Find the number of sides in the step of the shape sequence that leads to the Koch Snowflake.

Question

Homi · Accepted Answer

Understand the Question

The construction of the Koch Snowflake begins at Step 0 with an equilateral triangle having $3$ sides.
In each subsequent step, every straight segment is divided into three equal parts, and the middle part is replaced with two new segments of equal length, forming an outward point.
This means each existing side is replaced by $4$ smaller sides, multiplying the total number of sides by $4$ at every step.
By identifying this geometric pattern, we can express the total number of sides at the $n^{\text{th}}$ step as a general formula.

Step 1 · Count Sides at Step 0

The initial shape at Step 0 is an equilateral triangle. Diagram 1

$\text{Number of sides at Step 0} = 3$

Step 2 · Calculate Sides at Step 1

To get to Step 1, each of the $3$ sides is replaced by $4$ new segments. Diagram 2

\begin{aligned} \text{Number of sides at Step 1} &= \text{Number of sides at Step 0} \times 4 \\[0.6em] &= 3 \times 4 \\[0.6em] &= 12 \end{aligned}

Step 3 · Calculate Sides at Step 2

Applying the same rule to each of the $12$ sides of Step 1: Diagram 3

\begin{aligned} \text{Number of sides at Step 2} &= \text{Number of sides at Step 1} \times 4 \\[0.6em] &= 12 \times 4 \\[0.6em] &= 48 \end{aligned}

Step 4 · Generalize Pattern for Step $n$

Observing the sequence of the number of sides:

At each step, the total number of sides is multiplied by $4$ .

Therefore, the number of sides $N_n$ at Step $n$ is: $N_n = 3 \times 4^n$

Answer

$3 \times 4^n$

Common Mistakes

Base Step Confusion: Counting the initial equilateral triangle as Step 1 instead of Step 0, which incorrectly leads to the formula $3 \times 4^{n-1}$ .
Additive vs. Multiplicative Growth: Assuming that $4$ sides are merely added to the total count ( $+4$ ), rather than recognizing that every individual side splits into $4$ sides (multiplying the total by $4$ ).

Question 5