From 52e0e587ff774ec47a088432cdb5738a39fb3739 Mon Sep 17 00:00:00 2001 From: Karl Berry Date: Wed, 27 Feb 2008 01:41:10 +0000 Subject: new (and special install) pstricks package pst-cox (24feb08) git-svn-id: svn://tug.org/texlive/trunk@6759 c570f23f-e606-0410-a88d-b1316a301751 --- Master/texmf-dist/doc/generic/pst-cox/README | 54 + Master/texmf-dist/doc/generic/pst-cox/gpl.txt | 674 +++++++++ Master/texmf-dist/doc/generic/pst-cox/lgpl.txt | 165 ++ .../doc/generic/pst-cox/pst-coxcoor/Gallery.tex | 1573 ++++++++++++++++++++ .../pst-cox/pst-coxcoor/pst-coxcoor_doc.pdf | Bin 0 -> 895667 bytes .../pst-cox/pst-coxcoor/pst-coxcoor_doc.tex | 1118 ++++++++++++++ .../doc/generic/pst-cox/pst-coxeterp/Gallery.tex | 342 +++++ .../pst-cox/pst-coxeterp/pst-coxeterp_doc.pdf | Bin 0 -> 204208 bytes .../pst-cox/pst-coxeterp/pst-coxeterp_doc.tex | 470 ++++++ 9 files changed, 4396 insertions(+) create mode 100644 Master/texmf-dist/doc/generic/pst-cox/README create mode 100644 Master/texmf-dist/doc/generic/pst-cox/gpl.txt create mode 100644 Master/texmf-dist/doc/generic/pst-cox/lgpl.txt create mode 100644 Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/Gallery.tex create mode 100644 Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/pst-coxcoor_doc.pdf create mode 100644 Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/pst-coxcoor_doc.tex create mode 100644 Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/Gallery.tex create mode 100644 Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.pdf create mode 100644 Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.tex (limited to 'Master/texmf-dist/doc/generic') diff --git a/Master/texmf-dist/doc/generic/pst-cox/README b/Master/texmf-dist/doc/generic/pst-cox/README new file mode 100644 index 00000000000..f7622032c22 --- /dev/null +++ b/Master/texmf-dist/doc/generic/pst-cox/README @@ -0,0 +1,54 @@ +PST-Cox v1.0 +------------------------ +This project contains two LaTeX packages for drawing regular complex polytopes: + - pst-coxcoor: pre-calculated regular polytopes up to dimension 4. + - pst-coxeterp: infinite series of regular complex regular polytopes. + +See for more information the files pst-coxcoor_doc.ps and pst-coxeterp_doc.ps of the package. + +First check whether the project can be installed by the distribution that you use. The packages need PSTrick and xkeyval of Hendri Adriaens. + +The structure and location in the Tex-tree is shown below. + +README.txt: should be not installed + + +| - pst-coxeterp_doc.ps +| --- pst-coxeterp-| +|-doc-| - Gallery_doc.ps +| | +| | - pst-coxcoor_doc.ps +| --- pst-coxcoor-| +| - Gallery.ps +| +| +|-dvips-[-- pst-coxcoor-[ pst-coxeter.pro +| +| +| --- pst-coxeterp-[ pst-coxeterp.sty +| | +| |-latex-| +| | | +| | --- pst-coxcoor-[ pst-coxcoor.sty +|-tex-| +| | --- pst-coxeterp-[ pst_coxeterp.tex +| | | +| |-generic-| +| | | +| | --- pst-coxcoor-[ pst-coxcoor.tex + +Don't forget to update your filename database after installing the +files. + +Contact informations +Jean-Gabriel Luque +Institut Gaspard Monge +Université Paris-Est +77454 Marne-la-Vallée Cedex 2 +Jean-Gabriel.Luque@univ-mlv.fr +or +Jean-Gabriel.Luque@wanadoo.fr + +License +Copyright © 2008 Jean-Gabriel Luque, Manuel Luque. +This work may be distributed and/or modified under the condition of the Lesser GPL. diff --git a/Master/texmf-dist/doc/generic/pst-cox/gpl.txt b/Master/texmf-dist/doc/generic/pst-cox/gpl.txt new file mode 100644 index 00000000000..94a9ed024d3 --- /dev/null +++ b/Master/texmf-dist/doc/generic/pst-cox/gpl.txt @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. 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If the Library as you +received it does not specify a version number of the GNU Lesser +General Public License, you may choose any version of the GNU Lesser +General Public License ever published by the Free Software Foundation. + + If the Library as you received it specifies that a proxy can decide +whether future versions of the GNU Lesser General Public License shall +apply, that proxy's public statement of acceptance of any version is +permanent authorization for you to choose that version for the +Library. diff --git a/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/Gallery.tex b/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/Gallery.tex new file mode 100644 index 00000000000..03b334e9855 --- /dev/null +++ b/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/Gallery.tex @@ -0,0 +1,1573 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% polygonesCoordinate\Gallery.tex +% Authors: J.-G. Luque and M. Luque +% Purpose: Demonstration of 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 . +% + +\documentclass[a4paper]{article} +\usepackage[latin1]{inputenc}% +\usepackage[margin=2cm]{geometry} +\usepackage{pst-coxcoor} +\usepackage{multido} +\usepackage{graphics} +\newcount\ChoicePolytope + +\def\Titre#1{ +\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} + \or\def\polname{$2\{3\}2\{5\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{5\}2\{3\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{3\}2\{3\}2\{4\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{4\}2\{3\}2\{3\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{3\}2\{4\}2\{3\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{3\}2\{3\}2\{5\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{5\}2\{3\}2\{2\}2$} \def\ep{0.1mm} + \or\def\polname{$3\left\{5\over2\right\}3$} \def\ep{0.1mm} + \or\def\polname{$5\left\{5\over2\right\}5$} \def\ep{0.1mm} + \or\def\polname{$2\left\{5\over2\right\}3$} \def\ep{0.1mm} + \or\def\polname{$3\left\{5\over2\right\}2$} \def\ep{0.1mm} + \or\def\polname{$3\left\{10\over3\right\}2$} \def\ep{0.1mm} + \or\def\polname{$2\left\{10\over3\right\}3$} \def\ep{0.1mm} + \or\def\polname{$3\left\{8\over3\right\}2$} \def\ep{0.1mm} + \or\def\polname{$2\left\{8\over3\right\}3$} \def\ep{0.1mm} + \or\def\polname{$5\{6\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{6\}5$} \def\ep{0.1mm} + \or\def\polname{$4\left\{8\over3\right\}3$} \def\ep{0.1mm} + \or\def\polname{$3\left\{8\over3\right\}4$} \def\ep{0.1mm} + \or\def\polname{$5\{5\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{5\}5$} \def\ep{0.1mm} + \or\def\polname{$5\left\{10\over3\right\}2$} \def\ep{0.1mm} + \or\def\polname{$2\left\{10\over3\right\}5$} \def\ep{0.1mm} + \or\def\polname{$5\{3\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{3\}5$} \def\ep{0.1mm} + \or\def\polname{$5\{4\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{4\}5$} \def\ep{0.1mm} + \or\def\polname{$5\left\{10\over3\right\}3$} \def\ep{0.1mm} + \or\def\polname{$3\left\{10\over3\right\}5$} \def\ep{0.1mm} + \or\def\polname{$5\{4\}3$} \def\ep{0.1mm} + \or\def\polname{$3\{4\}5$} \def\ep{0.1mm} + \or\def\polname{$5\{3\}3$} \def\ep{0.1mm} + \or\def\polname{$3\{3\}5$} \def\ep{0.1mm} + \or\def\polname{$5\left\{5\over2\right\}3$} \def\ep{0.1mm} + \or\def\polname{$3\left\{5\over2\right\}5$} \def\ep{0.1mm} + \or\def\polname{$2\left\{5\over2\right\}2\{3\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{3\}2\left\{5\over 2\right\}2$} \def\ep{0.1mm} + \or\def\polname{$2\left\{5\over2\right\}2\{5\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{5\}2\left\{5\over 2\right\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{6\}3$} \def\ep{0.1mm} + \or\def\polname{$3\{6\}2$} \def\ep{0.1mm} + \or\def\polname{$2\{\frac52\}2\{3\}2\{3\}2$} \def\ep{0.1mm} + \fi + {\Huge Polytope \polname} +} +\def\demoPolytopes#1{%} +\begin{center} +\Titre{#1} +\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}} + +\def\demoPolytopesGrand#1{%} +\begin{center} +\Titre{#1} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=7cm,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=2cm} +\CoxeterCoordinates[drawvertices=false,choice=#1,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=2cm} +\CoxeterCoordinates[drawcenters=false,choice=#1,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=2cm} +\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}} + +% +\def\demoPolytopesPetit#1{%} +\begin{center} +\Titre{#1} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=0.7cm,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=0.2cm} +\CoxeterCoordinates[drawvertices=false,choice=#1,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.2cm} +\CoxeterCoordinates[drawcenters=false,choice=#1,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.2cm} +\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} +\author{Jean-Gabriel \textsc{Luque}\footnote{Jean-Gabriel.Luque@univ-mlv.fr} and Manuel + \textsc{Luque}\footnote{mluque5130@aol.com}} +\begin{document} +\maketitle\newpage +\section{Les polygons (dimension 2)} +\multido{\ichoice=1+1}{7}{% +\demoPolytopes{\ichoice}\newpage} +\begin{center} +{\huge Polytope $3\{8\}2$} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=12,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=12]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawvertices=false,choice=12,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawcenters=false,choice=12,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawedges=false,choice=12,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=12]} & +\texttt{[drawcenters=false,choice=12]} & +\texttt{[drawedges=false,choice=12]} +\end{tabular} +\end{center}\newpage + +\begin{center} +{\huge Polytope $2\{8\}3$} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=13,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=13]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawvertices=false,choice=13,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawcenters=false,choice=13,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawedges=false,choice=13,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=13]} & +\texttt{[drawcenters=false,choice=13]} & +\texttt{[drawedges=false,choice=13]} +\end{tabular} +\end{center}\newpage + + +\multido{\ichoice=14+1}{11}{% +\demoPolytopes{\ichoice}\newpage} + +\begin{center} +{\huge Polytope $2\{5\}3$} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=23,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=23]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawvertices=false,choice=23,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawcenters=false,choice=23,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawedges=false,choice=23,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=23]} +& +\texttt{[drawcenters=false,choice=23]} +& +\texttt{[drawedges=false,choice=23]} +\end{tabular} +\end{center} + +\newpage +\multido{\ichoice=24+1}{2}{% +\demoPolytopes{\ichoice}\newpage} %%%%%% + %ù%%%%%% +\begin{center} + {\huge Polytope $3\{\frac 52\}3$} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=37,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=37]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawvertices=false,choice=37,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawcenters=false,choice=37,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawedges=false,choice=37,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=37]} & +\texttt{[drawcenters=false,choice=37]} & +\texttt{[drawedges=false,choice=37]} +\end{tabular} +\end{center} +%%%%% +\newpage + +\multido{\ichoice=38+1}{13}{% +\demoPolytopes{\ichoice} +\newpage} +\multido{\ichoice=51+1}{3}{ +\begin{center} +\Titre{\ichoice} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=\ichoice,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=\ichoice]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawvertices=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawcenters=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawedges=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=\ichoice]} & +\texttt{[drawcenters=false,choice=\ichoice]} & +\texttt{[drawedges=false,choice=\ichoice]} +\end{tabular} +\end{center} +\newpage +} +\begin{center} + \multido{\ichoice=54+1}{1}{ +\Titre{\ichoice}} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=54,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$ +\texttt{CoxeterCoordinates[choice=54]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawvertices=false,choice=54,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawcenters=false,choice=54,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawedges=false,choice=54,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=54]} & +\texttt{[drawcenters=false,choice=54]} & +\texttt{[drawedges=false,choice=54]} +\end{tabular} +\end{center} +\newpage +\multido{\ichoice=55+1}{5}{% +\demoPolytopes{\ichoice} +\newpage} +\multido{\ichoice=60+1}{1}{ +\begin{center} +\Titre{\ichoice} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=\ichoice,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=\ichoice]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.2cm} +\CoxeterCoordinates[drawvertices=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.2cm} +\CoxeterCoordinates[drawcenters=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.2cm} +\CoxeterCoordinates[drawedges=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=\ichoice]} & +\texttt{[drawcenters=false,choice=\ichoice]} & +\texttt{[drawedges=false,choice=\ichoice]} +\end{tabular} +\end{center} +\newpage} +\multido{\ichoice=61+1}{3}{ +\begin{center} +\Titre{\ichoice} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=\ichoice,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=\ichoice]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawvertices=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawcenters=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawedges=false,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=\ichoice]} & +\texttt{[drawcenters=false,choice=\ichoice]} & +\texttt{[drawedges=false,choice=\ichoice]} +\end{tabular} +\end{center} +\newpage +} +\begin{center} + \multido{\ichoice=64+1}{1}{ +\Titre{\ichoice}} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=64,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$ \texttt{CoxeterCoordinates[choice=64]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawvertices=false,choice=64,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawcenters=false,choice=64,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawedges=false,choice=64,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=64]} & +\texttt{[drawcenters=false,choice=64]} & +\texttt{[drawedges=false,choice=64]} +\end{tabular} +\end{center} +\newpage + +\section{Polyhedron (dimension 3)} +\multido{\ichoice=8+1}{3}{% +\demoPolytopes{\ichoice} \begin{center} +\begin{pspicture}(-2,-5)(2,5) +\psset{unit=2} +\CoxeterCoordinates[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.3mm] +\end{pspicture} +{\tt +[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.3mm] +} +\end{center} +\newpage} +\multido{\ichoice=26+1}{2}{% +\demoPolytopes{\ichoice} \begin{center} +\begin{pspicture}(-2,-5)(2,5) +\psset{unit=2} +\CoxeterCoordinates[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.3mm] +\end{pspicture} +{\tt +[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.3mm] +} +\end{center} +\newpage} + +\multido{\ichoice=30+1}{2}{% +\demoPolytopes{\ichoice} \begin{center} +\begin{pspicture}(-2,-5)(2,5) +\psset{unit=2} +\CoxeterCoordinates[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.3mm] +\end{pspicture} +{\tt +[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.3mm] +} +\end{center} +\newpage} +\multido{\ichoice=65+1}{4}{% +\demoPolytopesGrand{\ichoice} \begin{center} +\begin{pspicture}(-2,-5)(2,5) +\psset{unit=3} +\CoxeterCoordinates[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.3mm] +\end{pspicture} +{\tt +[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.3mm] +} +\end{center} +\newpage} + +\section{Polytopes (dimension $4$)} + +\begin{center} +{\Huge Witting polytope}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=11,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=11]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawvertices=false,choice=11,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawcenters=false,choice=11,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawedges=false,choice=11,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=11]} & +\texttt{[drawcenters=false,choice=11]} & +\texttt{[drawedges=false,choice=11]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.4cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=11,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=11]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.4cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=11,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=11]} +\end{center}\newpage +\begin{center} +{\huge Polytope $3\{4\}2\{3\}2\{3\}2$} +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=28,linewidth=0.1mm] % par défaut choice=1 (332) +\end{pspicture} + +$\backslash$\texttt{CoxeterCoordinates[choice=28]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawvertices=false,choice=28,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawcenters=false,choice=28,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm} +\CoxeterCoordinates[drawedges=false,choice=28,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=28]} +& +\texttt{[drawcenters=false,choice=28]} +& +\texttt{[drawedges=false,choice=28]} +\end{tabular} +\end{center} +\newpage +\multido{\ichoice=29+1}{1}{% +\demoPolytopes{\ichoice} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.4cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=\ichoice]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.4cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=\ichoice]} +\end{center}\newpage} + +\multido{\ichoice=32+1}{3}{% +\demoPolytopes{\ichoice} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.4cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=\ichoice]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.4cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=\ichoice,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=\ichoice]} +\end{center}\newpage} +\begin{center} +{\Huge Le $600$-topes}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=35,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=35]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawvertices=false,choice=35,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawcenters=false,choice=35,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawedges=false,choice=35,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=35]} & +\texttt{[drawcenters=false,choice=35]} & +\texttt{[drawedges=false,choice=35]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.5cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=35,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=35]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.5cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=35,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=35]} +\end{center}\newpage + +\begin{center} +{\Huge The $120$-topes}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=3.