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authorKarl Berry <karl@freefriends.org>2008-02-27 01:41:10 +0000
committerKarl Berry <karl@freefriends.org>2008-02-27 01:41:10 +0000
commit52e0e587ff774ec47a088432cdb5738a39fb3739 (patch)
treedb08a7c283495c0bbdc3bf159b7e0b96f68a453b /Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp
parentf82487f7cb5a8a26f143589f509ed0a76b51b82f (diff)
new (and special install) pstricks package pst-cox (24feb08)
git-svn-id: svn://tug.org/texlive/trunk@6759 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp')
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-rw-r--r--Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.tex470
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diff --git a/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/Gallery.tex b/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/Gallery.tex
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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% pst-coxeter_parameter\Gallery.tex
+% Authors: J.-G. Luque and M. Luque
+% Purpose: Demonstration of the library pst-coxeterp
+% Created: 02/02/2008
+% License: LGPL
+% Project: PST-Cox V1.00
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Copyright © 2008 Jean-Gabriel Luque, Manuel Luque.
+% This work may be distributed and/or modified under the condition of
+% the Lesser GPL.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% This file is part of PST-Cox V1.00.
+%
+% PST-Cox V1.00 is free software: you can redistribute it and/or modify
+% it under the terms of the Lesser GNU General Public License as published by
+% the Free Software Foundation, either version 3 of the License, or
+% (at your option) any later version.
+%
+% PST-Cox V1.00 is distributed in the hope that it will be useful,
+% but WITHOUT ANY WARRANTY; without even the implied warranty of
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+% Lesser GNU General Public License for more details.
+%
+% You should have received a copy of the Lesser GNU General Public License
+% along with PST-Cox V1.00. If not, see <http://www.gnu.org/licenses/>.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\documentclass[a4paper]{article}
+\usepackage[latin1]{inputenc}%
+\usepackage[margin=2cm]{geometry}
+\usepackage{pst-coxeterp}
+\usepackage{multido}
+\usepackage{amssymb}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{graphics}
+% d\'emonstration
+% JG Luque 12 août 2003
+\newcount\ChoicePolytope
+\def\S{\mbox{\goth S}}
+\def\Sym{{\bf Sym}}
+\def\sym{{\sl Sym}}
+\def\QSym{{QSym}}
+\def\N{{\mathbb N}}\def\L{{\mathbb L}}
+\def\C{{\mathbb C}}
+\def\Z{{\mathbb Z}}
+\def\R{{\mathbb R}}
+\def\Q{{\mathbb Q}}
+\def\demoPolytopes#1{%}
+\begin{center}
+\ifcase\ichoice\or \def\polname{$2\{3\}3$}\def\ep{0.5mm}
+ \or \def\polname{$3\{3\}2$}\def\ep{0.3mm}\or
+\def\polname{$3\{3\}3$}\def\ep{0.3mm}\or
+ \def\polname{$3\{4\}2$}\def\ep{0.3mm}\or \def\polname{$3\{4\}4$}\def\ep{0.1mm}
+ \or \def\polname{$3\{4\}3$}\def\ep{0.1mm}\or \def\polname{$4\{3\}4$}\def\ep{0.1mm}\or
+\def\polname{$2\{4\}3\{3\}3$}\def\ep{0.1mm}\or \def\polname{ Hessien}\def\ep{0.1mm}
+ \or \def\polname{$3\{3\}3\{4\}2$}\def\ep{0.1mm}
+ \or \def\polname{de Witting} \def\ep{0.01mm} \or
+ \def\polname{$3\{8\}2$} \def\ep{0.1mm} \or
+ \def\polname{$2\{8\}3$} \def\ep{0.1mm} \or
+ \def\polname{$3\{5\}3$} \def\ep{0.1mm}
+ \or\def\polname{$4\{4\}3$} \def\ep{0.1mm}
+ \or\def\polname{$4\{3\}2$} \def\ep{0.1mm}
