A shape that multiplies
Fold a paper triangle into four small triangles. Cut out the middle one.
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- Stage 0 is one piece.
- Stage 1: every piece becomes three. Now there are 3.
- Stage 2: every piece becomes three again. Now there are 9.
- So the counts run 1, 3, 9, 27 — ratio 3, every stage.
The rule
Repeat the fold and cut on each triangle that is left. Each round is one stage. Stage 0 is the whole triangle.
The pattern is the Sierpiński triangle. The triangles that are left are shown black.
A fractal is a pattern made of smaller copies of itself.
Each black triangle becomes three at the next stage. So the count is a GP with r = 3.
Stage 0 has 1 triangle, which is 3⁰. Any number except 0, to the power 0, is 1. Stage n has 3ⁿ black triangles.
The black area
Take the area of the start triangle as 1. Each cut keeps 3 of the 4 equal small triangles. So the black area is multiplied by 3/4 each stage.
Stage 1: 3/4. Stage 2: 3/4 × 3/4 = 9/16. Stage 3: 9/16 × 3/4 = 27/64.
Stage n has (3/4)ⁿ of the first area. This is also a GP. It shrinks, because 3/4 is less than 1.
| Stage | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Triangles | 1 | 3 | 9 | 27 |
| Area | 1 | 3/4 | 9/16 | 27/64 |