Fractal Path
Consider the following fractal path from \((0,0)\) to \((1,0)\) in the Cartesian plane formed using the following recursive process. Start with the straight line segment from \((0,0\) to \((1,0)\). Then iteratively, for each line segment \(s\) in the current curve, replace \(s\) with a sub-path of four line segments such that
- each of the \(4\) line segments has length equal to \(2/5\) of the length of \(s\),
- the first of the four line segments is the first \(2/5\) of \(s\),
- the middle two of the four line segments together with the middle \(1/5\) of \(s\) form an outward pointing isosceles triangle, and
- the last of four line segments is the final \(2/5\) of \(s\).
After one iteration the fractal path looks like this:
\(\begin{picture} \draw (0,0) -- (0.4,0) -- (0.5,0.3873) -- (0.6,0) -- (1,0); \end{picture}\)After two iterations the fractal path looks like this:
\(\begin{picture} \draw (0,0) -- (0.4,0) -- (0.5,0.3873) -- (0.6,0) -- (1,0); \end{picture}\)After three iterations the fractal path looks like this:
\(\begin{picture} \draw (0,0) -- (0.4,0) -- (0.5,0.3873) -- (0.6,0) -- (1,0); \end{picture}\)| Part A | Determine the length of the fractal path after \(30\) iterations. Express your answer rounded to \(6\) digits past the decimal point. | |
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| Part B | Determine the area between the fractal path and the \(x\)-axis, after \(100^{100}\) iterations. |