Fractals , self-similar shapes

A fractal is a shape that looks the same at every scale. Zoom into a fern leaf, and you see smaller leaves with the same shape. Zoom into a coastline, and you see ridges and bays with the same kind of structure as the whole coast. These shapes have self-similarity , and they have surprising properties, like infinite perimeter inside finite area.

Concept

Self-similarity. A shape is self-similar if a magnified part looks like the whole shape (or one of its rotations/reflections).

Koch snowflake (Helge von Koch, 1904). Start with an equilateral triangle. On each side, replace the middle third by an outward "bump" , two segments making a small triangle. Repeat this process on the new shape's sides. Continue forever.

The starting triangle has $3$ sides. Each iteration replaces every straight side with $4$ segments. So after $n$ iterations the number of sides is $3 \cdot 4^n$. If the original side length is $L$, each new segment has length $L/3^n$. So the perimeter is

$\text{Perimeter}_n = 3 \cdot 4^n \cdot \frac{L}{3^n} = 3 L \left(\frac{4}{3}\right)^n.$

As $n \to \infty$, perimeter $\to \infty$. But the snowflake stays inside a fixed-size triangle, so its area is finite (it converges to $\tfrac{8}{5}$ of the original triangle's area). An infinite perimeter enclosing a finite area! Such are fractals.

Sierpinski triangle (Waclaw Sierpinski, 1915). Start with a filled equilateral triangle. Mark the midpoints of its three sides and remove the central upside-down triangle. You are left with three smaller triangles. Repeat on each. The limit is a "lacy" triangle with zero area and infinite perimeter.

Fractal dimension. Squares have dimension $2$ (you need two numbers to locate a point). Cubes have dimension $3$. Fractals have dimensions in between , for example, the Koch curve has dimension $\log 4 / \log 3 \approx 1.262$, the Sierpinski triangle has $\log 3 / \log 2 \approx 1.585$. They are "thicker" than a line but "thinner" than a plane.

Where fractals appear in nature.

• Coastlines, mountains, river networks.
• Trees, ferns, broccoli.
• Lung bronchi, blood vessels.
• Lightning bolts and frost patterns.

Why nature loves fractals. They are an efficient way to maximise surface area without taking up too much volume. The branching of bronchi packs the lung's huge inner area into a small chest cavity. The branching of trees collects light over a wide canopy with relatively little wood.

Worked examples

Example 1. Find the perimeter of the Koch snowflake after $2$ iterations, starting from a triangle with side $9$.

• Iteration $0$: $3 \cdot 9 = 27$.
• Iteration $1$: $3 \cdot 4 \cdot 3 = 36$.
• Iteration $2$: $3 \cdot 4^2 \cdot 1 = 48$.

Example 2. How many sides does the Koch snowflake have at iteration $4$?

• $3 \cdot 4^4 = 768$.

Example 3. A Sierpinski triangle after $3$ iterations: how many small triangles remain?

• After $1$ step: $3$ triangles. After $n$ steps: $3^n$.
• After $3$ steps: $27$ small triangles.

Example 4. Compute the area ratio of a Sierpinski triangle after $2$ steps compared with the original.

• Each iteration keeps $3/4$ of the area (one of four small triangles is removed).
• After $2$ steps: $(3/4)^2 = 9/16$.

Try it yourself

1. Compute the perimeter of a Koch snowflake (starting side $27$ cm) after $3$ iterations.
2. After how many iterations does the Sierpinski triangle have $81$ small triangles?
3. Area fraction of Sierpinski triangle remaining after $4$ steps?
4. Number of sides of Koch snowflake after $5$ iterations starting from a triangle?
5. Sketch a "Cantor set" by removing the middle third of $[0,1]$, then the middle third of the two remaining pieces, etc. How many segments after $4$ steps?
6. Look up "Pascal's triangle modulo $2$". The pattern of $1$'s forms a Sierpinski triangle. Verify this for the first $8$ rows.
7. The fractal dimension of the Koch curve is $\log 4 / \log 3$. Is this less than or greater than $1$? Less than or greater than $2$?
8. Why is the surface of the human lungs essentially fractal?

Activity / Insight

Draw your own fractal. With a ruler and pencil, draw the first three iterations of the Koch snowflake on an equilateral triangle of side $9$ cm. By iteration $3$ you will have $192$ tiny segments , and a beautiful star shape. Each iteration triples the work but the visual reward is immense. This is the same recursive construction that gives the natural world its rough beauty.