%% assoc-L3L.tex - an affine projection of the Loday associahedron similar to the drawing in Lodays paper %% Author: Markus Rost %% Created: 10 Mar 2026 %% to be compiled with pdflatex \documentclass{amsart} %%%%%%%% %%%%% Shows an affine projection of Loday's realization of the %%%%% 3-dimensional associahedron. It is close to the drawing at the %%%%% end of Loday's paper (Loday, Jean-Louis. Realization of the %%%%% Stasheff polytope. Arch. Math. 83, no. 3, 267-278, 2004. MR %%%%% 2108555.) %%%%% This file is edited from cube-3L.tex, which see for more comments. %%%%%%%% \newif\ifLABELS % %% uncomment to show the coordinates % \LABELStrue % \newif\ifNAMES % %% uncomment to show instead the internal names of the vertices %%% recommended if you read this file % \NAMEStrue % no effect with \LABELSfalse \ifNAMES \LABELStrue \fi \newif\ifPARENS % %% uncomment to get paren expressions as labels % \PARENStrue % \ifPARENS \LABELSfalse \fi % just in case \newif\ifGREEN % %% uncomment to draw the 2. dim subassociahedra in green % \GREENtrue % %%%%%%%% \ifPARENS % \DeclareFixedFont{\PARENfont}{OT1}{cmr}{b}{n}{5} % \fi % \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \thispagestyle{empty} % \centerline{% %%% projection parameters to match Lodays drawing, somewhat \tikzpicture[x={(1cm,-3cm/2)}, y={(2cm,0cm)}, z={(1cm,-2cm/3)}] %%%%%%%% Loday coordinates \def\LCOORDINATES{% A0/1/2/3/4/(((01)2)3)4, % ((( 01 )2)3)4 A1/2/1/3/4/((0(12))3)4, % (( 0(12) )3)4 X0/1/4/1/4/((01)(23))4, % ( (01)(23) )4 A2/3/2/1/4/(0(1(23)))4, % ( 0(1(23)) )4 A3/3/1/2/4/(0((12)3))4, % ( 0((12)3) )4 Y1/1/2/6/1/((01)2)(34), % ((01)2)(34) Y2/1/6/2/1/(01)(2(34)), % (01)(2(34)) B0/4/3/2/1/0(1(2(34))), % 0(1(2(34))) Z1/2/1/6/1/(0(12))(34), % (0(12))(34) B1/4/1/4/1/0((12)(34)), % 0((12)(34)) X2/1/6/1/2/(01)((23)4), % (01)((23)4) X3/4/3/1/2/0(1((23)4)), % 0(1((23)4)) B2/4/2/1/3/0((1(23))4), % 0((1(23))4) B3/4/1/2/3/0(((12)3)4)% % 0(((12)3)4) } %% drop the third coordinate \path foreach \NNN/\x/\y/\z/\t in \LCOORDINATES { % coordinate (\NNN) at ($(\x,\y,\t)$) } ; % %%%%%%%% %% placement in shipout is independent of labels \useasboundingbox (Z1) (B3) (Y2) (X2) ; % %%%%%%%% %%%% the 9 faces of the associahedron \def\faceQA{A2,A3,B3,B2} % square \def\faceQB{A1,A0,Y1,Z1} % rectangle at PA \def\faceQC{X3,B0,Y2,X2} % rectangle T(Q1) %% T=transpose %% 2-dim sub-associahedron + parallel \def\facePA{A2,A3,A1,A0,X0} % \def\facePF{B0,B1,Z1,Y1,Y2} % %% 2-dim sub-associahedron' + parallel \def\facePB{B2,B3,B1,B0,X3} % T(PA) \def\facePG{X0,A0,Y1,Y2,X2} % T(PF) %% pentagons with no parallel face \def\facePC{A3,B3,B1,Z1,A1} % \def\facePD{A2,B2,X3,X2,X0} % T(PC) %%%% the 21 edges of the associahedron %% \facePA completely in front \def\EDGESBACK{% Y2/Y1,Y2/X2,Y2/B0,B0/X3,B0/B1% } \def\EDGESFRONT{% A0/A1,A0/X0,X0/A2,A1/A3,A0/Y1, % A1/Z1,Z1/B1,A2/B2,A3/B3,A3/A2, % Y1/Z1,B2/X3,B3/B2,B3/B1,X2/X3,X2/X0% } %%%%%%%% \def\CYCLE#1{foreach\p[count=\i]in#1{\ifnum\i=1 \else--\fi(\p)}--cycle} % Back, Back R \fill[blue!15] \CYCLE\facePF \CYCLE\faceQC ; % % Bottom \ifGREEN % \fill[green!50] \CYCLE\facePB ; % \else % \fill[white] \CYCLE\facePB ; % \fi % \draw[blue] foreach \XXX/\YYY in \EDGESBACK { (\XXX) -- (\YYY) } ; % % Left up, Front R, Front L \fill[red!20,opacity=0.6] \CYCLE\faceQB \CYCLE\facePD \CYCLE\facePC ; % \ifGREEN % \fill[red!20,opacity=0.6] \CYCLE\facePG ; % Top \fill[green!50,opacity=0.6] \CYCLE\facePA ; % Front up \fill[red!20,opacity=0.6] \CYCLE\faceQA ; % Front \else % \fill[white,opacity=0.2] \CYCLE\facePG ; % Top \fill[red!20,opacity=0.6] \CYCLE\facePA ; % Front up \fill[red!20,opacity=0.4] \CYCLE\faceQA ; % Front \fi % \draw[blue] foreach \XXX/\YYY in \EDGESFRONT { (\XXX) -- (\YYY) } ; % %%%%%%%% \ifLABELS \begin{scope} [outer sep=0pt,inner sep=2pt] \tiny \tikzset{ % A0place/.style={above right,xshift=-2.5pt}, % X0place/.style={below right,xshift=-1pt,yshift=0.5pt}, % } % \def\PLACES{ % A0/A0place, A1/below left, A2/below left, A3/below right, % B0/above left, B1/below left, B2/below, B3/below, % X0/X0place, X2/right, X3/below right, % Y1/above, Y2/above, Z1/left} \ifNAMES %%%%%%%% show the names of the vertices \path foreach \XXX/\place in \PLACES { % node[\place] at (\XXX) {\XXX} } ; % \else % \ifNAMES %%%%%%%% show the coordinates \path foreach \XXX/\place in \PLACES { % node[\place] (N\XXX) at (\XXX) {\phantom{0000}} } % foreach \XXX/\x/\y/\z/\t in \LCOORDINATES { % node at (N\XXX) {\x\y\z\t} } ; % \fi % \ifNAMES \end{scope} \fi % \ifLABELS \ifPARENS \tikzset{ labelbox/.style={ % fill=white,fill opacity=0.6, % draw,text opacity=1, % inner sep=4/3pt}, % } % \def\Label#1{\scalebox{.5}{#1}} \def\Label#1{\PARENfont#1} \path foreach \XXX/\x/\y/\z/\t/\P in \LCOORDINATES { % node[labelbox] at (\XXX) {\Label{\P}} } ; % \fi \endtikzpicture } % \centerline \end{document} %%%%%%%% %% png creation (small size, transparent) %%% pdflatex cube-3LL.tex %%% pdftocairo -png -singlefile -r 97.5 -transp cube-3LL.pdf tmp %%% magick -define png:exclude-chunk=date,time tmp.png -trim +repage cube-3LL.png %%% optipng cube-3LL.png %%%% magick adds white (fallback) background to the transparent image %%%% and has the convenient option -trim. %%%% Size of trimmed image is 385x226. %% png creation (large size) %%% pdflatex cube-3LL.tex %%% pdftocairo -png -singlefile -r 361.35 -W 1625 -H 1033 -x 723 -y 481 cube-3LL.pdf cube-3LL-large %%% optipng cube-3LL-large.png %%%% The resolution 361.35 ppi translates 1pt to 5 pixels. Hence 1 %%%% pixel in the png corresponds to 0.2pt. A TeX rule becomes 2 %%%% pixels wide. %%%% trim parameters: %%% magick tmp.png -format "%@\n" info: %% 1425x833+823+581 %%%% padding is 100 pixels %% program versions %%% This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Debian) (preloaded format=pdflatex) %%% Document Class: amsart 2017/10/31 v2.20.4 %%% Package: tikz 2020/01/08 v3.1.5b (3.1.5b) %%% pdftocairo version 0.86.1 %%% ImageMagick 7.1.2-19 Q16-HDRI x86_64 23897 https://imagemagick.org %%% OptiPNG version 0.7.7 %%%%%%%% %%%% Created: %%% 2026-03-10 %%%% Latest change of an image: %%% 2026-04-29 % minor changes %%% 2026-05-02 % tiny changes %%%% Latest code change: %%% 2026-04-29 % draw edges only once %%% 2026-05-02 % new macro \CYCLE %%%%%%%%