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Barnsley's Fern

Run Barnsley's Fern MicroSim Fullscreen

About This MicroSim

The Barnsley fern is a fractal named after the British mathematician Michael Barnsley who first described it in his 1988 book Fractals Everywhere. He designed it to resemble the black spleenwort (Asplenium adiantum-nigrum), a real fern species. This fractal is a beautiful example of how simple mathematical rules can create remarkably lifelike organic forms.

How It Works

The Barnsley fern is created using an Iterated Function System (IFS), which applies one of four affine transformations repeatedly to generate points:

  1. f1 (Stem): Creates the stem of the fern (1% probability)
  2. f2 (Main shape): Creates the overall fern shape and successively smaller leaflets (85% probability)
  3. f3 (Left leaflet): Creates the largest left-hand leaflet (7% probability)
  4. f4 (Right leaflet): Creates the largest right-hand leaflet (7% probability)

Each transformation is chosen randomly based on these probabilities, and the system iterates thousands of times to build up the fern image point by point.

Controls

  • Iterations (10,000-200,000): Number of points to plot. More iterations create a denser, more detailed fern.
  • Scale (20-80): Controls the overall size of the fern display.
  • Leaf Angle (0-15°): Adjusts the angle at which leaflets branch from the stem.
  • Curl (0-10): Controls how much the fern tip curls over.
  • Black Spleenwort Preset: Checkbox that sets all parameters to values that best resemble the black spleenwort fern.

The Black Spleenwort

The black spleenwort (Asplenium adiantum-nigrum) is a species of fern native to Europe, Asia, and Africa. Key characteristics:

  • Dark, glossy stems (stipes)
  • Triangular fronds with pinnate leaflets
  • Found on rocks, walls, and shaded banks
  • Evergreen in mild climates

Barnsley specifically designed his fractal parameters to mimic this fern's distinctive appearance.

Mathematical Foundation

The Four Transformations

Each transformation has the form:

\[ \begin{pmatrix} x_{n+1} \\ y_{n+1} \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x_n \\ y_n \end{pmatrix} + \begin{pmatrix} e \\ f \end{pmatrix} \]
Function a b c d e f Probability
f1 (Stem) 0 0 0 0.16 0 0 1%
f2 (Main) 0.85 0.04 -0.04 0.85 0 1.6 85%
f3 (Left) 0.2 -0.26 0.23 0.22 0 1.6 7%
f4 (Right) -0.15 0.28 0.26 0.24 0 0.44 7%

Embedding This MicroSim

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

<iframe src="https://dmccreary.github.io/geometry-course/sims/barnsleys-fern/main.html" height="702px" width="100%" scrolling="no"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Understand how Iterated Function Systems create fractal patterns
  2. Explore the relationship between mathematical parameters and visual appearance
  3. Recognize self-similarity in fractal structures
  4. Connect mathematical abstractions to natural forms

Activity Suggestions

  1. Parameter Exploration: Adjust each slider one at a time and observe how it affects the fern's appearance. Which parameter has the most dramatic effect?

  2. Iteration Investigation: Start with low iterations (10,000) and gradually increase. How many iterations are needed before the fern looks "complete"?

  3. Create Your Own Fern: Try to create different fern species by adjusting the parameters. Can you make a fern that looks different from the black spleenwort?

  4. Nature Connection: Compare the fractal fern to photos of real ferns. Discuss what features are captured well and what is simplified.

Discussion Questions

  • Why do you think Barnsley chose a fern to model with fractals?
  • How does self-similarity appear in the Barnsley fern?
  • What other natural objects might be modeled with similar techniques?
  • How does randomness (the probability of each transformation) contribute to the natural appearance?

Historical Context

Michael Barnsley (born 1946) is a British mathematician who made significant contributions to fractal geometry and its applications. His work on Iterated Function Systems provided a mathematical framework for understanding and generating fractals. The Barnsley fern has become one of the most iconic examples of how mathematics can capture the essence of natural forms.

References