% Title: Demonstration sans parole : theoreme des calissons
% Author: Jacques Duma
% Site: http://math.et.info.free.fr/TikZ/index.html

\documentclass{article}


\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\pagestyle{empty}

\section*{Th\'eor\`eme des Calissons}

\vfill

%: \calih \calig \calid
\newcommand{\calih}[2][black!20]{\draw[fill=#1,shift={(#2)}] (0,0)--++(30:1)--++(150:1)--++(210:1)--cycle;}
\newcommand{\calig}[2][black!40]{\draw[fill=#1,shift={(#2)}] (0,0)--++(150:1)--++(270:1)--++(330:1)--cycle;}
\newcommand{\calid}[2][black!60]{\draw[fill=#1,shift={(#2)}] (0,0)--++(-90:1)--++(30:1)--++(90:1)--cycle;}
%: \dd = 1/2   \rr = sqrt(3)/2
\def\dd{0.5}
\def\rr{0.866025}

\begin{center}
% Hexagone
\begin{tikzpicture} [scale=0.75]
\draw[very thick,fill=yellow!25] (30:5)--(90:5)--(150:5)--(210:5)--(270:5)--(330:5)--cycle;
\calid[yellow!25]{$(-5*\rr,2+\dd)$}
\calid[yellow!25]{$(-5*\rr,1+\dd)$}
\calih[yellow!25]{$(-4*\rr,0)$}
\calig[yellow!25]{$(-4*\rr,0)$}
\calid[yellow!25]{$(-4*\rr,0)$}
\calig[yellow!25]{$(-4*\rr,-2*\dd)$}
\calig[yellow!25]{$(-4*\rr,-4*\dd)$}
\calih[yellow!25]{$(-3*\rr,5*\dd)$}
\calig[yellow!25]{$(-3*\rr,5*\dd)$}
\calig[yellow!25]{$(-3*\rr,3*\dd)$}
\calih[yellow!25]{$(-3*\rr,-3*\dd)$}
\calig[yellow!25]{$(-3*\rr,-3*\dd)$}
\calid[yellow!25]{$(-3*\rr,-3*\dd)$}
\calig[yellow!25]{$(-3*\rr,-5*\dd)$}
\calid[yellow!25]{$(-3*\rr,-5*\dd)$}
\calih[yellow!25]{$(-2*\rr,-8*\dd)$}
\calih[yellow!25]{$(-\rr,-9*\dd)$}
\calih[yellow!25]{$(0,-10*\dd)$}
\calid[yellow!25]{$(-3*\rr,5*\dd)$}
\calih[yellow!25]{$(-2*\rr,2*\dd)$}
\calig[yellow!25]{$(-2*\rr,2*\dd)$}
\calid[yellow!25]{$(-2*\rr,2*\dd)$}
\calig[yellow!25]{$(-2*\rr,0)$}
\calid[yellow!25]{$(-2*\rr,0)$}
\calid[yellow!25]{$(-2*\rr,-2*\dd)$}
\calih[yellow!25]{$(-\rr,-5*\dd)$}
\calig[yellow!25]{$(-\rr,-5*\dd)$}
\calid[yellow!25]{$(-\rr,-5*\dd)$}
\calih[yellow!25]{$(-2*\rr,6*\dd)$}
\calid[yellow!25]{$(-2*\rr,6*\dd)$}
\calih[yellow!25]{$(-\rr,3*\dd)$}
\calid[yellow!25]{$(-\rr,3*\dd)$}
\calid[yellow!25]{$(-\rr,1*\dd)$}
\calih[yellow!25]{$(0,-2*\dd)$}
\calig[yellow!25]{$(0,-2*\dd)$}
\calid[yellow!25]{$(0,-2*\dd)$}
\calid[yellow!25]{$(0,-4*\dd)$}
\calih[yellow!25]{$(0,-8*\dd)$}
\calih[yellow!25]{$(2*\rr,-8*\dd)$}
\calih[yellow!25]{$(\rr,-7*\dd)$}
\calih[yellow!25]{$(\rr,-9*\dd)$}
\calih[yellow!25]{$(-\rr,7*\dd)$}
\calih[yellow!25]{$(0,6*\dd)$}
\calig[yellow!25]{$(0,6*\dd)$}
\calid[yellow!25]{$(0,6*\dd)$}
\calid[yellow!25]{$(0,4*\dd)$}
\calih[yellow!25]{$(\rr,\dd)$}
\calig[yellow!25]{$(\rr,\dd)$}
\calih[yellow!25]{$(2*\rr,0)$}
\calig[yellow!25]{$(2*\rr,0)$}
\calid[yellow!25]{$(2*\rr,0)$}
\calig[yellow!25]{$(2*\rr,-2*\dd)$}
\calid[yellow!25]{$(2*\rr,-2*\dd)$}
\calig[yellow!25]{$(2*\rr,-4*\dd)$}
\calid[yellow!25]{$(2*\rr,-4*\dd)$}
\calih[yellow!25]{$(3*\rr,-7*\dd)$}
\calih[yellow!25]{$(0,8*\dd)$}
\calih[yellow!25]{$(\rr,7*\dd)$}
\calih[yellow!25]{$(2*\rr,6*\dd)$}
\calig[yellow!25]{$(2*\rr,6*\dd)$}
\calid[yellow!25]{$(2*\rr,6*\dd)$}
\calig[yellow!25]{$(2*\rr,4*\dd)$}
\calig[yellow!25]{$(4*\rr,6*\dd)$}
\calig[yellow!25]{$(5*\rr,5*\dd)$}
\calig[yellow!25]{$(5*\rr,3*\dd)$}
\calig[yellow!25]{$(5*\rr,\dd)$}
\calih[yellow!25]{$(3*\rr,3*\dd)$}
\calig[yellow!25]{$(3*\rr,3*\dd)$}
\calid[yellow!25]{$(3*\rr,3*\dd)$}
\calid[yellow!25]{$(3*\rr,\dd)$}
\calih[yellow!25]{$(4*\rr,-2*\dd)$}
\calig[yellow!25]{$(4*\rr,-2*\dd)$}
\calid[yellow!25]{$(4*\rr,-2*\dd)$}
\calig[yellow!25]{$(4*\rr,-4*\dd)$}
\calid[yellow!25]{$(4*\rr,-4*\dd)$}
\end{tikzpicture}
\end{center}
 
