%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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}