# Computation Workshop Solution Checker

## 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 zero iterations the fractal path looks like this:

$$\begin{picture} \draw (0,0) -- (1,0); \end{picture}$$

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. Determine the area between the fractal path and the $$x$$-axis, after $$100^{100}$$ iterations.