The chaos game: a fractal from chance
Goals
- Students find midpoints with coordinates.
- Students describe the Sierpiński triangle and its self-similarity.
- Students see how a random rule can still give an exact shape.
Before the film: predict
Pick one of a triangle’s corners at random, jump halfway towards it, mark the spot and repeat thousands of times. Will you get a random cloud, or something else?
The film
Pure Randomness Draws a Perfect Fractal — The Chaos Game · hypnoticpatterns.doner.cloud/video/pure-randomness-draws-a-perfect-fractal
The lesson, minute by minute
| 5 min | Predict; then everyone does five jumps by hand with a die on a printed triangle. |
|---|---|
| 3 min | Watch the film “Pure Randomness Draws a Perfect Fractal”. |
| 20 min | Worksheet at three levels. |
| 10 min | Discuss: the questions below. |
| 7 min | Live demo: the square with and without the restriction, then the pentagon at 0.618. |
The worksheet
Starter
The corners are A (0, 0), B (8, 0) and C (4, 8). Start at (4, 4) and jump halfway to A, then C, then B, then C. Write down each point.
Answer: (2, 2), (3, 5), (5.5, 2.5), (4.75, 5.25).
How do you find the point halfway between two points?
Answer: Take the average of the x values and the average of the y values.
Core
Each stage of the Sierpiński triangle removes the middle quarter of every triangle. What share of the area is left after 1, 2 and 5 stages?
Answer: 3/4 = 75 %, 9/16 ≈ 56.3 %, (3/4)⁵ ≈ 23.7 %.
How many small triangles are there after 5 stages?
Answer: 3⁵ = 243.
Each jump halves the distance to the fractal. After 10 jumps, what part of the starting distance is left?
Answer: (1/2)¹⁰ = 1/1,024, about a thousandth.
Challenge
The Sierpiński triangle is made of 3 copies of itself at half the size. Its dimension d solves 2ᵈ = 3. Find d.
Answer: d = log 3 ÷ log 2 ≈ 1.585: more than a line (1), less than a filled triangle (2).
Play the game in a square with jumps of one half. Why do the points just fill the square?
Answer: Four half-size copies of a square fill the square exactly, with no gaps, so the shape the rule leaves unchanged is the whole square.
Discuss
- How can random choices give a shape that is exactly the same every time?
- What changes if you never pick the same corner twice in a row?
- Where in nature do you see shapes made of smaller copies of themselves?
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