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author | Karl Berry <karl@freefriends.org> | 2008-02-27 01:41:10 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2008-02-27 01:41:10 +0000 |
commit | 52e0e587ff774ec47a088432cdb5738a39fb3739 (patch) | |
tree | db08a7c283495c0bbdc3bf159b7e0b96f68a453b /Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp | |
parent | f82487f7cb5a8a26f143589f509ed0a76b51b82f (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')
-rw-r--r-- | Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/Gallery.tex | 342 | ||||
-rw-r--r-- | Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.pdf | bin | 0 -> 204208 bytes | |||
-rw-r--r-- | Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.tex | 470 |
3 files changed, 812 insertions, 0 deletions
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 new file mode 100644 index 00000000000..7d47fd13eb0 --- /dev/null +++ b/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/Gallery.tex @@ -0,0 +1,342 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% 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 Binary files differnew file mode 100644 index 00000000000..efc7bfa09f5 --- /dev/null +++ b/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.pdf 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 new file mode 100644 index 00000000000..4cb0c32a187 --- /dev/null +++ b/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.tex @@ -0,0 +1,470 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% 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} |