+ \or\def\polname{$2\{3\}4$} \def\ep{0.1mm}
+ \or\def\polname{$2\{6\}4$} \def\ep{0.1mm}
+ \or\def\polname{$4\{6\}2$} \def\ep{0.1mm}
+ \or\def\polname{$5\{3\}5$} \def\ep{0.1mm}
+ \or\def\polname{$2\{10\}3$} \def\ep{0.1mm}
+ \or\def\polname{$3\{10\}2$} \def\ep{0.1mm}
+ \or\def\polname{$2\{5\}3$} \def\ep{0.1mm}
+ \or\def\polname{$3\{5\}2$} \def\ep{0.1mm}
+ \or\def\polname{$2\{4\}3$} \def\ep{0.1mm}
+ \or\def\polname{$2\{3\}2\{4\}3$} \def\ep{0.1mm}
+ \or\def\polname{$3\{4\}2\{3\}2$} \def\ep{0.1mm}
+ \or\def\polname{$3\{4\}2\{3\}2\{3\}2$} \def\ep{0.1mm}
+ \or\def\polname{$2\{3\}2\{3\}2\{4\}3$} \def\ep{0.1mm}
+ \fi {\Huge Polytope \polname}
+
+\begin{pspicture}(-9,-9)(9,9)
+\psset{unit=3cm,linewidth=0.01mm}
+\CoxeterCoordinates[choice=#1,linewidth=\ep] % par défaut choice=1 (332)
+\end{pspicture}
+
+$\backslash$\texttt{CoxeterCoordinates[choice=#1]}
+\end{center}
+\begin{center}
+\begin{tabular}{ccc}
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.7cm}
+\CoxeterCoordinates[drawvertices=false,choice=#1,linewidth=0.01mm] %
+\end{pspicture}
+&
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.7cm}
+\CoxeterCoordinates[drawcenters=false,choice=#1,linewidth=0.01mm] %
+\end{pspicture}
+&
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.7cm}
+\CoxeterCoordinates[drawedges=false,choice=#1,linewidth=0.01mm] %
+\end{pspicture}\\
+\texttt{[drawvertices=false,choice=#1]}
+&
+\texttt{[drawcenters=false,choice=#1]}
+&
+\texttt{[drawedges=false,choice=#1]}
+\end{tabular}
+\end{center}}
+%
+\title{The Gallery of Infinite Series}
+\author{Jean-Gabriel \textsc{Luque}\footnote{Jean-Gabriel.Luque@univ-mlv.fr},
+Manuel \textsc{Luque}\footnote{manuel.luque27@gmail.com}}
+\begin{document}
+\maketitle
+\newpage
+\section{Real polygons}
+There are the polytopes $2\{\frac pq\}2$ (with $p$ and $q$ in $\N$)
+in the notation of Coxeter. Use the command:
+\begin{verbatim}
+\psset{unit=1.5cm}\Polygon[P=p,Q=q]
+\end{verbatim}
+\[\begin{array}{|c|c|c|}
+\hline 2&3&4\\
+\hline \begin{pspicture}(-1.5,-3)(1.5,3)
+\psset{unit=1.5cm}\Polygon[P=2,Q=1]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Polygon[P=3]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Polygon[P=4]
+\end{pspicture}\\
+\hline 5&\frac52&6\\
+\hline \begin{pspicture}(-1.5,-3)(1.5,3)
+\psset{unit=1.5cm}\Polygon[P=5,Q=1]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Polygon[P=5,Q=2]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Polygon[P=6]
+\end{pspicture}\\
+\hline 7&\frac72&\frac73\\
+\hline \begin{pspicture}(-1.5,-3)(1.5,3)
+\psset{unit=1.5cm}\Polygon[P=7]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Polygon[P=7,Q=2]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Polygon[P=10,Q=3]
+\end{pspicture}\\
+\hline
+\end{array}
+\]
+\newpage
+\section{Simplices }
+There are the real polytopes $2\{3\}2\cdots2\{3\}2$ in dimension $n$
+(tetrahedron, pentatope, sextatope etc...) in the notation of
+Coxeter. Use the command:
+\begin{verbatim}
+\psset{unit=1.5cm}\Simplex[dimension=n]
+\end{verbatim}
+\[\begin{array}{|c|c|c|}
+\hline 2&3&4\\
+\hline \begin{pspicture}(-1.5,-3)(1.5,3)
+\psset{unit=1.5cm}\Simplex[dimension=2]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Simplex[dimension=3]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Simplex[dimension=4]
+\end{pspicture}\\
+\hline 5&6&7\\
+\hline \begin{pspicture}(-1.5,-3)(1.5,3)
+\psset{unit=1.5cm}\Simplex[dimension=5]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Simplex[dimension=6]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Simplex[dimension=7]
