Koch Snowflake
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The Koch snowflake (also called the Koch curve or Koch star) is a fractal curve constructed by recursively adding triangular bumps to a line segment.
Construction
- Start with an equilateral triangle $K_0$ of side length 1
- Replace the middle third of each edge with two sides of an equilateral triangle (pointing outward)
- Repeat step 2 for each edge of the resulting polygon
This produces a sequence of curves $K_0, K_1, K_2, \ldots$ converging to the Koch snowflake $K_\infty$.
Perimeter
Each iteration multiplies the number of segments by 4 and the length of each segment by $1/3$:
$$ \text{Perimeter}(K_n) = 3 \cdot \left(\frac{4}{3}\right)^n $$As $n \to \infty$:
$$ \text{Perimeter}(K_\infty) = \lim_{n \to \infty} 3 \cdot \left(\frac{4}{3}\right)^n = \infty $$Area
The area converges to a finite value:
$$ \text{Area}(K_\infty) = \frac{2\sqrt{3}}{5} $$(for initial triangle with side length 1)
Fractal Dimension
The Koch snowflake has Hausdorff dimension:
$$ d = \frac{\log 4}{\log 3} \approx 1.26186 $$This reflects that it is “more than” a 1-dimensional curve but “less than” a 2-dimensional surface.
Properties
- Infinite perimeter enclosing finite area
- Self-similar: each edge is a scaled copy of the whole boundary
- Continuous but nowhere differentiable