How do you find the length of the Koch curve?
The article by Sime Ungar provides a simple geometric proof. The length of the intermediate curve at the nth iteration of the construction is (4/3)n, where n = 0 denotes the original straight line segment. Therefore the length of the Koch curve is infinite.
What is the equation for the Koch snowflake?
3 s 2 4 ( 1 + ∑ k = 1 n 3 ⋅ 4 k − 1 9 k ) . Letting n go to infinity shows that the area of the Koch snowflake is 2√35s2 2 3 5 s 2 . Since the area of the original equilateral triangle is √34s2 3 4 s 2 , this means that the area of the snowflake is 8/5 times the area of the original equilateral triangle [Details].
What is the fractal dimension of the Koch curve?
1.26
The relation between log(L(s)) and log(s) for the Koch curve we find its fractal dimension to be 1.26.
What is meant by Koch curve?
A Koch curve is a fractal curve that can be constructed by taking a straight line segment and replacing it with a pattern of multiple line segments. Then the line segments in that pattern are replaced by the same pattern.
How do you make a Koch curve?
Construction
- Step1: Draw an equilateral triangle.
- Step2: Divide each side in three equal parts.
- Step3: Draw an equilateral triangle on each middle part.
- Step4: Divide each outer side into thirds.
- Step5: Draw an equilateral triangle on each middle part.
How do you measure fractal length?
D = log N/log S. This is the formula to use for computing the fractal dimension of any strictly self-similar fractals. The dimension is a measure of how completely these fractals embed themselves into normal Euclidean space.
How is the Koch curve formed?
The Von Koch curve is a fractal. The rule for generating this curve is to start with an equilateral triangle and to replace each line segment by a zig-zag curve (a generator) made up of 4 copies of the line segment it replaces, each reduced to one third of the original length.
How is fractal length calculated?
Who made Koch curve?
Swedish mathematician Niels von Koch published the fractal that bears his name in 1906. It begins with an equilateral triangle; three new equilateral triangles are constructed on each of its sides using the middle thirds as the bases, which are then removed to form a six-pointed star.
What is the dimension of the Koch snowflake?
infinite 1-dimensional
This value of D is called the Hausdorff dimension of S. For example the von Koch snowflake has infinite 1-dimensional measure (length) and zero 2-dimensional measure (area). So its Hausdorff dimension is somewhere in between – in fact it is the same as its box-counting dimension, namely 2 ln(2)/ ln(3).
What is the difference between Koch curve and snowflake?
The Koch snowflake (also known as the Koch curve, star, or island) is a mathematical curve and one of the earliest fractal curves to have been described. A Koch curve is a fractal generated by a replacement rule.
How do you draw a Koch curve?
How do you make a fractal formula?
Creating a new formula
- Create a new fractal.
- Click New on the File menu, and then click Fractal Formula File.
- Click New Formula on the Insert menu.
- After the init: label, insert the following line:
- After the loop: label, insert the following line:
- After the bailout: label, insert the following line:
What is the area of a Koch snowflake?
Area of the Koch Snowflake For our construction, the length of the side of the initial triangle is given by the value of s. By the result above, using a = s, the area of the initial triangle S(0) is therefore √34s2 3 4 s 2 .
What is the math behind fractals?
One of the more standard methods to measure fractals is to use the Hausdorff Dimension, which is D = log N / log s, where N is the number of parts a fractal produces from each segment, and s is the size of each new part compared to the original segment.