Circle Packing Calculator

Results:

Understanding Circle Packing

Basic Principles

Packing Density = Total Circle Area / Container Area

Maximum Density (Hexagonal) ≈ 0.9069

Lattice Arrangements

  • Square Lattice
  • Hexagonal Lattice
  • Triangular Lattice

Density Optimization

  • Space Utilization
  • Gap Minimization
  • Edge Effects

Boundary Conditions

  • Container Shape
  • Circle Size Ratio
  • Edge Constraints

Advanced Topics

Computational Methods

  • Greedy Algorithms
  • Genetic Algorithms
  • Force-Directed Methods

Multi-Size Packing

  • Size Distribution
  • Hierarchical Packing
  • Optimization Strategies

Applications

  • Material Science
  • Molecular Structure
  • Container Loading

Real-World Applications

Industrial Design

  • Container Optimization
  • Material Usage
  • Cost Reduction

Scientific Research

  • Crystal Structures
  • Particle Systems
  • Biological Systems

Optimization Problems

  • Resource Allocation
  • Space Planning
  • Network Design