Iteration
0
Segments
3
Perimeter
3.00
Area
0.433

Growth Comparison

Perimeter
Area
Click "Next" to add fractal detail to each edge

❄️ The Paradox

The Koch snowflake has infinite perimeter but encloses a finite area! Each iteration multiplies the perimeter by 4/3, growing without bound. Yet the area converges to exactly 8/5 of the original triangle.

📐 The Construction

Start with an equilateral triangle. For each edge:

Divide into 3 equal parts
Build triangle on middle third
Remove the base

Repeat forever on all new edges.

📏 The Mathematics

At iteration n:

Segments = 3 × 4ⁿ
Perimeter = 3 × (4/3)ⁿ
Area → 8/5 × (original)

Fractal dimension: log(4)/log(3) ≈ 1.26

🔬 Historical Context

Helge von Koch created this in 1904 to show a curve that is continuous everywhere but differentiable nowhere—you can't draw a tangent line at any point! It contradicted 19th century intuition about curves.