Koch snowflake
A triangle whose edges bend again and again: the perimeter grows without bound while the area stays finite.
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Formulas and key values
- Construction rule
Replace the middle third of every edge with two sides of an equilateral triangle pointing outward.
- PerimeterIn the graph ≈ 9.481
- AreaIn the graph ≈ 0.6827
- Fractal dimension
s is the side of the starting triangle (s = 1 in the graph) and n the number of iterations.
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Start the video01 · Definition
What is the Koch snowflake?
Start with an equilateral triangle. Cut every edge into thirds, and on the middle third raise a smaller equilateral triangle pointing outward. Repeat on every new edge, forever. Helge von Koch described the edge curve in 1904 as an example of a curve that is continuous everywhere yet has a tangent nowhere.
Each round multiplies the number of edges by 4 and shrinks them by 3, so the perimeter grows by a factor of 4/3 every time and has no limit. The area, though, only adds ever-smaller triangles and converges to 8/5 of the original triangle. A finite region with an infinitely long boundary is the snowflake’s whole lesson.
02 · Properties
Key properties
- Perimeter after n steps: 3s(4/3)ⁿ, which grows without bound.
- Area after n steps: (√3/4)s²(8/5 − (3/5)(4/9)ⁿ), converging to 2√3s²/5.
- Fractal dimension log 4 / log 3 ≈ 1.2619.
- Each step multiplies the number of edges by 4: 3·4ⁿ edges after n steps.
- The limit curve is continuous but has no tangent at any point.
03 · Questions
Koch snowflake: common questions
- How is the Koch snowflake constructed?
- Construction rule: Replace the middle third of every edge with two sides of an equilateral triangle pointing outward.. s is the side of the starting triangle (s = 1 in the graph) and n the number of iterations.
- What is the perimeter of the Koch snowflake?
- Perimeter: P_n = 3s(4/3)^n. s is the side of the starting triangle (s = 1 in the graph) and n the number of iterations.
- What is the area of the Koch snowflake?
- Area: A_n = (√3/4)s^2(8/5 - (3/5)(4/9)^n). s is the side of the starting triangle (s = 1 in the graph) and n the number of iterations.
04 · Keep exploring