Fractals

Look closely at a fern. Each leaf looks like a small copy of the whole fern, and each small leaf has even smaller copies on it. Shapes like this are called fractals: the same pattern repeats again and again at smaller and smaller scales. We call this being self-similar.

Nature is full of them. A tree has limbs, the limbs have branches, the branches have smaller branches. Clouds, coastlines, mountains and lightning all work the same way.

Mathematicians make fractals on purpose, by repeating one simple rule. The Polish mathematician Sierpinski found one now called the Sierpinski Carpet.

The rule: take a square and divide it into 9 equal smaller squares. Remove the middle one. Then do the same to each of the 8 squares that are left. And again, and again.

After every step there are two things to count: the squares that remain (smaller each time), and the square holes where squares were removed.

Squares first. Call the number of remaining squares at Step n Rₙ. Every remaining square turns into 8 at the next step, so Rₙ₊₁ = 8Rₙ.

So R₀ = 1 (the starting square), R₁ = 8 × 1 = 8, R₂ = 8 × 8 = 8². In general Rₙ = 8ⁿ: the same fast growth as folding paper.

Holes grow differently. Each remaining square makes one new hole at the next step, and all the old holes stay. So Hₙ₊₁ = Hₙ + Rₙ.

That gives H₀ = 0, H₁ = 1, H₂ = 1 + 8 = 9, H₃ = 1 + 8 + 8² = 73. Each step adds the next power of 8.

A shortcut, if you like: adding 1 + 8 + 8² + … up to 8ⁿ⁻¹ always gives (8ⁿ − 1)/7. Check: for n = 3, (512 − 1)/7 = 511/7 = 73.