%% assoc-L3A.tex - an affine projection of the Loday associahedron %% Author: Markus Rost %% Created: 8 Mar 2026 %% to be compiled with pdflatex \documentclass{amsart} %%%%%%%% %%%%% Shows an affine projection of Loday's realization of the %%%%% 3-dimensional associahedron (Loday, Jean-Louis. Realization of %%%%% the Stasheff polytope. Arch. Math. 83, no. 3, 267-278, 2004. MR %%%%% 2108555.) %%%%% Another drawing can be found at the end of Loday's paper %%%%% (p.277). %%%%%%%% %%%%% The realization lies in the hyperplane %%%%% %%%%% H = { x + y + z + t = 10 } %%%%% %%%%% in R^4. The drawing is the projection %%%%% %%%%% (x, y, z, t) %%%%% -> (x, z, t) %%%%% -> (x - 3ct, z - ct) c = 0.1925 %%%%% The plane { t = 4 } contains the 2-dimensional %%%%% sub-associahedron %%%%% %%%%% 3214 - 1414 - 1234 %%%%% \ / %%%%% 3124 -- 2134 %%%%% %%%%% the parallel plane { t = 1 } another pentagon face. %%%%%%%% %%%%% By the way: %%%%% The 3 quadrilaterals are parallelograms and exactly 2 of them %%%%% share (both) directions. From this one sees that the drawing %%%%% %%%%% https://commons.wikimedia.org/wiki/File:Associahedron_K5.svg %%%%% %%%%% is NOT an affine projection of Loday's realization. %%%%%%%% %%%%% A version of the macro \LCOORDINATES contains the coordinates of %%%%% Loday's realization of the 4-dimensional associahedron. I did %%%%% not dare to try for a nice plane projection. %%%%% Appended are python scripts to produce parenthetical expressions %%%%% and their Loday coordinates. %%%%%%%% \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 % %% this is cmb10 scaled 0.5 (there is no cmb5) %% cmb10 has same sizes and spacing as cmr10 \DeclareFixedFont{\PARENfont}{OT1}{cmr}{b}{n}{5} % \fi % \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \thispagestyle{empty} % \centerline{% \tikzpicture[x={(1cm,0cm)}, y={(0cm,1cm)}, z={(-3.85mm*3/2,-3.85mm/2)}] % %% a first version used the default tikz projection: %%% [x={(1cm,0cm)}, y={(0cm,1cm)}, z={(-3.85mm,-3.85mm)}] %%%%%%%% Loday coordinates %%% Computed from \VERTICES in cube-4-84.tex with a python script %%% (appended below). %%% The internal names (A0, etc.) could be eventually replaced by %%% the coordinates and dropped. However I find them simpler to %%% recognize. \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) } %%%% for the 4-dim. case %% \def\LCOORDINATES{% %% A0/1/2/3/4/5, % (((( 01 )2)3)4)5 %% A1/2/1/3/4/5, % ((( 0(12) )3)4)5 %% X0/1/4/1/4/5, % (( (01)(23) )4)5 %% A2/3/2/1/4/5, % (( 0(1(23)) )4)5 %% A3/3/1/2/4/5, % (( 0((12)3) )4)5 %% Y1/1/2/6/1/5, % ( ((01)2)(34) )5 %% Y2/1/6/2/1/5, % ( (01)(2(34)) )5 %% B0/4/3/2/1/5, % ( 0(1(2(34))) )5 %% Z1/2/1/6/1/5, % ( (0(12))(34) )5 %% B1/4/1/4/1/5, % ( 0((12)(34)) )5 %% X2/1/6/1/2/5, % ( (01)((23)4) )5 %% X3/4/3/1/2/5, % ( 0(1((23)4)) )5 %% B2/4/2/1/3/5, % ( 0((1(23))4) )5 %% B3/4/1/2/3/5, % ( 0(((12)3)4) )5 %% A0a/1/2/3/8/1, % (((01)2)3)(45) %% A0b/1/2/9/2/1, % ((01)2)(3(45)) %% A0c/1/8/3/2/1, % (01)(2(3(45))) %% A0d/5/4/3/2/1, % 0(1(2(3(45)))) %% A1a/2/1/3/8/1, % ((0(12))3)(45) %% A1b/2/1/9/2/1, % (0(12))(3(45)) %% A1d/5/1/6/2/1, % 0((12)(3(45))) %% X0a/1/4/1/8/1, % ((01)(23))(45) %% X0c/1/8/1/4/1, % (01)((23)(45)) %% X0d/5/4/1/4/1, % 