What is a fractal?
A fractal is a shape made by using one rule over and over. Each time, the rule is used on smaller and smaller pieces. So when you zoom in, you see the same pattern again.
This property has a name: self-similarity. It means a part looks like the whole. A real fractal in maths goes on forever. On paper, we can only draw a few levels.
Self-similar patterns in nature and art
- Fern leaf: every leaflet is a small fern.
- Tree: a trunk splits into branches, each branch splits again into smaller branches.
- Romanesco broccoli and cauliflower: each floret is a small copy of the whole head.
- Rivers, lightning and coastlines: big zig-zags are made of smaller zig-zags.
- Art: some rangoli, kolam and temple designs repeat one shape inside itself.
Nature stops after a few levels, so these are fractal-like, not perfect fractals.
How to build the Sierpinski triangle
- Draw an equilateral triangle and shade it. This is level 0.
- Find the midpoint of each side. Join the three midpoints. This makes 4 smaller triangles.
- Remove (unshade) the middle one. You have 3 shaded triangles: level 1.
- Repeat steps 2–3 on every shaded triangle.
The pattern is named after the Polish mathematician Wacław Sierpiński, who described it in 1915.
Counting patterns: triangles, side and area
At each level, every shaded triangle becomes 3 shaded triangles. So:
| Level n | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Shaded triangles (3ⁿ) | 1 | 3 | 9 | 27 | 81 |
| Side of each (×½) | 1 | 1/2 | 1/4 | 1/8 | 1/16 |
| Area left ((3/4)ⁿ) | 1 | 3/4 | 9/16 | 27/64 | 81/256 |
Each new level removes one quarter of what is left. If we went on forever, the shaded area would get closer and closer to 0, while the number of triangles grows without end.
Other famous fractals
Koch snowflake: on every straight side, replace the middle third with two sides of a small triangle that points outward. The edge gets longer every time.
Fractal tree: draw a line, then two shorter branches at its end, then two shorter branches on each of those.
Sierpinski carpet: cut a square into 9, remove the middle one, repeat on the other 8.
Key formulas and definitions
- Self-similar: a part looks like the whole
- Sierpinski: number of shaded triangles at level n = 3ⁿ
- Side of each small triangle at level n = (1/2)ⁿ × original side
- Shaded area at level n = (3/4)ⁿ × original area
- Triangles removed at level n (new holes) = 3ⁿ⁻¹
Worked examples
1. How many shaded triangles are there at level 2 of the Sierpinski triangle?
Level 0: 1. Level 1: 1 × 3 = 3. Level 2: 3 × 3 = 9. So 9 triangles (3² = 9).
2. The level-0 triangle has side 32 cm. What is the side of one small triangle at level 3?
Each level halves the side. 32 → 16 → 8 → 4. So 4 cm (32 × (1/2)³ = 4).
3. The level-0 triangle has area 64 cm². Find the shaded area at level 2.
Each level keeps 3/4 of the area. Level 1: 64 × 3/4 = 48 cm². Level 2: 48 × 3/4 = 36 cm².
4. At which level are there 243 shaded triangles?
Keep multiplying by 3: 1, 3, 9, 27, 81, 243. That is level 5 (3⁵ = 243).
5. How many triangles in total have been removed after level 3?
New holes at levels 1, 2, 3 are 1, 3 and 9. Total = 1 + 3 + 9 = 13.
6. Is a fern leaf a perfect fractal? Explain.
No. Each leaflet looks like the whole leaf, so it is self-similar, but the copying stops after about 3–4 levels. A maths fractal goes on forever. So a fern is fractal-like.
Common mistakes
- Thinking the count adds 3 each time (1, 4, 7 …). It multiplies by 3: 1, 3, 9, 27.
- Removing a corner triangle instead of the middle one. Only the middle (upside-down) triangle is taken out.
- Saying the area halves each step. Only one of the four pieces goes, so 3/4 of the area stays.
- Calling any repeated pattern (like a row of tiles) a fractal. A fractal repeats at smaller and smaller sizes, inside itself.