Question 22
A fractal is a pattern that repeats forever, and every part of it, regardless of how zoomed in or zoomed out you look at the picture, will pretty much look similar to the whole image. Koch’s curve is a fractal that has infinite length but no area, generated by the following process, as shown in the diagram below:
• At iteration 0, draw a horizontal line segment that has length 1.
• At each of iterations 1, 2, …, each line segment from the previous iteration is divided into three equal segments, and the middle line segment is replaced by the two sides of an equilateral triangle of the same length as the segment being removed.
Let $T_n$ denote the length of the curve obtained at iteration $n$. What is the least integer $n$ such that $T_n>5$?
• At iteration 0, draw a horizontal line segment that has length 1.
• At each of iterations 1, 2, …, each line segment from the previous iteration is divided into three equal segments, and the middle line segment is replaced by the two sides of an equilateral triangle of the same length as the segment being removed.
Let $T_n$ denote the length of the curve obtained at iteration $n$. What is the least integer $n$ such that $T_n>5$?