+\end{pspicture}\\
+\hline 8&9&10\\
+\hline \begin{pspicture}(-1.5,-3)(1.5,3)
+\psset{unit=1.5cm}\Simplex[dimension=8]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Simplex[dimension=9]
+\end{pspicture}&\begin{pspicture}(-3,-3)(3,3)
+\psset{unit=1.5cm}\Simplex[dimension=10]
+\end{pspicture}\\
+\hline
+\end{array}
+\]\newpage
+\section{The infinite series $\gamma_n^p$}
+It is an infinite series of polytopes with two parameters $p$ and
+$n$. The parameter $n$ is the dimension of the polytope. In the
+notation of Coxeter, its name reads $p\{4\}2\{3\}\dots\{3\}2$. In
+the case $p=2$, we recovers the family of the hypercubes. Use the
+command:
+ \begin{verbatim}
+ \gammapn[P=p,dimension=n]
+ \end{verbatim}
+\[\begin{array}{|c|c|c|}
+\hline \gamma_2^2&\gamma_2^3&\gamma_2^4\\
+\hline \begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.2cm}\gammapn[dimension=2,P=2,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.2cm}\gammapn[P=3,dimension=2,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1cm}\gammapn[P=4,dimension=2,linewidth=0.01mm]
+\end{pspicture}\\
+\hline \gamma_3^2&\gamma_3^3&\gamma_3^4\\ \hline
+\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1cm}\gammapn[P=2,dimension=3,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.8cm}\gammapn[P=3,dimension=3,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.7cm}\gammapn[P=4,dimension=3,linewidth=0.01mm]
+\end{pspicture}\\
+\hline \gamma_4^2&\gamma_4^3&\gamma_4^4\\
+\hline \begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.8cm}\gammapn[P=2,dimension=4,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.6cm}\gammapn[P=3,dimension=4,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.55cm}\gammapn[P=4,dimension=4,linewidth=0.01mm]
+\end{pspicture}\\
+\hline
+\end{array}
+\]%
+\newpage
+\section{The infinite series $\beta_n^p$}
+It is an infinite series of polytopes with two parameters $p$ and
+$n$ reciprocals of $\gamma_n^p$. The parameter $n$ is the dimension
+of the polytope. In the notation of Coxeter, its name reads
+$2\{3\}2\{3\}\dots\{3\}2\{4\}p$. In the case $p=2$, we recovers the
+family of the $2^n$-topes which generalizes the tetrahedron for
+higher dimension. Use the command:
+ \begin{verbatim}
+ \betapn[P=p,dimension=n]
+ \end{verbatim}
+\[\begin{array}{|c|c|c|}
+\hline \beta_2^2&\beta_2^3&\beta_2^4\\
+\hline \begin{pspicture}(-2,-3)(2,3)
+\psset{unit=2cm}\betapn[dimension=2,P=2]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betapn[P=3,dimension=2,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.4cm}\betapn[P=4,dimension=2,linewidth=0.01mm]
+\end{pspicture}\\
+\hline \beta_3^2&\beta_3^3&\beta_3^4\\ \hline
+\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=2cm}\betapn[P=2,dimension=3,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betapn[P=3,dimension=3,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.4cm}\betapn[P=4,dimension=3,linewidth=0.01mm]
+\end{pspicture}\\
+\hline \beta_4^2&\beta_4^3&\beta_4^4\\
+\hline \begin{pspicture}(-2,-3)(2,3)
+\psset{unit=2cm}\betapn[P=2,dimension=4,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betapn[P=3,dimension=4,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.4cm}\betapn[P=4,dimension=4,linewidth=0.01mm]
+\end{pspicture}\\
+\hline
+\end{array}
+\]%
+\newpage
+\section{The infinite series $\gamma_2^p$}
+It is a special case of the series $\gamma_n^p$ for $n=2$. In this
