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Sierpinski Triangles

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Description

The Sierpinski triangle is one of the most famous fractals in mathematics, named after Polish mathematician Wacław Sierpiński who described it in 1915. This MicroSim demonstrates how a simple recursive rule creates an infinitely complex geometric pattern through repeated subdivision.

Starting with a single equilateral triangle, the fractal is generated by:

  1. Finding the midpoint of each side of the triangle
  2. Connecting these midpoints to create four smaller triangles
  3. Removing (not drawing) the center triangle
  4. Recursively repeating this process on each of the three remaining triangles

At recursion depth 0, you see a single solid triangle. With each increase in depth, the pattern subdivides, creating a self-similar structure where each part resembles the whole. At higher recursion levels, the characteristic pattern emerges: a triangle composed of ever-smaller triangles with triangular "holes" at every scale.

Key Mathematical Properties

  • Self-similarity: The pattern looks the same at any scale
  • Fractal dimension: Approximately 1.585 (between 1D and 2D)
  • Area: Approaches zero as recursion depth increases to infinity
  • Perimeter: Grows infinitely as recursion depth increases
  • Triangle count: At depth n, there are 3^n filled triangles

Educational Applications

This MicroSim helps students understand:

  • Recursion: How simple rules applied repeatedly create complex patterns
  • Fractals: Self-similar structures that appear in nature and mathematics
  • Geometric patterns: How subdivision creates intricate designs
  • Mathematical infinity: How processes can continue indefinitely
  • Exponential growth: The number of triangles triples with each recursion level

Lesson Plan

Learning Objectives

Students will: - Understand the concept of recursive subdivision in geometry (Bloom's: Understanding) - Analyze how fractals exhibit self-similarity at different scales (Bloom's: Analyzing) - Explore the relationship between recursion depth and pattern complexity (Bloom's: Applying) - Evaluate how mathematical infinity appears in finite visualizations (Bloom's: Evaluating)

Activity Sequence

Introduction (5 minutes) 1. Show the MicroSim at recursion depth 0 2. Ask: "What happens if we divide each triangle into smaller triangles?" 3. Gradually increase the recursion depth while students observe

Exploration (10 minutes) 1. Have students experiment with different recursion depths (0-7) 2. Ask them to record the number of triangles at each depth 3. Guide them to discover the pattern: 1, 3, 9, 27, 81, 243, 729, 2187

Pattern Recognition (5 minutes) 1. Help students recognize the exponential growth: 3^0, 3^1, 3^2, etc. 2. Discuss self-similarity: zoom in on any part and see the same pattern 3. Compare to other fractals in nature (snowflakes, coastlines, ferns)

Mathematical Connection (5 minutes) 1. Discuss what happens at "infinite" recursion 2. Introduce the concept of fractal dimension 3. Connect to area and perimeter paradoxes

Extension Questions - What would happen if we used squares instead of triangles? - Can you find similar patterns in nature or art? - How is this related to computer programming and recursion? - What other fractals have you seen?

Assessment

Students can demonstrate understanding by: - Explaining the recursive rule in their own words - Predicting triangle counts at different depths - Identifying self-similar patterns at different scales - Drawing the first 2-3 recursion levels by hand

Standards Alignment

  • CCSS.MATH.HSG.MG.A.1: Use geometric shapes to model real-world situations
  • CCSS.MATH.PRACTICE.MP7: Look for and make use of structure
  • NGSS HS-ETS1-4: Use mathematics and computational thinking
  • Pascal's Triangle (the pattern appears in binomial coefficients)
  • Chaos Game (an alternative construction method using random points)
  • Koch Snowflake (another famous fractal)
  • Binary Trees (similar recursive structure)
  • Cellular Automata (Rule 90 generates Sierpinski pattern)

Technical Notes

This MicroSim uses recursive function calls to generate the fractal pattern. The drawSierpinski() function calls itself three times at each level (except at the base case), creating an elegant example of recursion in computer science.

The recursion is limited to depth 7 to maintain performance, as depth 8 would require drawing 6,561 triangles (3^8).

Credits

Created as part of the AI-Assisted Geometry Course educational materials.

Note: Remember to create a screenshot (sierpinski-triangles.png) of the MicroSim for social media preview images.