Repeat the process: take every straight line segment, divide it into three equal parts, and replace the middle third with two segments of that length.
This simplifies to: [ A_n = A_0 \left[ 1 + \sum_{k=1}^n \frac{3 \times 4^{k-1}}{9^k} \right] = A_0 \left[ 1 + \frac{1}{3} \sum_{k=1}^n \left(\frac{4}{9}\right)^{k-1} \right] ] snowflake by haese mathematics
For each side, remove the middle third and replace it with two segments of the same length (forming an equilateral "bump"). The number of sides increases. Repeat the process: take every straight line segment,
Since ( A_0 = \frac{\sqrt{3}}{4} ), the final area is: [ A_{\infty} = \frac{8}{5} \cdot \frac{\sqrt{3}}{4} = \frac{2\sqrt{3}}{5} ] divide it into three equal parts