+case, the polytopes are complex polygons. The projection used here
+is different than the projection used with {\tt gammapn}. Use the
+command:
+\begin{verbatim}
+\gammaptwo[P=p]
+\end{verbatim}
+\[\begin{array}{|c|c|c|}
+\hline \gamma_2^3&\gamma_2^4&\gamma_2^5\\
+\hline \begin{pspicture}(-2,-3)(2,3) \psset{unit=1cm}\gammaptwo[P=3]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1cm}\gammaptwo[P=4,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1cm}\gammaptwo[P=5,linewidth=0.01mm]
+\end{pspicture}\\
+\hline \gamma_2^6&\gamma_2^7&\gamma_2^8\\ \hline
+\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1cm}\gammaptwo[P=6,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.8cm}\gammaptwo[P=7,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.7cm}\gammaptwo[P=8,linewidth=0.01mm]
+\end{pspicture}\\
+\hline \gamma_2^9&\gamma_2^{10}&\gamma_2^{11}\\
+\hline \begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.8cm}\gammaptwo[P=9,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.7cm}\gammaptwo[P=10,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=0.7cm}\gammaptwo[P=11,linewidth=0.01mm]
+\end{pspicture}\\
+\hline
+\end{array}
+\]%
+\newpage
+\section{The infinite series $\beta_2^p$}
+It is a special case of the series $\beta_n^p$ for $n=2$. In this
+case, the polytopes are complex polygons. The projection used here
+is different than the projection used with {\tt betapn}. Use the
+command:
+\begin{verbatim}
+\betaptwo[P=p]
+\end{verbatim}
+\[\begin{array}{|c|c|c|}
+\hline \beta_2^3&\beta_2^4&\beta_2^5\\
+\hline \begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=3]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=4,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=5,linewidth=0.01mm]
+\end{pspicture}\\
+\hline \beta_2^6&\beta_2^7&\beta_2^8\\ \hline
+\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=6,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=7,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=8,linewidth=0.01mm]
+\end{pspicture}\\
+\hline \beta_2^9&\beta_2^{10}&\beta_2^{11}\\
+\hline \begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=9,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=10,linewidth=0.01mm]
+\end{pspicture}&\begin{pspicture}(-2,-3)(2,3)
+\psset{unit=1.5cm}\betaptwo[P=11,linewidth=0.01mm]
+\end{pspicture}\\
+\hline
+\end{array}
+\]%
+\begin{thebibliography}{ABC}
+%
+\bibitem{Cox1}
+H. S. M. Coxeter, {\em Regular Complex Polytopes}, Second Edition,
+Cambridge University Press, 1991 .
+%
+\end{thebibliography}
+\end{document}
diff --git a/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.pdf b/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.pdf
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Binary files differ
diff --git a/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.tex b/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.tex
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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% pst-coxeter_parameter\pst-coxeterp_doc.tex
+% Authors: J.-G. Luque and M. Luque
+% Purpose: Documentation for the library pst-coxcoor
+% Created: 02/02/2008
+% License: LGPL
+% Project: PST-Cox V1.00
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Copyright © 2008 Jean-Gabriel Luque, Manuel Luque.
+% This work may be distributed and/or modified under the condition of
+% the Lesser GPL.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% This file is part of PST-Cox V1.00.
