The carpet, and fractals around us
Show on a big screenTake a square. Cut it into 9 equal squares. Each one is a ninth. Cut out the middle ninth. Do it again on every square left. What remains?
- One square.
- Cut it into 9. Take out the middle one.
- Do it again on each of the 8 squares left.
After one cut, 8 squares remain. After two cuts, 64 squares remain.
Rule: Cut the square into 9 equal squares. Take out the middle one. Repeat on every square that is left.
| Step | Squares left | Holes | Area left |
|---|---|---|---|
| 0 | 1 | 0 | 1 |
| 1 | 8 | 1 | 8/9 |
| 2 | 64 | 9 | 64/81 |
| 3 | 512 | 73 | 512/729 |
Every square left is cut into 9. The middle one is taken out. So 8 are left from each. The count is 8 times bigger each step.
Take the first square's area as 1. Each step leaves 8/9 of the area before it. The shape this rule builds is the Sierpinski carpet.
Some things in nature are roughly fractals too. Romanesco looks like a green cauliflower. Its small buds are shaped like the whole head. A river network splits into small streams. The small streams look like the big river.
Slide Step. Watch the squares left, the holes and the area.
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Try the lab to finish this lesson.