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=36,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=36]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawvertices=false,choice=36,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawcenters=false,choice=36,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} +\CoxeterCoordinates[drawedges=false,choice=36,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=36]} & +\texttt{[drawcenters=false,choice=36]} & +\texttt{[drawedges=false,choice=36]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=36,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=36]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=36,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=36]} +\end{center}\newpage + +%\multido{\ichoice=71+1}{1}{% +%\demoPolytopesPetit{\ichoice} \begin{center} +%\begin{pspicture}(-2,-5)(2,5) +%\psset{unit=0.5} +%\CoxeterCoordinates[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.1mm] +%\end{pspicture} +%{\tt +%[drawvertices=false,drawcentersfaces=true,drawcenters=false,choice=\ichoice,linewidth=0.1mm] +%} +%\end{center} +%\newpage} +\begin{center} +{\Huge The great starry $600$-topes}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=0.9cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=71,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=71]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.2cm} +\CoxeterCoordinates[drawvertices=false,choice=71,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.2cm} +\CoxeterCoordinates[drawcenters=false,choice=71,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.2cm} +\CoxeterCoordinates[drawedges=false,choice=71,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=71]} & +\texttt{[drawcenters=false,choice=71]} & +\texttt{[drawedges=false,choice=71]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=71,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=71]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=71,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=71]} +\end{center}\newpage +% +% +\begin{center} +{\Huge The great starry $120$-topes}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=72,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=72]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.5cm} +\CoxeterCoordinates[drawvertices=false,choice=72,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.5cm} +\CoxeterCoordinates[drawcenters=false,choice=72,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.5cm} +\CoxeterCoordinates[drawedges=false,choice=72,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=72]} & +\texttt{[drawcenters=false,choice=72]} & +\texttt{[drawedges=false,choice=72]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=5cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=72,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=72]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=10cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=72,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=72]} +\end{center}\newpage +% +% +% +\begin{center} +{\Huge $2\{3\}2\{\frac52\}2\{5\}2$}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=73,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=73]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawvertices=false,choice=73,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawcenters=false,choice=73,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm} +\CoxeterCoordinates[drawedges=false,choice=73,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=73]} & +\texttt{[drawcenters=false,choice=73]} & +\texttt{[drawedges=false,choice=73]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=73,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=73]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=73,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=73]} +\end{center}\newpage +% +% +% +\begin{center} +{\Huge $2\{3\}2\{5\}2\{\frac52\}2$}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=74,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=74]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.5cm} +\CoxeterCoordinates[drawvertices=false,choice=74,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.5cm} +\CoxeterCoordinates[drawcenters=false,choice=74,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.5cm} +\CoxeterCoordinates[drawedges=false,choice=74,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=74]} & +\texttt{[drawcenters=false,choice=74]} & +\texttt{[drawedges=false,choice=74]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=3cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=74,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=74]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=3cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=74,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=74]} +\end{center}\newpage +% +% +\begin{center} +{\Huge $2\{\frac52\}2\{3\}2\{5\}2$}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=75,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=75]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawvertices=false,choice=75,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawcenters=false,choice=75,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawedges=false,choice=75,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=75]} & +\texttt{[drawcenters=false,choice=75]} & +\texttt{[drawedges=false,choice=75]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2.5cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=75,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=75]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=3cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=75,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=75]} +\end{center}\newpage +% +% +\begin{center} +{\Huge $2\{\frac52\}2\{5\}2\{3\}2$}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=76,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=76]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawvertices=false,choice=76,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawcenters=false,choice=76,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawedges=false,choice=76,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=76]} & +\texttt{[drawcenters=false,choice=76]} & +\texttt{[drawedges=false,choice=76]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2.5cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=76,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=76]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=3cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=76,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=76]} +\end{center}\newpage +% +\begin{center} +{\Huge $2\{5\}2\{3\}2\{\frac52\}2$}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=77,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=77]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawvertices=false,choice=77,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawcenters=false,choice=77,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawedges=false,choice=77,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=77]} & +\texttt{[drawcenters=false,choice=77]} & +\texttt{[drawedges=false,choice=77]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2.5cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=77,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=77]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=3cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=77,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=77]} +\end{center}\newpage +% +\begin{center} +{\Huge $2\{5\}2\{\frac52\}2\{3\}2$}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=78,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=78]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawvertices=false,choice=78,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawcenters=false,choice=78,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawedges=false,choice=78,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=78]} & +\texttt{[drawcenters=false,choice=78]} & +\texttt{[drawedges=false,choice=78]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2.5cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=78,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=78]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=3cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=78,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=78]} +\end{center}\newpage + +% +\begin{center} +{\Huge $2\{5\}2\{\frac52\}2\{5\}2$}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=79,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=79} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawvertices=false,choice=79,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawcenters=false,choice=79,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[drawedges=false,choice=79,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=79]} & +\texttt{[drawcenters=false,choice=79]} & +\texttt{[drawedges=false,choice=79]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=2.5cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=79,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=79]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=3cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=79,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=79]} +\end{center}\newpage +% +\begin{center} +{\Huge $2\{\frac525\}2\{5\}2\{\frac52\}2$}\\ +\begin{pspicture}(-9,-9)(9,9) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=80,linewidth=0.01mm] % par défaut choice=1 (332) +\end{pspicture} +$\backslash$\texttt{CoxeterCoordinates[choice=80]} +\end{center} +\begin{center} +\begin{tabular}{ccc} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawvertices=false,choice=80,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawcenters=false,choice=80,linewidth=0.01mm] % +\end{pspicture} +& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm} +\CoxeterCoordinates[drawedges=false,choice=80,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,choice=80]} & +\texttt{[drawcenters=false,choice=80]} & +\texttt{[drawedges=false,choice=80]} +\end{tabular} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=80,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawvertices=false,drawedges=false,drawcenters=false,drawcentersfaces=true,choice=80]} +\end{center} +\begin{center} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1cm} +\CoxeterCoordinates[drawvertices=false,drawedges=false,drawcenters=false,drawcenterscells=true,choice=80,linewidth=0.01mm] % +\end{pspicture}\\ +\texttt{[drawcenters=false,choice=80]} +\end{center}\newpage + + +\section{Examples} +\begin{center} +\begin{pspicture}(-5,-5)(5,5) +\psset{unit=2}%\rotatebox{90}{ +\CoxeterCoordinates[drawcenters=false,drawcentersfaces=true, +choice=8,linewidth=0.3mm,linecolor=green,linecolor=blue,sizeCentersFaces=0.15, +colorCentersFaces=red,styleCentersFaces=pentagon] +%} +% \psset{unit=1} +\CoxeterCoordinates[drawvertices=false,choice=9,linewidth=0.01mm, +styleCenters=+,sizeCenters=0.1,colorCenters=black] +\end{pspicture}\\ +%Les centres des ar\^etes d'un polytope Hessien sont les sommets +%d'un polytope $2\{4\}3\{3\}3$. +The centers of the edges of an Hessian are the vertices of a +$2\{4\}3\{3\}3$. +\begin{verbatim} +\psset{unit=2}%\rotatebox{90}{ +\CoxeterCoordinates[drawcenters=false,drawcentersfaces=true, +choice=8,linewidth=0.3mm,linecolor=green,linecolor=blue,sizeCentersFaces=0.15, +colorCentersFaces=red,styleCentersFaces=pentagon] +%} + \psset{unit=1} +\CoxeterCoordinates[drawvertices=false,choice=9,linewidth=0.01mm, +styleCenters=+,sizeCenters=0.1,colorCenters=black] +\end{verbatim} +\end{center} +\newpage +\begin{center} +\begin{pspicture}(-2,-7)(2,10) +\psset{unit=2} \rotatebox{90}{ +\CoxeterCoordinates[drawcenters=false,choice=10,linewidth=0.3mm,linecolor=green,sizeVertices=0.2, +colorVertices=magenta]} \psset{unit=1.725} +\CoxeterCoordinates[drawvertices=false,choice=9,linewidth=0.01mm, +styleCenters=+,sizeCenters=0.1,colorCenters=black] +\end{pspicture}\\ +%Les centres des ar\^etes d'un polytope Hessien sont les sommets +%d'un polytope $2\{4\}3\{3\}3$. +The centers of the edges of an Hessian are a the vertices of a +$3\{3\}3\{4\}2$. +\begin{verbatim} +\psset{unit=2} \rotatebox{90}{ +\CoxeterCoordinates[drawcenters=false,choice=10,linewidth=0.3mm,linecolor=green,sizeVertices=0.2, +colorVertices=magenta]} \psset{unit=1.725} +\CoxeterCoordinates[drawvertices=false,choice=9,linewidth=0.01mm, +styleCenters=+,sizeCenters=0.1,colorCenters=black]\end{verbatim} +\end{center} +\newpage +\begin{center} +\begin{pspicture}(-2,-7)(2,10) +\psset{unit=3} +\CoxeterCoordinates[drawcenters=false,choice=8,linewidth=1mm,sizeVertices=0.1,colorVertices=magenta] +\psset{unit=1} +\CoxeterCoordinates[drawcenters=false,choice=9,linewidth=0.01mm,linecolor=blue, +styleVertices=triangle,sizeVertices=0.07,colorVertices=blue] +\rotatebox{180} +{\CoxeterCoordinates[drawcenters=false,choice=9,linewidth=0.01mm,linecolor=green, +styleVertices=triangle,sizeVertices=0.07,colorVertices=green]}\end{pspicture}\\ +%Les sommets d'un polytope $2\{4\}3\{3\}3$ sont les sommets de deux +%polytopes Hessien r\'eciproques. +The vertices of a polytope $2\{4\}3\{3\}3$ are the vertices of two +reciprocal Hessien polytopes +\begin{verbatim} +\psset{unit=3} +\CoxeterCoordinates[drawcenters=false,choice=8,linewidth=1mm, +sizeVertices=0.1,colorVertices=magenta] +\psset{unit=1} +\CoxeterCoordinates[drawcenters=false,choice=9,linewidth=0.01mm,linecolor=blue, +styleVertices=+,sizeVertices=0.1,colorVertices=blue] \rotatebox{180} +{\CoxeterCoordinates[drawcenters=false,choice=9,linewidth=0.01mm,linecolor=green, +styleVertices=+,sizeVertices=0.1,colorVertices=green]} +\end{verbatim} +\end{center}\newpage +\begin{center} +\begin{pspicture}(-2,-8)(2,8) +\psset{unit=2} \rotatebox{180} +{\CoxeterCoordinates[drawcenters=false,choice=9, +linewidth=0.3mm,linecolor=green,styleVertices=triangle,sizeVertices=0.1,colorVertices=blue]} +\psset{unit=2} +\CoxeterCoordinates[drawvertices=false,drawcenters=false,drawcentersfaces=true +,choice=9,linewidth=0.01mm] +\end{pspicture}\\ +%Les centres des faces d'un polytope Hessien sont les sommets d'un +%polytope Hessien (r\'eciproque du premier). +The centers of the faces of an Hessian are the vertices of its +reciprocal. +\begin{verbatim} +\psset{unit=2} \rotatebox{180} +{\CoxeterCoordinates[drawcenters=false,choice=9, +linewidth=0.3mm,linecolor=green,styleVertices=triangle,sizeVertices=0.1,colorVertices=blue]} +\psset{unit=2} +\CoxeterCoordinates[drawvertices=false,drawcenters=false,drawcentersfaces=true +,choice=9,linewidth=0.01mm] +\end{verbatim} +\end{center} +\newpage +\begin{center} +\begin{pspicture}(-2,-8)(2,8) +\psset{unit=1.5} \rotatebox{180} +{\CoxeterCoordinates[drawcenters=false,choice=8, +linewidth=0.3mm,linecolor=green,styleVertices=triangle,sizeVertices=0.1,colorVertices=blue]} +\psset{unit=2} +\CoxeterCoordinates[drawvertices=false,drawcenters=false,drawcentersfaces=true +,choice=10,linewidth=0.01mm] +\end{pspicture}\\ +%Les centres des faces d'un polytope $2\{4\}3\{3\}3$ sont les +%sommets d'un polytope $3\{3\}3\{4\}2$ (r\'eciproque du premier). +The centers of the faces of a $2\{4\}3\{3\}3$ are the vertices of a +$3\{3\}3\{4\}2$. +\begin{verbatim} +\psset{unit=1.5} \rotatebox{180} +{\CoxeterCoordinates[drawcenters=false,choice=8, +linewidth=0.3mm,linecolor=green,styleVertices=triangle,sizeVertices=0.1,colorVertices=blue]} +\psset{unit=2} +\CoxeterCoordinates[drawvertices=false,drawcenters=false,drawcentersfaces=true +,choice=10,linewidth=0.01mm] +\end{verbatim} +\end{center} +\newpage +\begin{center} +\begin{pspicture}(-2,-8)(2,8) + \psset{unit=2.5} +\CoxeterCoordinates[drawvertices=false,drawcenters=false,choice=11,linewidth=0.01mm, +linecolor=black] \psset{unit=0.575} \reflectbox +{\CoxeterCoordinates[drawcenters=false,choice=11, +linewidth=0.01mm,linecolor=yellow,styleVertices=triangle,sizeVertices=0.1,colorVertices=blue]} +\psset{unit=1.73} +\CoxeterCoordinates[drawedges=false,drawvertices=false,drawcenters=false, +drawcenterscells=true ,choice=11] +\end{pspicture}\\ +The centers of the cells of a Witting polytope are the vertices of +its reciprocal. +\begin{verbatim} + \psset{unit=2.5} +\CoxeterCoordinates[drawvertices=false,drawcenters=false,choice=11,linewidth=0.01mm, +linecolor=black] \psset{unit=0.575} \reflectbox +{\CoxeterCoordinates[drawcenters=false,choice=11, +linewidth=0.01mm,linecolor=yellow,styleVertices=triangle,sizeVertices=0.1,colorVertices=blue]} +\psset{unit=1.73} +\CoxeterCoordinates[drawedges=false,drawvertices=false,drawcenters=false, +drawcenterscells=true ,choice=11] +\end{verbatim} +\end{center} +\newpage +\begin{center} +\begin{pspicture}(-2,-8)(2,8) +\psset{unit=3} +\CoxeterCoordinates[drawedges=false,drawcenters=false, + choice=10,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.1] +\CoxeterCoordinates[drawvertices=false,drawcenters=false,choice=27] +\end{pspicture}\\ One of the eight $3\{4\}2\{3\}2$ contained in a $3\{3\}3\{4\}2$. +\begin{verbatim} +\psset{unit=3} +\CoxeterCoordinates[drawedges=false,drawcenters=false, + choice=10,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.1] +\CoxeterCoordinates[drawvertices=false,drawcenters=false,choice=27] +\end{verbatim} +\end{center}\newpage +\begin{center} +\begin{pspicture}(-2,-8)(2,8) +\psset{unit=3} +\CoxeterCoordinates[drawedges=false,drawcenters=false, + choice=10,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.1] +\rotatebox{10.5}{\psset{unit=1.75}\CoxeterCoordinates[drawvertices=false,drawcenters=false,choice=26]} +\end{pspicture}\\ One of the $8$ $2\{3\}2\{4\}3$ contained in a $3\{3\}3\{4\}2$. +\begin{verbatim} +\psset{unit=3} +\CoxeterCoordinates[drawedges=false,drawcenters=false, + choice=10,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.1] +\rotatebox{10.5} {\psset{unit=1.75} +\CoxeterCoordinates[drawvertices=false,drawcenters=false,choice=26]} +\end{verbatim} +\end{center} +\newpage +\begin{center} +\begin{pspicture}(-2,-8)(2,8) +\psset{unit=3} \CoxeterCoordinates[drawcenters=false, + choice=32,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true, + sizeCentersCells=0.05,styleCentersCells=o] + \CoxeterCoordinates[drawcenters=false, + choice=33,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true] + \psset{unit=0.25}\CoxeterCoordinates[drawcenters=false, + choice=33,linecolor=blue,linewidth=0.6mm,colorVertices=blue,sizeVertices=0.1,drawcenterscells=true] +\CoxeterCoordinates[drawcenters=false, + choice=32,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true, + sizeCentersCells=0.05,styleCentersCells=o] + \psset{unit=0.25}\CoxeterCoordinates[drawcenters=false, + choice=33,linecolor=blue,linewidth=0.6mm,colorVertices=blue,sizeVertices=0.1,drawcenterscells=true] +\CoxeterCoordinates[drawcenters=false, + choice=32,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true, + sizeCentersCells=0.05,styleCentersCells=o] + \psset{unit=0.25}\CoxeterCoordinates[drawcenters=false, + choice=33,linecolor=blue,linewidth=0.6mm,colorVertices=blue,sizeVertices=0.1,drawcenterscells=true] +\CoxeterCoordinates[drawcenters=false, + choice=32,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true, + sizeCentersCells=0.05,styleCentersCells=o] +\end{pspicture} +\end{center} +A 16-tope in an hypercube in a 16-tope in an hypercube in ... +\begin{verbatim} +\psset{unit=3} \CoxeterCoordinates[drawcenters=false, + choice=32,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true, + sizeCentersCells=0.05,styleCentersCells=o] + \CoxeterCoordinates[drawcenters=false, + choice=33,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true] + \psset{unit=0.25}\CoxeterCoordinates[drawcenters=false, + choice=33,linecolor=blue,linewidth=0.6mm,colorVertices=blue,sizeVertices=0.1,drawcenterscells=true] +\CoxeterCoordinates[drawcenters=false, + choice=32,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true, + sizeCentersCells=0.05,styleCentersCells=o] + \psset{unit=0.25}\CoxeterCoordinates[drawcenters=false, + choice=33,linecolor=blue,linewidth=0.6mm,colorVertices=blue,sizeVertices=0.1,drawcenterscells=true] +\CoxeterCoordinates[drawcenters=false, + choice=32,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true, + sizeCentersCells=0.05,styleCentersCells=o] + \psset{unit=0.25}\CoxeterCoordinates[drawcenters=false, + choice=33,linecolor=blue,linewidth=0.6mm,colorVertices=blue,sizeVertices=0.1,drawcenterscells=true] +\CoxeterCoordinates[drawcenters=false, + choice=32,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,drawcenterscells=true, + sizeCentersCells=0.05,styleCentersCells=o]\end{verbatim} +\newpage +\newpage +\begin{center} +\begin{pspicture}(-2,-4)(2,4) +\psset{unit=2} +\CoxeterCoordinates[drawcenters=false,drawcentersfaces=false,drawvertices=false, + choice=30,linewidth=0.1mm,linecolor=blue] + \CoxeterCoordinates[drawcenters=false, + choice=31,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=magenta] + \CoxeterCoordinates[drawcenters=false,drawcentersfaces=true,drawvertices=false, + drawedges=false,choice=30] +\end{pspicture} +\begin{pspicture}(-2,-4)(2,4) +\psset{unit=2} \CoxeterCoordinates[drawcenters=false, + choice=31,linewidth=0.1mm,drawvertices=false,linecolor=magenta] +{\psset{unit=0.635}\CoxeterCoordinates[drawcenters=false,drawcentersfaces=false, + choice=30,linewidth=0.1mm,linecolor=blue,colorVertices=blue,sizeVertices=0.1] + } + \CoxeterCoordinates[drawcenters=false,drawedges=false,drawcentersfaces=true, + choice=31,linewidth=0.1mm,drawvertices=false] +\end{pspicture} + +\end{center} +A dodec\ae dron in an ikos\ae dron and an iko\ae dron in a dodec\ae +dron. +\begin{verbatim} +\begin{pspicture}(-2,-4)(2,4) +\psset{unit=2} +\CoxeterCoordinates[drawcenters=false,drawcentersfaces=false,drawvertices=false, + choice=30,linewidth=0.1mm,linecolor=blue] + \CoxeterCoordinates[drawcenters=false, + choice=31,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=magenta] + \CoxeterCoordinates[drawcenters=false,drawcentersfaces=true,drawvertices=false, + drawedges=false,choice=30] +\end{pspicture} +\begin{pspicture}(-2,-4)(2,4) +\psset{unit=2} \CoxeterCoordinates[drawcenters=false, + choice=31,linewidth=0.1mm,drawvertices=false,linecolor=magenta] +{\psset{unit=0.635}\CoxeterCoordinates[drawcenters=false,drawcentersfaces=false, + choice=30,linewidth=0.1mm,linecolor=blue,colorVertices=blue,sizeVertices=0.1] + } + \CoxeterCoordinates[drawcenters=false,drawedges=false,drawcentersfaces=true, + choice=31,linewidth=0.1mm,drawvertices=false] +\end{pspicture} +\end{verbatim} +\newpage +%%%% +\begin{center} +\begin{pspicture}(-7,-7)(7,7) +\psset{unit=5} +\CoxeterCoordinates[drawcenters=false,drawcentersfaces=false,drawvertices=false, + choice=35,linewidth=0.1mm,linecolor=blue] + {\psset{unit=0.86} + \CoxeterCoordinates[drawcenters=false, + choice=36,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=red] + } + \CoxeterCoordinates[drawcenters=false,drawcenterscells=true,drawvertices=false, + drawedges=false,choice=35] +\end{pspicture} +\end{center} +A $120$-tope in a $600$-tope. +\begin{verbatim} +\begin{pspicture}(-7,-7)(7,7) +\psset{unit=5} +\CoxeterCoordinates[drawcenters=false,drawcentersfaces=false,drawvertices=false, + choice=35,linewidth=0.1mm,linecolor=blue] + {\psset{unit=0.86} + \CoxeterCoordinates[drawcenters=false, + choice=36,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=red] + } + \CoxeterCoordinates[drawcenters=false,drawcenterscells=true,drawvertices=false, + drawedges=false,choice=35] +\end{pspicture} +\end{verbatim} +\newpage +%%%% +\begin{center} +\begin{pspicture}(-7,-7)(7,7) +\psset{unit=5} +\CoxeterCoordinates[drawcenters=false,drawcentersfaces=false,drawvertices=false, + choice=36,linewidth=0.1mm,linecolor=blue] + \CoxeterCoordinates[drawcenters=false, + choice=35,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=red] + \CoxeterCoordinates[drawcenters=false,drawcenterscells=true,drawvertices=false, + drawedges=false,choice=36] +\end{pspicture} +\end{center} +A $600$-tope in a $120$-tope. +\begin{verbatim} +\begin{pspicture}(-7,-7)(7,7) +\psset{unit=5} +\CoxeterCoordinates[drawcenters=false,drawcentersfaces=false,drawvertices=false, + choice=36,linewidth=0.1mm,linecolor=blue] + \CoxeterCoordinates[drawcenters=false, + choice=35,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=red] + \CoxeterCoordinates[drawcenters=false,drawcenterscells=true,drawvertices=false, + drawedges=false,choice=36] +\end{pspicture} +\end{verbatim} +\newpage +%%%% +%%%% +\begin{center} +\begin{pspicture}(-7,-7)(7,7) +\psset{unit=0.5} \CoxeterCoordinates[drawcenters=false, + choice=71,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=magenta] + \rotatebox{4.5}{\psset{unit=6.9}\CoxeterCoordinates[drawcenters=false, +choice=36,styleVertices=*,linewidth=0.1mm,linecolor=blue]} +\end{pspicture} +\end{center} +A starry $120$-tope in a $120$-tope. +\begin{verbatim} +\psset{unit=0.5} \CoxeterCoordinates[drawcenters=false, + choice=71,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=magenta] + \rotatebox{4.5}{\psset{unit=6.9}\CoxeterCoordinates[drawcenters=false, +choice=36,styleVertices=*,linewidth= +\end{verbatim} +%%%% +%\begin{center} +%\begin{pspicture}(-7,-7)(7,7) +%{\psset{unit=5} \CoxeterCoordinates[drawcenters=false, +% choice=72,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=magenta]} +% \rotatebox{5}{\CoxeterCoordinates[unit=4.98,drawcenters=false, +%choice=35,styleVertices=*,linewidth=0.1mm,linecolor=blue]} +%\end{pspicture} +%\end{center} +%A starry $600$-tope in a $600$-tope. +%\begin{verbatim} +%\begin{pspicture}(-7,-7)(7,7) +%\psset{unit=0.5} \CoxeterCoordinates[drawcenters=false, +% choice=72,linewidth=0.1mm,colorVertices=blue,sizeVertices=0.05,linecolor=magenta] +% \CoxeterCoordinates[unit=7,drawcenters=false, +%choice=35,styleVertices=*,linewidth=0.1mm,linecolor=blue] +%\end{pspicture}\end{verbatim} +\end{document} diff --git a/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/pst-coxcoor_doc.pdf b/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/pst-coxcoor_doc.pdf new file mode 100644 index 00000000000..4b24fd50775 Binary files /dev/null and b/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/pst-coxcoor_doc.pdf differ diff --git a/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/pst-coxcoor_doc.tex b/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/pst-coxcoor_doc.tex new file mode 100644 index 00000000000..f346c7e410a --- /dev/null +++ b/Master/texmf-dist/doc/generic/pst-cox/pst-coxcoor/pst-coxcoor_doc.tex @@ -0,0 +1,1118 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% polygonesCoordinate\pst-coxcoor_doc.tex +% 7 Authors: J.-G. Luque and M. Luque +% 8 Purpose: Documentation for pst-coxcoor +% 9 Created: 02/02/2008 +% 10 License: LGPL +% 11 Project: PST-Cox V1.00 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% polygonesCoordinate\Gallery.tex +% Authors: J.