+%
+% PST-Cox V1.00 is free software: you can redistribute it and/or modify
+% it under the terms of the Lesser GNU General Public License as published by
+% the Free Software Foundation, either version 3 of the License, or
+% (at your option) any later version.
+%
+% PST-Cox V1.00 is distributed in the hope that it will be useful,
+% but WITHOUT ANY WARRANTY; without even the implied warranty of
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+% Lesser GNU General Public License for more details.
+%
+% You should have received a copy of the Lesser GNU General Public License
+% along with PST-Cox V1.00. If not, see <http://www.gnu.org/licenses/>.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\documentclass[a4paper]{article}
+\usepackage[latin1]{inputenc}%
+\usepackage[margin=2cm]{geometry}
+\usepackage{pst-coxeterp}
+\usepackage{multido}
+\usepackage{amssymb}
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{graphics}
+% d\'emonstration
+% JG Luque 12 août 2003
+\newtheorem{example}{Example}[section]
+\newcount\ChoicePolytope
+\def\C{{\mathbb C}}
+
+\title{The Library {\tt pst-coxeterp}}
+\author{Jean-Gabriel \textsc{Luque}\footnote{Universit\'e Paris-Est, Laboratoire d'informatique
+de l'Institut-Gaspard Monge, Jean-Gabriel.Luque@univ-mlv.fr} and
+Manuel
+ \textsc{Luque}\footnote{mluque5130@aol.com}}
+\begin{document}
+\maketitle
+ \begin{abstract}
+ We describe the {\tt LaTex} library {\tt pst-coxeterp} devoted to
+ draw regular complex polytopes belonging in the infinite series.
+ \end{abstract}
+ \section{Introduction}
+ Inspired by the dissertation of G.C. Shephard \cite{Sh}, Coxeter
+ toke twenty years to write his most famous book {\em Regular Complex Polytopes} \cite{Cox}. But its
+ interest for the polytope dates from the beginning of his career as
+ shown his numerous publications on the subject (reader can refer to
+ \cite{Reg} or \cite{Kalei}). According to the preface of
+ \cite{Cox}, the term of complex polytopes is due to D.M.Y.
+ Sommerville \cite{Som}. A complex polytope may have more than two
+ vertices on an edge (and in particular the polygons may have more
+ than two edges at a vertice). It is a finite set of flags of subspaces in $\C^n$
+ with certain constraints
+ which will be not explain here \footnote{For a precise
+ definition, see \cite{Cox} Ch12}.
+ In fact, a complex polytope can be generated from one vertice by a finite number of pseudo-reflections.
+ More precisely, as for the classical solids, it
+ can be constructed from an arrangement of mirrors,
+ considering a point in the intersection of all but one the mirrors
+ and computing the orbit of this point by the pseudo-reflections generated by the mirrors. In the
+ case of the real polytopes, one uses classical reflections which are
+ involutions. It is not the case for general complex polytopes, since
+ a reflection may include a component which is a rotation.
+The classification of the complex polytopes is due to G.C. Shephard
+\cite{Sh} and is closely related to the classification of the
+complex unitary reflection groups \cite{ST}. This classification
+includes four infinite series of polytopes: the well-known real
+polygons (including the starry polygon) which have two parameters,
+the series of simplices (triangle, tetrahedron, pentatope, sextatope
+etc...) which have only one parameter, the dimension and to
+reciprocal series $\gamma_n^p$ and $\beta_n^p$. The library
+described here is a {\tt LaTex} package for drawing the polytopes of
+these infinite series.
+\section{Install {\tt pst-coxeterp}}
+The package contains two files: A latex style file {\tt
+pst-coxeterp.sty} which call the latex file {\tt pst-coxeterp.tex}
+containing the description of the macros. The installation is very
+simple. It suffices to copy the files {\tt pst-coxeterp.sty} and
+{\tt pst-coxeterp.tex} in the appropriate directories.
