\documentclass[12pt]{article}
\usepackage[Mickael]{ammaths}
\begin{document}
\module{Constructible polygons}{3}{07}{2 periods}
\en\nocouleur
\prereq{Ruler and compass rules and methods}
\object{\begin{itemize}
\item study the constructibility of regular polygons.
\end{itemize}}
\mater{\begin{itemize}
\item Ruler
\item Compas
\item Task sheet
\item Hints for the pentagon, 15-gon and 16-gon.
\item Beamer
\end{itemize}}
\modpart{The easy ones}{20 mins}
Students work in pairs. They have to find the minimal number of actions needed to draw an equilateral triangle, a
square, a regular hexagona, and a regular octogon. This part is marked over 10.
\modpart{The regular pentagon}{20 mins}
The second task is to construct a regular pentagon. This part is marked over 10. Progressive hints are available on
demand. Every hint asked by a group takes one point of the final mark.
\modpart{The regular pentadecagon}{15 mins}
The third task is to construct a regular 15-gon. One hint may be given : use a pentagon and an equilateral triangle.
This par is not marked
\modpart{The regular heptadecagon}{20 mins}
The third task is to construct a regular 17-gon. A construction protocol is given to each student and has to be carried out. The end result is also marked over 10.
\modpart{Constructible polygons}{20 mins}
Each group has to list all constructible polygons with a number of sides less than or equal to $20$.
\modpart{Lecture : Some results about constructible polygons}{15 mins}
A quick history of the problem of constructible polygons, including Euclid's methods, Gauss' results and Gardner's link with the Sierpinski's binary sieve.
\pagebreak
\moddocdis{Constructible polygons}{3}{07}{Answer sheet}
\sectionblack{\Large Part 1 -- Construct some easy regular polygons}
\begin{multicols}{2}
\begin{center}
\begin{tabularx}{7cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Equilateral triangle}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{7cm}{|X|}
\hline
\\[7cm]
\hline
\end{tabularx}
\end{center}
\begin{center}
\begin{tabularx}{7cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Square}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{7cm}{|X|}
\hline
\\[7cm]
\hline
\end{tabularx}
\end{center}
\end{multicols}
\begin{multicols}{2}
\begin{center}
\begin{tabularx}{7cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Regular Octogon}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{7cm}{|X|}
\hline
\\[7cm]
\hline
\end{tabularx}
\end{center}
\begin{center}
\begin{tabularx}{7cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Regular hexagon}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{7cm}{|X|}
\hline
\\[7cm]
\hline
\end{tabularx}
\end{center}
\end{multicols}
\newpage
\sectionblack{Construction of the pentagon}
Try to find a way to construct a regular pentagon. If you don't manage, go to the teacher's desk and ask for a hint. The hints are progressive and cost 1 point over 10 each.
\vspace*{10cm}
\sectionblack{Construction of the pentadecagon}
Try to find a way to construct a regular pentadecagon (with 15 equal sides). If you don't manage, go to the teacher's desk and ask for a hint. There is only one available hint and this part is not marked.
\newpage
\sectionblack{The heptadecagon : a construction protocol}
Follow the following instructions to construct a regular heptadecagon.
\begin{enumerate}
\item Given an arbitrary point $O$, draw a circle centered on $O$ and a horizontal diameter drawn through $O$.
\item Call the right end of the diameter dividing the circle into a semicircle $P_1$.
\item Construct the diameter perpendicular to the original diameter by finding the perpendicular bisector $OB$, with $B$ at the top of the circle.
\item Construct $J$ a quarter of the way up $OB$.
\item Join $JP_1$ and find $E$ on line segment $OP_1$ so that $\angle OJE$ is a quarter of $\angle OJP_1$.
\item Find $F$ on line $OP_1$, but on the other side of $O$, so that $\angle EJF$ is 45 degrees.
\item Construct the semicircle with diameter $FP_1$, on the same side as $J$. This semicircle cuts $OB$ at $K$.
\item Draw a semicircle with center $E$ and radius $EK$, on the same side as $B$ and with both endpoints on $OP_1$. This cuts the line segment $OP_1$ at $N_4$.
\item Construct a line perpendicular to $OP_1$ through $N_4$. This line meets the original semicircle at $P_4$.
\item You now have points $P_1$ and $P_4$ of a heptadecagon. Use $P_1$ and $P_4$ to get the remaining 15 points of the heptadecagon around the original circle by constructing $P_1$, $P_4$, $P_7$, $P_{10}$, $P_{13}$, $P_{16}$, $P_2$ and so on.
\item Connect the adjacent points $P_i$ for $i=1$ to $17$, forming the heptadecagon.
\end{enumerate}
\pagebreak
\sectionblack{The constructible polygons}
\begin{enumerate}
\item List all regular polygons with $20$ or less sides that you think are constructible withe ruler and compass. Explain in a few words each construction.
\vspace*{8cm}
\item A Fermat prime is a prime number of the form $F_n=2^{2^n}+1$, where $n$ is a nonnegative integer. Compute the first five Fermat primes.
\vspace*{3cm}
\item There is a connection between the constructible polygons and the Fermat prime. Try to find it.
\vspace*{5cm}
\end{enumerate}
\pagebreak
\moddoc{Hints for the construction of the regular pentagon}
{\Large
\begin{enumerate}
\hrule
\bigskip
\item Draw a circle in which to inscribe the pentagon and mark the center point $O$.
\bigskip
\hrule
\bigskip
\item Choose a point $A$ on the circle that will serve as one vertex of the pentagon. Draw a line through $O$ and $A$.
\bigskip
\hrule
\bigskip
\item Construct a line perpendicular to the line $OA$ passing through point $O$. Mark its intersection with one side of the circle as the point $B$.
\bigskip
\hrule
\bigskip
\item Construct the point $C$ as the midpoint of $O$ and $B$.
\bigskip
\hrule
\bigskip
\item Draw a circle centered at $C$ through the point $A$. Mark its intersection with the line $OB$ (inside the original circle) as the point $D$.
\bigskip
\hrule
\bigskip
\item Draw a circle centered at $A$ through the point $D$. Mark its intersections with the original circle as the points $E$ and $F$.
\bigskip
\hrule
\bigskip
\item Draw a circle centered at $E$ through the point $A$. Mark its other intersection with the original circle as the point $G$.
\bigskip
\hrule
\bigskip
\item Draw a circle centered at $F$ through the point $A$. Mark its other intersection with the original circle as the point $H$.
\bigskip
\hrule
\end{enumerate}
}
\end{document}