-G. Luque and M. Luque +% Purpose: Demonstration of 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 . +% + +\documentclass[a4paper]{article} +\usepackage[latin1]{inputenc}% +\usepackage[margin=2cm]{geometry} +\usepackage{pst-coxcoor} +\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-coxcoor}} +\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-coxcoor} devoted to + draw regular complex polytopes. + \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 developed 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}. Many of these groups +are fundamental in geometry. For example, the polytope Hessian is a +$3$-dimensional polytope whose symmetry group is generated by $3$ +pseudo-reflections $s_1$, $s_2$ and $s_3$ verifying +$s_1^3=s_2^3=s_3^3=Id$, $s_1s_2s_1=s_2s_1s_2$, $s_2s_3s_2=s_3s_2s_3$ +and $s_1s_3=s_3s_1$ and which is related to the determination of the +nine inflection points of a cubic curve and the 27 lines in a cubic +plane.\\ +The library described here is a {\tt LaTex} package for drawing two +dimensional projections of regular complex polytopes. The +coordinates of the vertices, edges, faces... of the projections have +been pre-calculated using a formal computer system.\\ +The polytopes considered are exceptional polytopes, for drawing +infinite series use the package {\it pst-coxeterp}.\\ + Note that this package have already been used by one of the author + to illustrate an article \cite{qutrit} in collaboration with E. Briand, + J.-Y. Thibon and F. Verstraete and in his ``{\it habilitation \`a + diriger les recherches}'' \cite{Luque}. +\section{Install {\tt pst-coxcoor}} +The package contains three files: A latex style file {\tt +pst-coxcoor.sty} which call the latex file {\tt pst-coxcoor.tex} +containing the description of the macro {\tt +$\setminus$CoxeterCoordinates} and a data file {\tt pst-coxcoor.pro} +which contains the list of the coordinates of each polytope.\\ The +installation is very simple. It suffices to +copy the files {\tt pst-coxcoor.sty}, {\tt pst-coxcoor.tex} and\\ +{\tt pst-coxcoor.pro} in the appropriate directories. +\begin{example}\rm +The file {\tt pst-coxcoor.sty} may be copy in the directory \\ {\tt +c:/texmf/tex/latex/pst-coxcoor},\\ + the file {\tt pst-coxcoor.tex} in\\ +{\tt c:/texmf/tex/generic/pst-coxcoor}\\ and the file {\tt +pst-coxcoor.pro} in\\ {\tt c:/texmf/tex/dvips/pst-coxcoor}. +\end{example} +To use the package add the code +\begin{verbatim} +\usepackage{pst-coxcoor}} +\end{verbatim} +in the beginning of your LaTex-file. +\begin{example}\rm +\begin{verbatim} +\documentclass[a4paper]{article} +... +\usepackage{pst-coxcoor} +.... +\end{verbatim} +\end{example} +The library needs the packages {\tt PSTrick} and {\tt pst-xkey}. + +\section{Characteristics of the polytopes} + The polytope considered here are two, three or four + ($\C$)-dimensional objects which generalizes the classical platonic + solids. They are constituted of vertices, edges, faces and cells + (four dimensional faces). The package contains only one macro {\tt $\setminus$CoxeterCoordinates} + which draws the vertices, + the edges, the centers of the edges\footnote{In general, for a complex polytope, the edges are + polygonal.}, the centers of the faces and the centers of the cells. + All the coordinates of the polytopes have been pre-computed and + stored in the file {\tt pst-coxcoor.pro}. +\subsection{List of the polytopes} +The parameter {\tt ichoice} contains the number identifying the +polytope. +\begin{example} +\rm Setting {\tt choice=9} makes the macro draw the (3 dimensional) +Hessian polytope which has $27$ vertices, $72$ triangular edges and +$27$ faces. +\[ +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=9] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-4,-4)(4,4) +\psset{unit=1.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=9] % +\end{pspicture} +\end{verbatim} +\end{example} +There is $80$ pre-calculated polytopes in the file {\tt +pst-coxcoor.pro}. Almost all the complex regular polytopes up to the +dimension four have been computed. Only some starry polytopes in +dimension $4$ are not in the list. The following tableau contains +the list of the polytopes with their names in the notation of +Coxeter \cite{Cox}. +\[ +\begin{array}{|c|c|c|} +\hline 2\{3\}3&3\{3\}3&3\{3\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=1] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=2] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=3] % +\end{pspicture}\\\hline +{\tt choice=1}&{\tt choice=2}&{\tt choice=3}\\\hline +\end{array}\] + %%%% +\[ +\begin{array}{|c|c|c|}\hline + 3\{4\}2&3\{4\}4&3\{4\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.8cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=4] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=5] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=6] % +\end{pspicture}\\\hline +{\tt choice=4}&{\tt choice=5}&{\tt choice=6}\\\hline + %%%% +\end{array}\] + %%%% +\[ +\begin{array}{|c|c|c|}\hline +4\{3\}4&2\{4\}3\{3\}3&3\{3\}3\{3\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=7] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.8cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=8] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=9] % +\end{pspicture}\\\hline +{\tt choice=7}&{\tt choice=8}&{\tt choice=9}\\\hline +\end{array} +\] +%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{3\}3\{4\}2&3\{3\}3\{3\}3\{3\}3&3\{8\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=10] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=11,drawcenters=false] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=12] % +\end{pspicture}\\\hline +{\tt choice=10}&{\tt choice=11}&{\tt choice=12}\\\hline +\end{array} +\] +%%%% +%%%% +\[ +\begin{array}{|c|c|c|}\hline +2\{8\}3&3\{5\}3&4\{4\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=13] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=14] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=15] % +\end{pspicture}\\\hline +{\tt choice=13}&{\tt choice=14}&{\tt choice=15}\\\hline +\end{array} +\] +%%% +%%% +\[ +\begin{array}{|c|c|c|}\hline +4\{3\}2&2\{3\}4&2\{6\}4\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=16] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=17] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=18] % +\end{pspicture}\\\hline +{\tt choice=16}&{\tt choice=17}&{\tt choice=18}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +4\{6\}2&5\{3\}5&2\{10\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=19] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=20] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=21] % +\end{pspicture}\\\hline +{\tt choice=19}&{\tt choice=20}&{\tt choice=21}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{10\}2&2\{5\}3&3\{5\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=22] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=23] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=24] % +\end{pspicture}\\\hline +{\tt choice=22}&{\tt choice=23}&{\tt choice=24}\\\hline +\end{array} +\] +%%%% +\[ +\begin{array}{|c|c|c|}\hline +2\{4\}3&2\{3\}2\{4\}3&3\{4\}2\{3\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=25] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.8cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=26] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=27] % +\end{pspicture}\\\hline +{\tt choice=25}&{\tt choice=26}&{\tt choice=27}\\\hline +\end{array} +\] +%%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{4\}2\{3\}2\{3\}2&2\{3\}2\{3\}2\{4\}3&2\{3\}2\{5\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=28] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=29] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=30] % +\end{pspicture}\\\hline +{\tt choice=28}&{\tt choice=29}&{\tt choice=30}\\\hline +\end{array} +\] +%%%% +\[ +\begin{array}{|c|c|c|}\hline +2\{5\}2\{3\}2&2\{3\}2\{3\}2\{4\}2&2\{4\}2\{3\}2\{3\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=31] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=32] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=33] % +\end{pspicture}\\\hline +{\tt choice=31}&{\tt choice=32}&{\tt choice=33}\\\hline +\end{array} +\] +%%%% +\[ +\begin{array}{|c|c|c|}\hline +2\{3\}2\{4\}2\{3\}2&2\{3\}2\{3\}2\{5\}2&2\{5\}2\{3\}2\{3\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=34] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=35,drawcenters=false] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=36,drawcenters=false] % +\end{pspicture}\\\hline +{\tt choice=34}&{\tt choice=35}&{\tt choice=36}\\\hline +\end{array} +\] +%%%% +%%%% +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{\frac52\}3&5\{\frac52\}5&2\{\frac52\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=37] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=38] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.8cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=39] % +\end{pspicture}\\\hline +{\tt choice=37}&{\tt choice=38}&{\tt choice=39}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{\frac52\}2&3\{\frac{10}3\}2&2\{\frac{103}\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=40] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.4cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=41] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=42] % +\end{pspicture}\\\hline +{\tt choice=40}&{\tt choice=41}&{\tt choice=42}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{\frac83\}2&2\{\frac83\}3&5\{6\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.1 cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=43] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=44] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=45] % +\end{pspicture}\\\hline +{\tt choice=43}&{\tt choice=44}&{\tt choice=45}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +2\{6\}5&4\{\frac83\}3&3\{\frac83\}4\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=46] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=47] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=48] % +\end{pspicture}\\\hline +{\tt choice=46}&{\tt choice=47}&{\tt choice=48}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +5\{5\}2&2\{5\}5&5\{\frac{10}3\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=49] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=50] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=51] % +\end{pspicture}\\\hline +{\tt choice=49}&{\tt choice=50}&{\tt choice=51}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +2\{\frac{10}3\}5&5\{3\}2&2\{3\}5\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=52] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=53] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=54] % +\end{pspicture}\\\hline +{\tt choice=52}&{\tt choice=53}&{\tt choice=54}\\\hline +\end{array} +\] +%%% +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +5\{4\}2&2\{4\}5&5\{\frac{10}3\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.8cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=55] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=56] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=57] % +\end{pspicture}\\\hline +{\tt choice=55}&{\tt choice=56}&{\tt choice=57}\\\hline +\end{array} +\] +%%%% +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{\frac{10}3\}5&5\{4\}3&3\{4\}5\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=58] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=59] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.25cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=60] % +\end{pspicture}\\\hline +{\tt choice=58}&{\tt choice=59}&{\tt choice=60}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +5\{3\}3&3\{3\}5&5\{\frac52\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=61] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=62] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.6cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=63] % +\end{pspicture}\\\hline +{\tt choice=61}&{\tt choice=62}&{\tt choice=63}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{\frac52\}5&2\{\frac52\}2\{3\}2&2\{3\}2\{\frac52\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=64] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=2cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=65] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=66] % +\end{pspicture}\\\hline +{\tt choice=64}&{\tt choice=65}&{\tt choice=66}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +2\{\frac52\}2\{3\}2&2\{5\}2\{\frac52\}2&2\{6\}3\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=2.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=67] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=68] % +\end{pspicture} + &\begin{pspicture}(-2,-2)(2,2) +\psset{unit=2.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=69] % +\end{pspicture}\\\hline +{\tt choice=67}&{\tt choice=68}&{\tt choice=69}\\\hline +\end{array} +\] +%%%%% +\[ +\begin{array}{|c|c|c|}\hline +3\{6\}2&2\{\frac52\}2\{3\}2\{3\}2&2\{3\}2\{3\}2\{\frac52\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.5cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=70] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.17cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=71] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.4cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=72] % +\end{pspicture}\\ + \hline +{\tt choice=70}&{\tt choice=71}&{\tt choice=72}\\\hline +\end{array} +\] +\[ +\begin{array}{|c|c|c|}\hline +2\{3\}2\{\frac52\}2\{5\}2&2\{3\}2\{5\}2\{\frac52\}2&2\{\frac52\}2\{3\}2\{5\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=73] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=74] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=75] % +\end{pspicture}\\ + \hline +{\tt choice=73}&{\tt choice=74}&{\tt choice=75}\\\hline +\end{array} +\] +% +\[ +\begin{array}{|c|c|c|}\hline +2\{\frac52\}2\{5\}2\{3\}2&2\{5\}2\{3\}2\{\frac52\}2&2\{5\}2\{\frac52\}2\{3\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=76] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=77] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1.3cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=78] % +\end{pspicture}\\ + \hline +{\tt choice=76}&{\tt choice=77}&{\tt choice=78}\\\hline +\end{array} +\] +\[ +\begin{array}{|c|c|c|}\hline +2\{5\}2\{\frac52\}2\{5\}2&2\{\frac525\}2\{5\}2\{\frac52\}2\\\hline + \begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=79] % +\end{pspicture}& +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.4cm,linewidth=0.01mm} +\CoxeterCoordinates[choice=80] % +\end{pspicture}\\ + \hline +{\tt choice=79}&{\tt choice=80}\\\hline +\end{array} +\] + + + +\subsection{The components of a polytope} + The library {\tt pst-coxcoor.sty} contains a macro for +drawing the vertices, the edges, the centers of the edges, the +centers of the faces and the centers of the cells of many +pre-calculated 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}, {\tt drawcenters}, {\tt drawcentersface}, and +{\tt drawcenterscells}. + + By default the values of the parameters {\tt +drawedges}, {\tt drawvertices}, {\tt drawcenters} are set to {\tt +true} and the values of {\tt drawcentersface} and {\tt +drawcenterscells} are set to {\tt false}. +\begin{example} +\rm By default, the vertices, the edges and the centers of the edges +are drawn but not the centers of the faces and the cells. +\[ +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[choice=28] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} + \CoxeterCoordinates[choice=28] +\end{pspicture} +\end{verbatim} +The macro does not draw the edges +\[ +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[choice=28,drawedges=false] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} + \CoxeterCoordinates[choice=28,drawedges=false] +\end{pspicture} +\end{verbatim} +or the vertices +\[ +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[choice=28,drawvertices=false] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} + \CoxeterCoordinates[choice=28,drawvertices=false] +\end{pspicture} +\end{verbatim} +or the centers of the edges. +\[ +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[choice=28,drawcenters=false] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} + \CoxeterCoordinates[choice=28,drawcenters=false] +\end{pspicture} +\end{verbatim} +Furthermore, one can draw the centers of the faces (when the +dimension of the polytope is at least 3) +\[ +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[choice=28,drawvertices=false,drawcenters=false,drawcentersfaces=true] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} + \CoxeterCoordinates[choice=28,drawvertices=false,drawcenters=false,drawcentersfaces=true] +\end{pspicture} +\end{verbatim} +and the centers of the cells (when the dimension of the polytope is +at least 4). +\[ +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.7cm} +\CoxeterCoordinates[choice=28,drawvertices=false,drawcenters=false,drawcenterscells=true] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=1cm} + \CoxeterCoordinates[choice=28,drawvertices=false,drawcenters=false,drawcenterscells=true] +\end{pspicture} +\end{verbatim} +\end{example} + +\section{Graphical parameters} +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 + \[ + \begin{pspicture}(-1,-1)(1,1) +\CoxeterCoordinates[choice=4,unit=0.3cm] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,unit=0.8cm] % +\end{pspicture} + \begin{pspicture}(-4,-4)(4,4) +\CoxeterCoordinates[choice=4,unit=2cm] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-1,-1)(1,1) +\CoxeterCoordinates[choice=4,unit=0.3cm] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,unit=0.8cm] % +\end{pspicture} + \begin{pspicture}(-4,-4)(4,4) +\CoxeterCoordinates[choice=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 {\it linewidth}. +\begin{example} +\rm + \[ +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.8,linewidth=0.01,linecolor=red} +\CoxeterCoordinates[choice=4] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,linewidth=0.1] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\psset{unit=0.8,linewidth=0.01,linecolor=red} +\CoxeterCoordinates[choice=4] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,linewidth=0.1] % +\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 + \[ +\begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorVertices=blue,styleVertices=pentagon,sizeVertices=0.2] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorVertices=magenta,sizeVertices=0.1,styleVertices=triangle] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorVertices=red,styleVertices=+,sizeVertices=0.2] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorVertices=blue,styleVertices=pentagon,sizeVertices=0.2] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorVertices=magenta,sizeVertices=0.1,styleVertices=triangle] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorVertices=red,styleVertices=+,sizeVertices=0.2] % +\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 + \[ +\begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorCenters=blue,styleCenters=pentagon,sizeCenters=0.2] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorCenters=magenta,sizeCenters=0.1,styleCenters=triangle] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorCenters=red,styleCenters=+,sizeCenters=0.2] % +\end{pspicture} +\] +\begin{verbatim} +\begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorCenters=blue,styleCenters=pentagon,sizeCenters=0.2] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorCenters=magenta,sizeCenters=0.1,styleCenters=triangle] % +\end{pspicture} + \begin{pspicture}(-2,-2)(2,2) +\CoxeterCoordinates[choice=4,colorCenters=red,styleCenters=+,sizeCenters=0.2] % +\end{pspicture} +\end{verbatim} +\end{example} + +The color, the style and the size of the centers of the faces can be +modify using the parameters {\tt colorCentersFaces}, {\tt +styleCentersFaces} and {\tt sizeCentersFaces}. +\begin{example} +\rm + \[\psset{unit=0.8cm,drawcentersfaces=true} +\begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,styleCentersFaces=pentagon,sizeCentersFaces=0.2] % +\end{pspicture} + \begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,colorCentersFaces=magenta,sizeCentersFaces=0.1] % +\end{pspicture} + \begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,colorCentersFaces=red,styleCentersFaces=+] % +\end{pspicture} +\] +\begin{verbatim} +\psset{unit=0.8cm,drawcentersfaces=true} +\begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,styleCentersFaces=pentagon,sizeCentersFaces=0.2] % +\end{pspicture} + \begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,colorCentersFaces=magenta,sizeCentersFaces=0.1] % +\end{pspicture} + \begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,colorCentersFaces=red,styleCentersFaces=+] % +\end{pspicture}\end{verbatim} +\end{example} + +The color, the style and the size of the centers of the cells can be +modify using the parameters {\tt colorCentersCells}, {\tt +styleCentersCells} and {\tt sizeCentersCells}. +\begin{example} +\rm + \[\psset{unit=0.8cm,drawcenterscells=true,drawcentersfaces=false} +\begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,styleCentersCells=pentagon,sizeCentersCells=0.2] % +\end{pspicture} + \begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,colorCentersCells=magenta,sizeCentersCells=0.1] % +\end{pspicture} + \begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,colorCentersCells=red,styleCentersCells=+] % +\end{pspicture} +\] +\begin{verbatim} +\psset{unit=0.8cm,drawcenterscells=true,drawcentersfaces=false} +\begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,styleCentersCells=pentagon,sizeCentersCells=0.2] % +\end{pspicture} + \begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,colorCentersCells=magenta,sizeCentersCells=0.1] % +\end{pspicture} + \begin{pspicture}(-3,-3)(3,3) +\CoxeterCoordinates[choice=33,colorCentersCells=red,styleCentersCells=+] % +\end{pspicture}\end{verbatim} +\end{example} +\section{How to modify or add a polytope to the Library} +The polytopes described in this library are the regular complex +polytopes as considered by Coxeter \cite{Cox}. But, in fact, the +same library can be used to draw any kind of polytopes (not +necessarily regular) if the user add the datas corresponding to the +vertices, the edges, the faces and the cells of the polytopes. + +To add a polytope, one has to modify the file {\tt +pst-coxeterp.pro}. This file contains the list of the polytopes +which can be drawn with the macro {\tt CoxeterCoordinates}. + For each polytope, the datas are organized as follows + \begin{verbatim} + /cox+name+datas{% The name of the Polytope + /ListePoints [ + % List of the edges + ] def + /ListeFaces [ + % List of the centers of the faces + ] def + /ListeCells [ + % List of the centers of the cells + ] def + /NbrFaces nf def % nb of faces + /NbrCells nc def % nb of cells + /NbrEdges ne def % nb of edges + /NbrVerticesInAnEdge nv def % nb of vertices per edge + } def + \end{verbatim} + The list {\tt /ListePoints} contains the description of the edges + of the polytope. The variable {\tt /NbrEdges} contains the number + of edges and the variables {\tt /NbrVerticesInAnEdges} contains the + number of vertices by edges. An edge is defined by its {\tt + /NbrVerticesInAnEdges} vertices. The list {\tt /ListePoints} of the + edges is the list of all edges described by the sequence of their + vertices. + \begin{example}\rm + Let us explain the structure on the example of the complex polytope + $3\{4\}2$. +\begin{verbatim} +/cox342datas{% + /ListePoints [ + [-1.054405725 .6087614291] + [-1.717232873 -.9914448614] + [0 -.7653668647] + [1.054405725 .6087614291] + [1.717232873 -.9914448614] + [0 -.7653668647] + [-.6628271482 .3826834323] + [0 -1.217522858] + [-1.717232873 -.9914448614] + [0 1.982889723] + [.6628271482 .3826834323] + [-1.054405725 .6087614291] + [.6628271482 .3826834323] + [0 -1.217522858] + [1.717232873 -.9914448614] + [0 1.982889723] + [-.6628271482 .3826834323] + [1.054405725 .6087614291] + ] def + /ListeFaces [ + [0 0] + ] def + /NbrFaces 1 def + /ListeCells [ + [0 0] + ] def + /NbrCells 1 def + /NbrEdges 6 def + /NbrVerticesInAnEdge 3 def + } def +\end{verbatim} + This is a complex polygon and the number $3$ indicates + that each edges is triangular and contains $3$ vertices. Hence, the + list {\tt /ListePoints} is a sequence of triplet of points. + For example, the first edge is constituted by the three vertices {\tt [-1.054405725 .6087614291] [-1.717232873 -.9914448614] + [0 -.7653668647]}. +Here, since there is $6$ edges of $3$ vertices, the list {\tt +/ListePoints} contains $18$ points with two coordinates.\\ Note +that, since $3\{4\}2$ is a polygon, it has neither faces nor cells. +In such a case, the variables {\tt ListeFaces} and {\tt ListeCells} +must contain only one point {\tt [0 0]} and the variables {\tt +/NbrFaces} and {\tt /NbrCells} contain $1$. + \end{example} +When the polytope has more than two dimensions, it has faces. The +number of faces is given by the variable {\tt /NbrFaces} and the +variable {\tt /ListeFaces} contains the list of the centers of the +faces.\\ +If the polytope has four dimensions, it has cells. The number of +cells is given by the variable {\tt /NbrCells} and the variable {\tt +/ListeCells} contains the list of the centers of the cells.\\ \\ +To add a polytope, add the datas in the files {\tt pst-coxeter.pro} +and modify the file {\tt pst-coxeter.tex} as follows. Change the +numbers of the polytopes at the line 26 of the file + \begin{verbatim} + %%% Parameter choice. Allows to choice the polytope. To each integer + %%% 080 \typeout{choice < or = 80 and + not `\the\pst@cntg'. Value 1 forced.} \pst@cntg=1 + \fi + \edef\psk@pstCoxeter@choice{#1}} + \end{verbatim} +Here, the number of polytope is $80$, if your add other datas you +must increase this number. + \begin{verbatim} + %%% Parameter choice. Allows to choice the polytope. To each integer + %%% 081 \typeout{choice < or = 81 and + not `\the\pst@cntg'. Value 1 forced.} \pst@cntg=1 + \fi + \edef\psk@pstCoxeter@choice{#1}} + \end{verbatim} + Hence, you must add the polytope to the list of polytopes (line 169 +- 251 of the file {\tt pst-coxcoor.tex}. +\begin{verbatim} + /choice \the\pst@cntg\space def + choice 1 eq {cox233datas} if + ... + choice 78 eq {cox362datas} if + choice 79 eq {cox25223232datas} if + choice 80 eq {cox23232522datas} if +%%% <-- add new polytope here + \end{verbatim} + For example, add the line + \begin{verbatim} + choice 81 eq {coxNEWdatas} if + \end{verbatim} + \begin{thebibliography}{ABC} +\bibitem{qutrit} E. Briand, J.-G. Luque, J.-Y. Thibon and F. Verstrate, {\it the +moduli space of the three qutrit states},Journal of Mathematical +Physics, vol. 45, num. 12, pp. 4855--4867, 2004. +% +\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{Luque} J.-G. Luque, {\em Invariants des hypermatrices}, +habilitation \`a diriger les recherches, Université Paris-Est, +Décembre 2007. +% +\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} 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 . +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\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 new file mode 100644 index 00000000000..efc7bfa09f5 Binary files /dev/null and b/Master/texmf-dist/doc/generic/pst-cox/pst-coxeterp/pst-coxeterp_doc.pdf 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 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 . +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\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} -- cgit v1.2.3