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Peano-Gosper Fractal

Run the Peano-Gosper Fractal MicroSim Fullscreen Edit the Peano-Gosper Fractal MicroSim Using the p5.js Editor

About This MicroSim

The Peano-Gosper curve (also known as the Gosper curve or flowsnake) is a space-filling fractal curve discovered by Bill Gosper. It is based on hexagonal geometry and creates beautiful, organic-looking patterns that tile the plane. The name "flowsnake" is a play on "snowflake" due to its visual similarity to a flowing, snake-like snowflake pattern.

How It Works

The Gosper curve is generated using an L-system (Lindenmayer system), a formal grammar that produces complex patterns through simple string rewriting rules:

  1. Start with a Symbol: Begin with the symbol 'A'

  2. Apply Rewriting Rules: For each iteration:

  3. Replace 'A' with: A-B--B+A++AA+B-

  4. Replace 'B' with: +A-BB--B-A++A+B

  5. Interpret as Turtle Graphics:

  6. 'A' or 'B': Move forward
  7. '+': Turn left 60 degrees

  8. '-': Turn right 60 degrees

  9. Repeat: Each recursion level applies these rules again, creating more intricate patterns.

Controls

  • Recursion Slider (1-5): Controls the depth of the fractal. Higher values create more detailed, space-filling patterns.
  • Size Slider (30-100%): Adjusts the overall size of the curve within the canvas.

Mathematical Properties

The Peano-Gosper curve has fascinating mathematical properties:

  • Space-Filling: As recursion increases, the curve fills a hexagonal region
  • Growth Factor: The curve grows by a factor of √7 with each iteration
  • 60-Degree Angles: All turns are multiples of 60 degrees, creating hexagonal symmetry
  • Fractal Dimension: Exactly 2 (it is a space-filling curve)
  • Self-Similarity: The curve is made of 7 smaller copies of itself

L-System Specification

Component Value
Axiom A
Rule A A-B--B+A++AA+B-
Rule B +A-BB--B-A++A+B
Angle 60°

Comparison with Other Space-Filling Curves

Curve Base Shape Angle Growth Factor
Peano-Gosper Hexagonal 60° √7 ≈ 2.646
Hilbert Square 90° 2
Peano Square 90° 3

Embedding This MicroSim

You can include this MicroSim on your website using the following iframe:

<iframe src="https://dmccreary.github.io/geometry-course/sims/peano-gosper-fractal/main.html" height="562px" width="100%" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Understand how L-systems generate complex fractals from simple rules
  2. Explore the relationship between recursion and visual complexity
  3. Recognize hexagonal symmetry in mathematical patterns
  4. Understand the concept of space-filling curves

Activity Suggestions

  1. Trace the Pattern: At low recursion levels, have students trace the path of the curve with their finger. How does the pattern repeat?

  2. Count the Segments: At level 1, count the line segments. At level 2, count again. Can students find the pattern? (Segments multiply by 7 each level)

  3. Hexagonal Tiling: Discuss how the Gosper curve can tile a plane using hexagonal shapes. What other shapes tile the plane?

  4. L-System Exploration: Write out the first few iterations of the L-system string by hand to understand how the rules work.

Discussion Questions

  • Why is this called a "flowsnake"?
  • How does hexagonal geometry differ from square geometry in fractals?
  • What would happen if we used 90-degree angles instead of 60-degree angles?
  • Where might you see similar patterns in nature (honeycomb, crystals)?

Historical Context

Bill Gosper discovered this curve in the 1970s while working on computer graphics and recreational mathematics. The curve is part of a family of space-filling curves that includes the Hilbert curve and Peano curve. Gosper was also known for his work on continued fractions and the game of Life.

References