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Learn moreFun with Fractals and Visualising a GP
Patterns that repeat at smaller and smaller scales are common in nature. The branching of a fern leaf, the veins carrying blood in the human body, the network of a river and its tributaries, and the outline of a rugged mountain range all show a rough version of the same idea, a whole made of smaller copies of itself. Studying a simple fractal rule like the one above builds intuition for these more complex natural patterns.
A fractal is a shape built by repeating one simple rule again and again. Every time the rule is applied, the shape grows more detailed, yet a small piece of it still looks like the whole. This pattern inside a pattern is called self-similarity.
| Koch snowflake | Sierpinski square | Cantor set | Sierpinski Triangle |
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Consider an equilateral triangle with side length \(L\), called Stage \(0\).
To move to Stage \(1\), every side of the triangle is treated the same way.
Each side is divided into three equal parts, the middle part is removed, and in its place two new sides of a small equilateral triangle are attached, pointing outward, so the missing middle part is replaced by a bump.
Once this is carried out on all three sides, the shape becomes a six pointed star.
Stage \(2\) is obtained by applying the exact same rule, dividing into three parts, removing the middle part, and adding a bump, on every straight segment of the Stage \(1\) shape.
Continuing in this manner produces Stage \(3\), Stage \(4\), and so on.
At Stage \(0\) the shape has \(3\) sides.
When the rule is applied once, each side turns into \(4\) smaller segments, since two straight pieces remain from the division and two new sides come from the bump.
So the number of sides becomes \(3 \times 4 = 12\) at Stage \(1\).
Applying the rule again on every one of these \(12\) sides gives \(12 \times 4 = 48\) sides at Stage \(2\).
The sequence formed by the number of sides at each stage is
\(3,\ 12,\ 48,\ 192,\ 768,\ \dots\)
Each term is obtained by multiplying the previous term by \(4\), so this constant multiplier is the common ratio, and the sequence is a geometric progression with first term \(a = 3\) and common ratio \(r = 4\).
Writing \(t_n\) for the number of sides at Stage \(n\), starting the count from \(n = 0\),
The explicit formula is \(t_n = 3 \times 4^{n}\)
The recursive formula is \(t_0 = 3, \qquad t_n = 4 \times t_{n-1} \text{ for } n \geq 1\)
The recursive formula is \(t_0 = 3, \qquad t_n = 4 \times t_{n-1} \text{ for } n \geq 1\)
Therefore coordinates are \((0, 3)\), \((1, 12)\), \((2, 48)\), \((3, 192)\), \((4, 768)\)
Important!
This shows the steep, upward curving shape typical of a GP with common ratio \(r = 4\), greater than \(1\), the points stay close to the \(x\)-axis at first, then rise sharply from Stage \(2\) onward.
Now let's discuss the area of this pattern
At Stage \(0\), the triangle has area \(A = 1\) square unit.
At Stage \(1\), a bump is added on each of the \(3\) sides, so \(3\) new triangles are added.
Each new triangle has area \(\dfrac{1}{9}\) of \(A\), so the total new area added at Stage \(1\) is
\(3 \times \dfrac{1}{9} \times 1 = \dfrac{1}{3}\)
At Stage \(2\), the number of new triangles becomes \(4\) times the previous step's count, and each new triangle's area is \(\dfrac{1}{9}\) of the previous step's triangle area. So the total new area added at each step, from Stage \(1\) onward, changes by a combined factor of
\(4 \times \dfrac{1}{9} = \dfrac{4}{9}\)
This means the areas added at each step form a GP with first term \(\dfrac{1}{3}\) and common ratio \(\dfrac{4}{9}\):
\(s_1 = \dfrac{1}{3}\)
\(s_2 = \dfrac{1}{3} \times \dfrac{4}{9} = \dfrac{4}{27}\)
\(s_3 = \dfrac{1}{3} \times \left(\dfrac{4}{9}\right)^{2} = \dfrac{16}{243}\)
\(s_4 = \dfrac{1}{3} \times \left(\dfrac{4}{9}\right)^{3} = \dfrac{64}{2187}\)
\(s_5 = \dfrac{1}{3} \times \left(\dfrac{4}{9}\right)^{4} = \dfrac{256}{19683}\)
\(s_6 = \dfrac{1}{3} \times \left(\dfrac{4}{9}\right)^{5} = \dfrac{1024}{177147}\)
\(s_7 = \dfrac{1}{3} \times \left(\dfrac{4}{9}\right)^{6} = \dfrac{4096}{1594323}\)
So, the coordinates \((n, s_n)\) are
\((1, \dfrac{1}{3})\), \((2, \dfrac{4}{27})\), \((3, \dfrac{16}{243})\), \((4, \dfrac{64}{2187})\), \((5, \dfrac{256}{19683})\), \((6, \dfrac{1024}{177147})\), \((7, \dfrac{4096}{1594323})\)
Important!
Since the common ratio \(\dfrac{4}{9}\) is less than \(1\), the area added at each stage keeps shrinking, getting closer and closer to \(0\) as the stage number increases, even though the number of sides (and number of new triangles) keeps growing rapidly.
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