\vfill

{\large Quelle que soit la fa\c{c}on de ranger des Calissons dans une bo\^{\i}te hexagonale, il y a toujours un nombre identique de Calissons rang\'es dans chacune des trois directions possibles.}

\vfill

\begin{center}
% Hexagone
\begin{tikzpicture} [scale=0.75]
\draw[very thick,fill=yellow!25] (30:5)--(90:5)--(150:5)--(210:5)--(270:5)--(330:5)--cycle;
\calid{$(-5*\rr,2+\dd)$}
\calid{$(-5*\rr,1+\dd)$}
\calih{$(-4*\rr,0)$}
\calig{$(-4*\rr,0)$}
\calid{$(-4*\rr,0)$}
\calig{$(-4*\rr,-2*\dd)$}
\calig{$(-4*\rr,-4*\dd)$}
\calih{$(-3*\rr,5*\dd)$}
\calig{$(-3*\rr,5*\dd)$}
\calig{$(-3*\rr,3*\dd)$}
\calih{$(-3*\rr,-3*\dd)$}
\calig{$(-3*\rr,-3*\dd)$}
\calid{$(-3*\rr,-3*\dd)$}
\calig{$(-3*\rr,-5*\dd)$}
\calid{$(-3*\rr,-5*\dd)$}
\calih{$(-2*\rr,-8*\dd)$}
\calih{$(-\rr,-9*\dd)$}
\calih{$(0,-10*\dd)$}
\calid{$(-3*\rr,5*\dd)$}
\calih{$(-2*\rr,2*\dd)$}
\calig{$(-2*\rr,2*\dd)$}
\calid{$(-2*\rr,2*\dd)$}
\calig{$(-2*\rr,0)$}
\calid{$(-2*\rr,0)$}
\calid{$(-2*\rr,-2*\dd)$}
\calih{$(-\rr,-5*\dd)$}
\calig{$(-\rr,-5*\dd)$}
\calid{$(-\rr,-5*\dd)$}
\calih{$(-2*\rr,6*\dd)$}
\calid{$(-2*\rr,6*\dd)$}
\calih{$(-\rr,3*\dd)$}
\calid{$(-\rr,3*\dd)$}
\calid{$(-\rr,1*\dd)$}
\calih{$(0,-2*\dd)$}
\calig{$(0,-2*\dd)$}
\calid{$(0,-2*\dd)$}
\calid{$(0,-4*\dd)$}
\calih{$(0,-8*\dd)$}
\calih{$(2*\rr,-8*\dd)$}
\calih{$(\rr,-7*\dd)$}
\calih{$(\rr,-9*\dd)$}
\calih{$(-\rr,7*\dd)$}
\calih{$(0,6*\dd)$}
\calig{$(0,6*\dd)$}
\calid{$(0,6*\dd)$}
\calid{$(0,4*\dd)$}
\calih{$(\rr,\dd)$}
\calig{$(\rr,\dd)$}
\calih{$(2*\rr,0)$}
\calig{$(2*\rr,0)$}
\calid{$(2*\rr,0)$}
\calig{$(2*\rr,-2*\dd)$}
\calid{$(2*\rr,-2*\dd)$}
\calig{$(2*\rr,-4*\dd)$}
\calid{$(2*\rr,-4*\dd)$}
\calih{$(3*\rr,-7*\dd)$}
\calih{$(0,8*\dd)$}
\calih{$(\rr,7*\dd)$}
\calih{$(2*\rr,6*\dd)$}
\calig{$(2*\rr,6*\dd)$}
\calid{$(2*\rr,6*\dd)$}
\calig{$(2*\rr,4*\dd)$}
\calig{$(4*\rr,6*\dd)$}
\calig{$(5*\rr,5*\dd)$}
\calig{$(5*\rr,3*\dd)$}
\calig{$(5*\rr,\dd)$}
\calih{$(3*\rr,3*\dd)$}
\calig{$(3*\rr,3*\dd)$}
\calid{$(3*\rr,3*\dd)$}
\calid{$(3*\rr,\dd)$}
\calih{$(4*\rr,-2*\dd)$}
\calig{$(4*\rr,-2*\dd)$}
\calid{$(4*\rr,-2*\dd)$}
\calig{$(4*\rr,-4*\dd)$}
\calid{$(4*\rr,-4*\dd)$}
\end{tikzpicture}
\end{center}
 

\vfill

\end{document}