+\begin{example}\rm
+The file {\tt pst-coxeterp.sty} may be copy in the directory \\ {\tt
+c:/texmf/tex/latex/pst-coxeterp},\\
+ the file {\tt pst-coxeterp.tex} in\\
+{\tt c:/texmf/tex/generic/pst-coxeterp}
+\end{example}
+To use the package add the code
+\begin{verbatim}
+\usepackage{pst-coxeterp}
+%\end{verbatim}
+in the beginning of your LaTex-file.
+\begin{example}\rm
+\begin{verbatim}
+\documentclass[a4paper]{article}
+...
+\usepackage{pst-coxeterp}
+....
+\end{verbatim}
+\end{example}
+The library needs the packages {\tt PSTrick} and {\tt pst-xkey}.%
+
+\section{The different families}
+This library contains six macros for drawing polytopes belonging in
+a infinite series.\\
+The first macro, {\tt Polygon}, draws real (starry or not) polygon.
+The polygon is defined by two parameters {\tt P} and {\tt Q} which
+defines the angle $2\frac QP\Pi $ between the segment from the
+center to the first vertices and the segment from the center to the
+second vertices. By default the value of {\tt Q} is $1$.
+\begin{example}
+\begin{pspicture}(-2,-2)(2,2)
+\Polygon[P=11,Q=1] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\Polygon[P=11,Q=3]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \Polygon[P=11,Q=4]
+\end{pspicture}
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\Polygon[P=11,Q=1] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\Polygon[P=11,Q=3]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \Polygon[P=11,Q=4]
+\end{pspicture}
+\end{verbatim}
+\end{example}
+
+The macro {\tt Simplex} draws simplices in dimension $n$. The
+simplices are the real polytopes whose automorphism groups are the
+symmetric groups. The dimension of the polytope can be chosen using
+the parameter {\tt dimension}.
+\begin{example}
+\begin{pspicture}(-2,-2)(2,2)
+\Simplex[dimension=2] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\Simplex[dimension=3]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \Simplex[dimension=5]
+\end{pspicture}
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\Simplex[dimension=2] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\Simplex[dimension=3]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \Simplex[dimension=5]
+\end{pspicture}
+\end{verbatim}
+\end{example}
+
+The polytopes $\gamma_n^p$ forms a two parameters family which
+contains as special case the hypercubes. The parameter $n$ is the
+dimension of the polytope and the parameter $p$ is the number of
+vertices per edge. Use the macro {\tt gammapn} and the parameters
+{\tt dimension} and {\tt P} to chose the characteristics of the
+polytope.
+\begin{example}
+\begin{pspicture}(-2,-2)(2,2)
+\gammapn[dimension=2,P=4] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\gammapn[dimension=3,P=3,unit=0.7cm]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \gammapn[dimension=5,P=2,unit=0.55cm]
+\end{pspicture}
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\gammapn[dimension=2,P=4] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\gammapn[dimension=3,P=3,unit=0.7cm]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \gammapn[dimension=5,P=2,unit=0.55cm]
+\end{pspicture}
+\end{verbatim}
+\end{example}
+
+The polytopes $\beta_n^p$ forms a two parameters family which
+contains as special case the hyperoctahedra. The parameter $n$ is
+the dimension of the polytope and the parameter $p$ is the number of
+cells of dimension $n-1$ containing a cell of dimension $n-2$. Use
+the macro {\tt betapn} and the parameters {\tt dimension} and {\tt
+P} to chose the characteristics of the polytope.
+\begin{example}
+\begin{pspicture}(-2,-2)(2,2)
+\betapn[dimension=2,P=4] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\betapn[dimension=3,P=3]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \betapn[dimension=5,P=2]
+\end{pspicture}
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\betapn[dimension=2,P=4] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\betapn[dimension=3,P=3]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \betapn[dimension=5,P=2]
+\end{pspicture}
+\end{verbatim}
+\end{example}
+
+The macro {\tt gammaptwo} draw the regular complex polytope
+$\gamma_2^p$ which is a special case of $\gamma_n^p$ for an other
+projection. Use the parameter {\tt P} for setting the number of
+vertices by edge.
