A shape inside itself
A fern has small leaves that look like the whole fern. A rule can build a shape like that. The midpoint of a side is the point halfway along it.
- One triangle.
- Join the midpoints. Take out the middle triangle.
- Do it again on each triangle left.
- One corner, enlarged, looks like the shape after one cut.
Rule: Join the midpoints of the sides. Take out the middle triangle. Repeat on every triangle that is left.
| Step | Triangles left | Holes | Area left |
|---|---|---|---|
| 0 | 1 | 0 | 1 |
| 1 | 3 | 1 | 3/4 |
| 2 | 9 | 4 | 9/16 |
| 3 | 27 | 13 | 27/64 |
Step 0 is the plain triangle, with area 1. Each hole is a middle triangle taken out.
Every triangle left splits into 3. So the count is 3 times bigger each step. Each triangle left also gets a new hole in its middle.
The middle triangle is 1/4 of the triangle around it. So each step leaves 3/4 of the area it had.
A fractal is a shape that contains smaller copies of itself. The copies keep getting smaller. This shape is the Sierpinski triangle.
Slide Step. Watch the triangles left, the holes and the area.
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