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\partie{Homework \#2}

\setcounter{probpart}{0}

In this exercise, we study \emph{Koch's snowflake}, a figure with surprising properties.

\probpart{-- Construction of the figure}

This construction can be done by hand or using a software such as
Geogebra. Draw each new figure as a new one, and not on the previous
one.

To create a Koch snowflake, start with an equilateral triangle of
side $9$cm, then recursively alter each line segment as follows:

\begin{enumerate}
\item Divide the line segment into three segments of equal length.
\item Draw an equilateral triangle that has the middle segment from
step 1 as its base and points outward.
\item Remove the line segment that is the base of the triangle from step 2.
\end{enumerate}

This process can theoretically go on ad infinitum, but you will stop
after two replacement steps. In this way, you should end up with
$3$ different figures, including the initial triangle.

\probpart{-- Study of the perimeter}

\renewcommand{\P}{\mathscr{P}}
\renewcommand{\A}{\mathscr{A}}

We call $\P(n)$ the perimeter of the figure after $n$ steps. For
example, $\P(0)$ is the perimeter of the initial triangle, and
$\P(2)$ is the perimeter of the figure after $2$ steps, the last one
you drew.

\begin{enumerate}
\item Compute $\P(0)$, the perimeter of the initial triangle.
Explain your computation.
\item Compute $\P(1)$ justifying every part of your computation.
\item Compute $\P(2)$ justifying every part of your computation.
\item Compute the ratios $\frac{\P(1)}{\P(0)}$ and
$\frac{\P(2)}{\P(1)}$. What do you notice ?
\item Prove that each replacement step multiplies the perimeter of
the figure by $\frac43$.
\item Deduce the perimeters of $\P(3)$, $\P(4)$, $\P(5)$ and
$\P(6)$. Give the answers as exact values and rounded values to 3
DP.
\item Use your calculator to find a value of $n$ such that
$\P(n)>250$.
\item What can you say about $\P(n)$ when $n$ gets greater and
greater ? Do you think there is a maximum value for $\P(n)$ ?
\end{enumerate}


\probpart{-- Study of the area}

We call $\A(n)$ the area of the figure after $n$ steps. For example,
$\A(0)$ is the area of the initial triangle, and $\A(2)$ is the area
of the figure after $2$ steps, the last one you drew.

\begin{enumerate}
\item Find the formula for the area of an equilateral triangle of
side $s$ cm. Explain your method.
\item Use the formula to compute $\A(0)$, the area of the initial
triangle. Give the exact value and a rounded value to 3 DP.
\item Use the formula to compute the area of an equilateral triangle
of side $3$ cm and deduce the value of $\A(1)$. Give the exact value
and a rounded value to 3 DP.
\item Use the formula to compute the area of an equilateral triangle
of side $1$ cm and deduce the value of $\A(2)$. Give the exact values
and rounded values to 3 DP.
\item Compute the differences $\A(1)-\A(0)$ and
$\A(2)-\A(1)$. Give the exact value and a rounded value to 3 DP.
What do you notice ?
\item Do you think the area of the figure will grow like its
perimeter ?
\item Find on the internet a proof of the fact that the area of the figure stays
finite.
\item What is the most surprising feature of about Koch's snowflake ?
\end{enumerate}




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