+\begin{example}
+\begin{pspicture}(-2,-2)(2,2)
+\gammaptwo[P=3] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\gammaptwo[P=4]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \gammaptwo[P=5]
+\end{pspicture}
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\gammaptwo[P=3] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\gammaptwo[P=4]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \gammaptwo[P=5]
+\end{pspicture}
+\end{verbatim}
+\end{example}
+
+The macro {\tt betaptwo} draw the regular complex polytope
+$\beta_2^p$ which is a special case of $\beta_n^p$ for an other
+projection (the same than for {\tt gammaptwo}). Use the parameter
+{\tt P} for setting the number of vertices by edge.
+\begin{example}
+\begin{pspicture}(-2,-2)(2,2)
+\betaptwo[P=3] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\betaptwo[P=4]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \betaptwo[P=5]
+\end{pspicture}
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\betaptwo[P=3] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\betaptwo[P=4]
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+ \betaptwo[P=5]
+\end{pspicture}
+\end{verbatim}
+\end{example}
+
+\section{Graphical parameters}
+\subsection{The components of a polytope}
+ The library {\tt pst-coxeterrep.sty} contains macros for
+drawing the vertices, the edges and the centers of the edges of
+polytopes of the infinite series of regular complex polytopes.
+
+It is possible to choice which components of the polytope will be
+drawn. It suffices to use the boolean parameters {\tt drawedges},
+{\tt drawvertices} and {\tt drawcenters}.
+
+ By default the values of the parameters {\tt
+drawedges}, {\tt drawvertices}, {\tt drawcenters} are set to {\tt
+true}.
+\begin{example}
+\rm
+\[
+\begin{pspicture}(-2,-2)(2,2)
+\Polygon[P=5,Q=2,drawcenters=false] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\Simplex[dimension=3,drawvertices=false] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.5}
+ \gammapn[P=4,dimension=4,drawedges=false]
+\end{pspicture}
+\]
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\Polygon[P=5,Q=2,drawcenters=false] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\Simplex[dimension=3,drawvertices=false] %
+\end{pspicture}
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.5}
+ \gammapn[P=4,dimension=4,drawedges=false]
+\end{pspicture}\end{verbatim}
+\end{example}
+\section{Graphical properties}
+It is possible to change the graphical characteristics of a
+polytope.\\
+The size of the polytope depends on the parameter {\tt unit}.
+\begin{example}
+\rm
+ \[
+ \begin{pspicture}(-1,-1)(1,1)
+\gammaptwo[P=4,unit=0.5cm] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\gammaptwo[P=4,unit=1cm] %
+\end{pspicture}
+ \begin{pspicture}(-4,-4)(4,4)
+\gammaptwo[P=4,unit=2cm] %
+\end{pspicture}
+\]
+\begin{verbatim}
+ \begin{pspicture}(-1,-1)(1,1)
+\gammaptwo[P=4,unit=0.5cm] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\gammaptwo[P=4,unit=1cm] %
+\end{pspicture}
+ \begin{pspicture}(-4,-4)(4,4)
+\gammaptwo[P=4,unit=2cm] %
+\end{pspicture}
+\end{verbatim}
+\end{example}
+Classically, one can modify the color and the width of the edges
+using the parameter {\tt linecolor} and {\tt linewidth}.
+\begin{example}
+\rm
+ \[
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.8,linewidth=0.01,linecolor=red}
+\betaptwo[P=5] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\betaptwo[P=5] %
+\end{pspicture}
+\]
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.8,linewidth=0.01,linecolor=red}
+\betaptwo[P=5] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\betaptwo[P=5] %
+\end{pspicture}
+\end{verbatim}
+\end{example}
+The color, the style and the size of the vertices can be modify
+using the parameters {\tt colorVertices}, {\tt styleVertices} and
+{\tt sizeVertices}. The style of the vertices can be chosen in the
+classical dot styles.
