The Helge von Koch fractal is one of the most famous fractals because it looks exactly like a snowflake! The fractal is one of the closest ones relating to nature because of its similarity to snow!
History
In 1904, Neils Fabian Helge von Koch discovered the von Koch curve which lead to his discovery of the von Koch snowflake which is made up of three of these curves put together. He discovered it while he was trying to find a way that was unlike Weierstrass’s to prove that functions are not differentiable, or do not curve. Also, he wrote a paper about the von Koch snowflake that proved that a figure can be continuous everywhere, but not differential.
Furthermore, before the von Koch fractal was created, mathematicians thought that if a function is continuous, it would have tangent lines everywhere. They were saying that a function could not be all corners. However, every iteration of the von Koch snowflake, a corner is added so von Koch disproved this idea.
Furthermore, before the von Koch fractal was created, mathematicians thought that if a function is continuous, it would have tangent lines everywhere. They were saying that a function could not be all corners. However, every iteration of the von Koch snowflake, a corner is added so von Koch disproved this idea.
How is it made?
The von Koch snowflake is made starting with a triangle as its base. Each iteration, each side is divided into thirds and the central third is turned into a triangular bump, therefore the perimeter increases. However, the same area is contained in the shape. That’s crazy right?! Also, you can just put a triangle 1/3 of the size of the first equilateral triangle and stick it onto the center of the sides of the triangle! Keep putting the triangles on 1/3 of the size from the previous iteration to make the fractal!
Also, it can be created starting with a straight line. Then, it is divided into three equal segments and these segments form an equilateral triangle with the middle segment as the base. Then the base is removed which creates the first iteration of the von Koch fractal.
Also, it can be created starting with a straight line. Then, it is divided into three equal segments and these segments form an equilateral triangle with the middle segment as the base. Then the base is removed which creates the first iteration of the von Koch fractal.
Side Length and Perimeter
In every iteration, one side is transformed into four sides. Therefore the equation for the number of sides of the snowflake is n = 3 * 4^a, which uses a as the number of iterations.
Also, for every iteration, the length of a side is shorter by ⅓. If x is the length of the snowflake from the previous iteration, use the equation length = x * 3^-a to find the length of the side of the snowflake.
However, for every iteration, the curve of the fractal grows to be 4/3 of its previous length, since new segments are added to the fractal every iteration.
Since the perimeter is the number of sides multiplied by the length of the sides, it can be found using the equation p = (3*4^a)(4/3)^a. So as the iteration increases toward infinity, the perimeter increase by no bound.
Also, for every iteration, the length of a side is shorter by ⅓. If x is the length of the snowflake from the previous iteration, use the equation length = x * 3^-a to find the length of the side of the snowflake.
However, for every iteration, the curve of the fractal grows to be 4/3 of its previous length, since new segments are added to the fractal every iteration.
Since the perimeter is the number of sides multiplied by the length of the sides, it can be found using the equation p = (3*4^a)(4/3)^a. So as the iteration increases toward infinity, the perimeter increase by no bound.
Since the perimeter of the fractal grows faster than it would linearly, the perimeter gets larger faster compared to the rest of the shape. A real world example would be when Charless Fowlkes made a pecan pie in the shape of the von Koch snowflake. This pie provided more crust to the large shape than a circular pie would have of the same size because the circumference scales linearly.