0(1((23)(45))) %% Y1b/1/2/9/1/2, % ((01)2)((34)5) %% Y1c/1/8/3/1/2, % (01)(2((34)5)) %% Y1d/5/4/3/1/2, % 0(1(2((34)5))) %% A2a/3/2/1/8/1, % (0(1(23)))(45) %% A2d/5/2/1/6/1, % 0((1(23))(45)) %% A3a/3/1/2/8/1, % (0((12)3))(45) %% A3d/5/1/2/6/1, % 0(((12)3)(45)) %% Y2c/1/8/2/1/3, % (01)((2(34))5) %% Y2d/5/4/2/1/3, % 0(1((2(34))5)) %% Z1b/2/1/9/1/2, % (0(12))((34)5) %% Z1d/5/1/6/1/2, % 0((12)((34)5)) %% X2c/1/8/1/2/3, % (01)(((23)4)5) %% X2d/5/4/1/2/3, % 0(1(((23)4)5)) %% B0d/5/3/2/1/4, % 0((1(2(34)))5) %% B1d/5/1/4/1/4, % 0(((12)(34))5) %% X3d/5/3/1/2/4, % 0((1((23)4))5) %% B2d/5/2/1/3/4, % 0(((1(23))4)5) %% B3d/5/1/2/3/4% % 0((((12)3)4)5) %% } %% one has \x+\y+\z+\t=10 %% drop the second coordinate \path foreach \NNN/\x/\y/\z/\t in \LCOORDINATES { % coordinate (\NNN) at ($(\x,\z,\t)$) } ; % %%%%%%%% %% placement in shipout is independent of labels \useasboundingbox (X0) (Z1) (B0) ; % %%%%%%%% %%%% 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 %% faces in back: at X2,Y2 : QC,PD,PF,PG %% faces in front: at A1,A3,B3 : QA,QB,PA,PB,PC \def\EDGESBACK{% 5 edges B0/Y2,Y2/X2,X2/X3, % QC X0/X2,X2/X3,Y2/Y1% PD,PF,PG } \def\EDGESFRONT{% 16 edges A2/A3,A3/B3,B3/B2,B2/A2, % QA A1/A0,A0/Y1,Y1/Z1,Z1/A1, % QB A0/X0,X0/A2,A1/A3, % PA B1/B0,B0/X3,X3/B2, % PB B3/B1,B1/Z1% PC } % \edef\EDGES{\EDGESBACK,\EDGESFRONT} %%%%%%%% \def\CYCLE#1{foreach\p[count=\i]in#1{\ifnum\i=1 \else--\fi(\p)}--cycle} \fill[white] \CYCLE\facePD ; % Bottom \fill[blue!30] \CYCLE\faceQC ; % Bottom-Back \fill[blue!30] \CYCLE\facePF ; % Back \fill[red!10] \CYCLE\facePG ; % Left \draw[blue] foreach \XXX/\YYY in \EDGESBACK { (\XXX) -- (\YYY) } ; % \ifGREEN % \def\SCOLOR{green!50} \else % \def\SCOLOR{red!20} \fi % \fill[\SCOLOR,opacity=0.6] \CYCLE\facePA ; % Front \fill[red!20,opacity=0.8] \CYCLE\faceQA ; % Front Right \fill[\SCOLOR,opacity=0.8] \CYCLE\facePB ; % Right \fill[red!20,opacity=0.8] \CYCLE\facePC ; % Top Right \fill[red!20,opacity=0.8] \CYCLE\faceQB ; % Top \draw[blue] foreach \XXX/\YYY in \EDGESFRONT { (\XXX) -- (\YYY) } ; % %%%%%%%% \ifLABELS \begin{scope} [outer sep=0pt,inner sep=2pt] \tiny %%%% ad hoc hacking... \tikzset{ % B3place/.style={below left,yshift=-1.8pt}, % } % \def\PLACES{ % A0/left, A1/above left, A2/below, A3/below right, % B0/right, B1/right, B2/below, B3/B3place, % X0/below, X2/above left, X3/right, % Y1/above, Y2/above right, Z1/above} \ifNAMES %%%%%%%% show the names of the vertices \path foreach \XXX/\place in \PLACES { % node[\place] at (\XXX) {\XXX} } ; % \else % \ifNAMES %%%%%%%% show the coordinates %%%% kinda merging \LCOORDINATES \PLACES \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 %%%% cf. cube-4-84.tex \tikzset{ labelbox/.style={ % fill=white,fill opacity=0.6, % draw,text opacity=1, % inner sep=4/3pt}, % } % \def\Label#1{\scalebox{.5}{#1}} %% let's be bold, not boring \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} %%%%%%%% %%%% python3 script to get paren expressions %%% import itertools %%% %%% def parens(n,s): %%% if n == 1: %%% return [s] %%% return itertools.chain.from_iterable( %%% [itertools.product(parens(k,s),parens(n-k,s+k)) for k in range(1,n)]) %%% %%% for n in range(7): %%% for p in parens(n,0): %%% print (p) %%%% python3 script