s snowflake and curve. Including looking at the perimeter and the area of the curve. This investigation is continued by looking at the square curve as well as the triangle’s curve. The Von Koch’s snowflake is constructed by starting with an equilateral triangle. Then remove inner third of each side and build another equilateral triangle at the same place the side was removed. This process is followed over and over again, indefinitely until it creates a snowflake shape.
The process described is shown in this picture:
The Koch curve starts with a line segment with zero separated lines. Then the single line segment …show more content…
The main difference is that instead of an equilateral triangle, it is an equilateral square. To create the square curve, the sides of the square are removed and are replaced by another equilateral square. This process is followed indefinitely. After one segment of an equilateral square is formed into three segments, the equilateral square is formed into five segments. This is different than the segments from the snowflake because three segments became four, not five. Both the length of the perimeter and the total area are determined by geometric progressions. The progression of the area converges to two while the progression of the perimeter goes to infinity, just like the Koch snowflake. Another similarity to the Koch snowflake is that there is a finite area and an infinite fractal curve. The total area to the nth iteration is shown by An= (⅕)+ (⅘) (the sum of k=0 to the n terms) (5/9)^k. The perimeter is