+\begin{example}
+\rm
+ \[
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=1.5cm,colorVertices=blue,styleVertices=pentagon,sizeVertices=0.2}
+\betapn[P=5,dimension=4] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\psset{unit=1.5cm,colorVertices=magenta,sizeVertices=0.1,styleVertices=triangle} %
+\betapn[P=5,dimension=4]
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\psset{unit=1.5cm,colorVertices=red,styleVertices=+,sizeVertices=0.2} %
+\betapn[P=5,dimension=4]
+\end{pspicture}
+\]
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=1.5cm,colorVertices=blue,styleVertices=pentagon,sizeVertices=0.2}
+\betapn[P=5,dimension=4] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\psset{unit=1.5cm,colorVertices=magenta,sizeVertices=0.1,styleVertices=triangle} %
+\betapn[P=5,dimension=4]
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\psset{unit=1.5cm,colorVertices=red,styleVertices=+,sizeVertices=0.2} %
+\betapn[P=5,dimension=4]
+\end{pspicture}
+\end{verbatim}
+\end{example}
+The color, the style and the size of the centers of the edges can be
+modify using the parameters {\tt colorCenters}, {\tt styleCenters}
+and {\tt sizeCenters}.
+\begin{example}
+\rm
+ \[
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.5cm,colorCenters=blue,styleCenters=pentagon,sizeCenters=0.2} %
+\gammapn[P=5,dimension=4] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.5cm,colorCenters=magenta,sizeCenters=0.1,styleCenters=triangle} %
+\gammapn[P=5,dimension=4] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.5cm,colorCenters=red,styleCenters=+,sizeCenters=0.2} %
+\gammapn[P=5,dimension=4] %
+\end{pspicture}
+\]
+\begin{verbatim}
+\begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.5cm,colorCenters=blue,styleCenters=pentagon,sizeCenters=0.2} %
+\gammapn[P=5,dimension=4] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.5cm,colorCenters=magenta,sizeCenters=0.1,styleCenters=triangle} %
+\gammapn[P=5,dimension=4] %
+\end{pspicture}
+ \begin{pspicture}(-2,-2)(2,2)
+\psset{unit=0.5cm,colorCenters=red,styleCenters=+,sizeCenters=0.2} %
+\gammapn[P=5,dimension=4] %
+\end{pspicture}
+\end{verbatim}
+\end{example}
+
+ \begin{thebibliography}{ABC}
+
+\bibitem{Reg} H. S. M. Coxeter, {\em Regular polytopes}, Third
+Edition, Dover Publication Inc., New-York, 1973.
+%
+\bibitem{Cox}
+H. S. M. Coxeter, {\em Regular Complex Polytopes}, Second Edition,
+Cambridge University Press, 1991 .
+%
+\bibitem{Kalei}
+ H.S.M. Coxeter, {\em Kaleidoscopes, selected writing of H.S.M.
+ Coxeter by F.A. Sherk, P. McMullen, A.C. Thompson, A. Ivi\'c Weiss}, Canadian Mathematical Society Series of Monographs and
+ Advanced texts, Published in conjunction with the fiftieth anniversary of
+ the canadian mathematical society, J. M. Borwein and P. B. Borwein
+ Ed., A Wiley-Interscience publication, 1995.
+%
+\bibitem{Sh} G.C. Shephard, {\em Regular Complex Polytopes},
+Proceeding of the London Mathermatical Society (3), 2 82-97.
+%
+\bibitem{ST} G.C. Shephard and J.A. Todd, {\it Finite unitary
+reflection groups}, Canadian Journal of Mathematics 6, 274-304,
+1954.
+%
+\bibitem{Som} M.Y. Sommerville, {\it Geometry of $n$ dimension},
+Methuen, Lodon, 1929.
+\end{thebibliography}
+
+ \end{document}