Tools to calculate the area and perimeter of the Koch flake (or Koch curve), the curve representing a fractal snowflake from Koch.
Koch Flake - dCode
Tag(s) : Geometry
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Koch Flake
Area Calculator
Perimeter Calculator
Answers to Questions (FAQ)
How to calculate the Koch Flake Perimeter?
The length of the border of the flake is infinite. At each iteration, a border of length 1 become 4/3.
Starting from a straight line segment divided by 3, a broken line of 4 segments os obtained: the length is therefore increased by 4/3 (increase of 33%).
Example: After 2 iterations, a line of initial length $ l $ has a new length of $ l \times \frac43 \times \frac43 = l \times \frac{16}{9} $.
If the number of iterations is infinite, the length is infinitely times increased by 4/3. The total length of this fractal curve is infinite.
$$ \lim\limits_{n \to +\infty} \left( \frac43 \right)^n = +\infty $$
How to calculate the area of the Koch flake?
The area of the flake is finite and equals $ 8/5 $ of the area of the initial triangle.
For a side of the triangle $ a $, the final area of the flake is $ \frac{2a^2\sqrt{3}}{5} $
How to draw a Koch flake?
The algorithm is as follows:
0 - Draw an isosceles triangle and for each side (segment)
1 - Calculate the points at 1/3 and 2/3 of the segment
2 - Draw the isosceles triangle based on the segment formed with the 2 points found
3 - Remove the base of this new triangle
4 - Repeat from step 1 for each segment of the new figure
Mathematically speaking, the final drawing is called the Koch curve, and its base is a set of Cantor.
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Koch Flake on dCode.fr [online website], retrieved on 2024-09-10,
