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

Question

Homi · Accepted Answer

We will observe how the number of sides changes from one step to the next in the Koch Snowflake construction.

Step 1 — Counting sides in the initial steps

Let us start by counting the number of sides for the first few steps shown in the diagram.

Step 0 is a simple equilateral triangle.

The number of sides in Step 0 is 3.

Diagram 1

Step 2 — Observing the change from Step 0 to Step 1

To get from Step 0 to Step 1, each side of the triangle is modified. Each straight line segment is divided into three equal parts. The middle part is removed, and two new segments are added to form an outward-pointing equilateral triangle. This means one original side becomes four new sides.

The number of sides in Step 1 is the number of sides in Step 0 multiplied by 4.

$\text{Number of sides in Step 1} = \text{Number of sides in Step 0} \times 4$

$= 3 \times 4$

$\boxed{12}$

Diagram 2

Step 3 — Observing the change from Step 1 to Step 2

To get from Step 1 to Step 2, the same modification rule is applied to each of the 12 sides of Step 1. Each straight line segment again becomes four new segments.

The number of sides in Step 2 is the number of sides in Step 1 multiplied by 4.

$\text{Number of sides in Step 2} = \text{Number of sides in Step 1} \times 4$

$= 12 \times 4$

$\boxed{48}$

Diagram 3

Step 4 — Finding the general pattern for Step n

We can see a clear pattern in the number of sides: At Step 0, the number of sides is $3$ . At Step 1, the number of sides is $3 \times 4^1 = 12$ . At Step 2, the number of sides is $3 \times 4^2 = 48$ .

Each step multiplies the previous number of sides by 4. This means the number of sides at Step n will be $3$ multiplied by $4$ raised to the power of $n$ .

Let $N_n$ be the number of sides at Step n.

$N_n = 3 \times 4^n$

Answer

Step 0: $3 \times 4^0 = 3$ Step 1: $3 \times 4^1 = 12$ Step 2: $3 \times 4^2 = 48$ Step 3: $3 \times 4^3 = 192$ Step n: $3 \times 4^n$

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