to compute Lodays coordinates %%% ## see \VERTICES in cube-4-84.tex %%% DATA=[ %%% ("A0", "Co", "(((( 01 )2)3)4)5"), %%% ("A1", "Ca", "((( 0(12) )3)4)5"), %%% ("X0", "Cb", "(( (01)(23) )4)5"), %%% ("A2", "Cd", "(( 0(1(23)) )4)5"), %%% ("A3", "Cd", "(( 0((12)3) )4)5"), %%% ("Y1", "Cc", "( ((01)2)(34) )5"), %%% ("Y2", "Cd", "( (01)(2(34)) )5"), %%% ("B0", "Ce", "( 0(1(2(34))) )5"), %%% ("Z1", "Cd", "( (0(12))(34) )5"), %%% ("B1", "Ce", "( 0((12)(34)) )5"), %%% ("X2", "Cd", "( (01)((23)4) )5"), %%% ("X3", "Ce", "( 0(1((23)4)) )5"), %%% ("B2", "Ce", "( 0((1(23))4) )5"), %%% ("B3", "Ce", "( 0(((12)3)4) )5"), %%% ("A0a", "Co", "(((01)2)3)(45)"), %%% ("A0b", "Co", "((01)2)(3(45))"), %%% ("A0c", "Co", "(01)(2(3(45)))"), %%% ("A0d", "Co", "0(1(2(3(45))))"), %%% ("A1a", "Ca", "((0(12))3)(45)"), %%% ("A1b", "Ca", "(0(12))(3(45))"), %%% ("A1d", "Ca", "0((12)(3(45)))"), %%% ("X0a", "Cb", "((01)(23))(45)"), %%% ("X0c", "Cb", "(01)((23)(45))"), %%% ("X0d", "Cb", "0(1((23)(45)))"), %%% ("Y1b", "Cc", "((01)2)((34)5)"), %%% ("Y1c", "Cc", "(01)(2((34)5))"), %%% ("Y1d", "Cc", "0(1(2((34)5)))"), %%% ("A2a", "Cd", "(0(1(23)))(45)"), %%% ("A2d", "Cd", "0((1(23))(45))"), %%% ("A3a", "Cd", "(0((12)3))(45)"), %%% ("A3d", "Cd", "0(((12)3)(45))"), %%% ("Y2c", "Cd", "(01)((2(34))5)"), %%% ("Y2d", "Cd", "0(1((2(34))5))"), %%% ("Z1b", "Cd", "(0(12))((34)5)"), %%% ("Z1d", "Cd", "0((12)((34)5))"), %%% ("X2c", "Cd", "(01)(((23)4)5)"), %%% ("X2d", "Cd", "0(1(((23)4)5))"), %%% ("B0d", "Ce", "0((1(2(34)))5)"), %%% ("B1d", "Ce", "0(((12)(34))5)"), %%% ("X3d", "Ce", "0((1((23)4))5)"), %%% ("B2d", "Ce", "0(((1(23))4)5)"), %%% ("B3d", "Ce", "0((((12)3)4)5)"), %%% ] %%% %%% import re %%% %%% for d in DATA: %%% name, color, S = d %%% %%% parens = re.split('[0-9]',S.replace(' ',''))[1:-1] %%% N = len(parens) %%% dp = [parens[h].count('(') - parens[h+1].count(')') for h in range(N-1)] %%% %%% LC = [] %%% for k in range(N): %%% %%% a = 1 %%% d = 0 %%% for h in range(k,N-1): %%% d += dp[h] %%% if d <= 0: break %%% a += 1 %%% %%% b = 1 %%% d = 0 %%% for h in range(k): %%% d -= dp[k-h-1] %%% if d <= 0: break %%% b += 1 %%% %%% LC.append(str(a*b)) %%% %%% print (name + '/' + '/'.join(LC) + ', % ' + S) %%%%%%%% %% png creation (normal size, transparent) %%% pdflatex cube-3L.tex %%% pdftocairo -png -singlefile -transp cube-3L.pdf tmp %%% magick -define png:exclude-chunk=date,time tmp.png -trim +repage cube-3L.png %%% optipng cube-3L.png %%%% magick adds white (fallback) background to the transparent image %%%% and has the convenient option -trim. %%%% Default resolution of pdftocairo is 150 ppi. Size of trimmed %%%% image is 281x331. %% png creation (large size) %%% pdflatex cube-3L.tex %%% pdftocairo -png -singlefile -r 361.35 -W 876 -H 996 -x 1098 -y 481 cube-3L.pdf cube-3L-large %%% optipng cube-3L-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: %% 676x796+1198+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: %%% 2026-03-08 % first version %%%% Latest change of an image: %%% 2026-03-10 % second version %%% 2026-04-29 % minor changes %%% 2026-05-02 % tiny change for small version %%%% Latest code change: %%% 2026-04-29 % draw edges only once %%% 2026-05-02 % new macro \CYCLE %%%%%%%%