Gosper Curve Generator

The Gosper curve, also known as the flowsnake or Peano-Gosper curve, is a space-filling curve discovered by Bill Gosper. It fills space using a hexagonal tiling pattern, creating a distinctive shape that resembles a snowflake or flowing organic form.

Properties

• Hausdorff dimension: 2 -- like other space-filling curves, the Gosper curve fills the plane.
• L-system definition: Axiom: A. Rules: A -> A-B--B+A++AA+B-, B -> +A-BB--B-A++A+B. Angle: 60 degrees.
• Scaling factor: Each iteration scales by 1/sqrt(7), and each segment is replaced by 7 sub-segments.
• Hexagonal symmetry: The curve naturally tiles the plane with hexagonal regions.

The Gosper Island

The boundary of the region filled by the Gosper curve forms what is known as the Gosper island. This boundary is itself a fractal, with a fractal dimension of approximately 1.129. The Gosper island can tile the plane, and its shape emerges naturally from the recursive construction of the curve.

Applications

• Antenna design: Fractal antenna patterns based on the Gosper curve can achieve compact wideband performance.
• Tiling and tessellation: The curve provides a natural hexagonal space-filling pattern useful in computational geometry.
• Art and design: The visually striking shape is used in generative art and decorative design.