Koch Snowflake Calculator
Dimensions:
Properties:
Understanding the Koch Snowflake
Fractal Properties
Perimeter = L₀ × (4/3)ⁿ
Area = (8/5) × A₀ × (1 - (3/4)ⁿ)
Fractal Dimension = log(4)/log(3) ≈ 1.2619
Self-Similarity
Each edge contains infinite copies of itself at different scales
Infinite Perimeter
As n → ∞, perimeter → ∞
Finite Area
Area converges to 8/5 times the initial triangle area
Construction Process
Base Triangle
Start with an equilateral triangle
Edge Division
Divide each edge into three equal parts
Triangle Addition
Replace middle segment with two segments forming an equilateral triangle
Mathematical Analysis
Geometric Series
Area follows a geometric series with ratio 4/9
Hausdorff Dimension
Measures the local size of space considering scaling
Symmetry
Exhibits 6-fold rotational symmetry
Applications and Significance
Natural Phenomena
- Coastline Patterns
- Crystal Growth
- Snowflake Formation
- Plant Structures
Mathematics
- Fractal Geometry
- Infinite Series
- Geometric Progression
- Topology Studies
Computer Graphics
- Procedural Generation
- Pattern Design
- Terrain Modeling
- Artistic Visualization