# How to draw the Cayley diagram of $A_{5}$?

It is so complex that I don't even know how to deal with it? Any idea?

Thanks so much.

• Welcome to TeX.SX. Questions about how to draw specific graphics that just post an image of the desired result are really not reasonable questions to ask on the site. Please post a minimal compilable document showing that you've tried to produce the image and then people will be happy to help you with any specific problems you may have. See minimal working example (MWE) for what needs to go into such a document. – Stefan Pinnow May 5 '18 at 3:26
• This may be reasonable starting point. But I'd go for a asymptote solution or at least base the graph on asymptote, like here. – marmot May 5 '18 at 3:41

An option using macros, although there are many manual things to do, and the result is a 2d approach to what is required, it is possible to rotate the icosahedron, thanks to the macro of Tom Bombadil answer in - Tikz:: shift and rotate in 3d?, the theory for the coordinates was consulted in:Truncated Icosahedron in TikZ?

Result: MWE:

\documentclass[border=20pt]{standalone}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{tikz}
\usetikzlibrary{arrows,3D}
\definecolor{WIRE}{HTML}{002FA7} % Klein Blue
\newcommand{\savedx}{0}
\newcommand{\savedy}{0}
\newcommand{\savedz}{0}

\newcommand{\somedrawing} {
% Definig all the coordinates from basic position.
% From theory from Paulo Ney in https://tex.stackexchange.com/q/269108/154390
\coordinate (1) at (0, 1,  3*\phi);
\coordinate (2) at (0,  -1,  3*\phi);
\coordinate (3) at (0,  1, -3*\phi);
\coordinate (4) at (0, -1, -3*\phi);
\coordinate (5) at (1, 2+\phi,  2*\phi);
\coordinate (6) at (-1, 2+\phi,  2*\phi);
\coordinate (7) at (1, -2-\phi,  2*\phi);
\coordinate (8) at (1, 2+\phi,  -2*\phi);
\coordinate (9) at (-1, -2-\phi,  2*\phi);
\coordinate (10) at (1, -2-\phi,  -2*\phi);
\coordinate (11) at (-1, 2+\phi,  -2*\phi);
\coordinate (12) at (-1, -2-\phi,  -2*\phi);
\coordinate (13) at (\phi, 2, \phi^3);
\coordinate (14) at (-\phi, 2, \phi^3);
\coordinate (15) at (\phi, -2, \phi^3);
\coordinate (16) at (\phi, 2, -\phi^3);
\coordinate (17) at (-\phi, -2, \phi^3);
\coordinate (18) at (\phi, -2, -\phi^3);
\coordinate (19) at (-\phi, 2, -\phi^3);
\coordinate (20) at (-\phi, -2, -\phi^3);
\coordinate (21) at (3*\phi, 0, 1);
\coordinate (22) at (-3*\phi, 0, 1);
\coordinate (23) at (3*\phi, 0, -1);
\coordinate (24) at (-3*\phi, 0, -1);
\coordinate (25) at (2*\phi,1, 2+\phi);
\coordinate (26) at (-2*\phi,1, 2+\phi);
\coordinate (27) at (2*\phi,-1, 2+\phi);
\coordinate (28) at (2*\phi,1, -2-\phi);
\coordinate (29) at (-2*\phi,-1, 2+\phi);
\coordinate (30) at (2*\phi,-1, -2-\phi);
\coordinate (31) at (-2*\phi,1, -2-\phi);
\coordinate (32) at (-2*\phi,-1, -2-\phi);
\coordinate (33) at (\phi^3,\phi, 2);
\coordinate (34) at (-\phi^3,\phi, 2);
\coordinate (35) at (\phi^3,-\phi, 2);
\coordinate (36) at (\phi^3,\phi, -2);
\coordinate (37) at (-\phi^3,-\phi, 2);
\coordinate (38) at (\phi^3,-\phi, -2);
\coordinate (39) at (-\phi^3,\phi, -2);
\coordinate (40) at (-\phi^3,-\phi, -2);
\coordinate (41) at (1,3*\phi, 0);
\coordinate (42) at (-1,3*\phi, 0);
\coordinate (43) at (1,-3*\phi, 0);
\coordinate (44) at (-1,-3*\phi, 0);
\coordinate (45) at (2+\phi,2*\phi,1);
\coordinate (46) at (-2-\phi,2*\phi,1);
\coordinate (47) at (2+\phi,-2*\phi,1);
\coordinate (48) at (2+\phi,2*\phi,-1);
\coordinate (49) at (-2-\phi,-2*\phi,1);
\coordinate (50) at (2+\phi,-2*\phi,-1);
\coordinate (51) at (-2-\phi,2*\phi,-1);
\coordinate (52) at (-2-\phi,-2*\phi,-1);
\coordinate (53) at (2,\phi^3,\phi);
\coordinate (54) at (-2,\phi^3,\phi);
\coordinate (55) at (2,-\phi^3,\phi);
\coordinate (56) at (2,\phi^3,-\phi);
\coordinate (57) at (-2,-\phi^3,\phi);
\coordinate (58) at (2,-\phi^3,-\phi);
\coordinate (59) at (-2,\phi^3,-\phi);
\coordinate (60) at (-2,-\phi^3,-\phi);

%Drawing background  vertices.
\foreach \n in {
44,57,43,58,18,49,37,22,24,31,19,3,4,10,52,60,12,20,32,40}{
\node[Vertb node] at (\n) {};
}

% Drawing all group arrows in the background
\foreach \n/\m  in {
43/55,58/43/,50/58,
60/52,52/49,49/57,57/44,44/60,
11/19,19/3,3/16,
10/18,18/4,4/20,20/12,12/10,
40/32,32/31,31/39,39/24,24/40,
29/37,37/22,22/34}{
\draw[grob] (\n)--(\m);
}

%Drawing all the relative lines in the background
\foreach \n/\m  in {
43/44,9/57,49/37,58/10,30/18,12/60,4/3,19/31,32/20,
52/40,24/34,39/51}{
\draw[relb] (\n)--(\m);
}

%Drawing middle perspective vertices.
\foreach \n in {
7,9,17,29,26,34,46,51,59,11,8,16,28,30,50,47,55}{
\node[Verf node] at (\n) {};
}
% Drawing all group arrows
\foreach \n/\m  in {
55/47,47/50,
15/7,7/9,9/17,17/2,2/15,
27/25,25/33,33/21,21/35,35/27,
30/38,38/23,23/36,36/28,28/30,
48/45,45/53,53/41,41/56,56/48,
16/8,8/11,
5/13,13/1,1/14,14/6,6/5,
34/26,26/29,
42/54,54/46,46/51,51/59,59/42}{
\draw[gro] (\n)--(\m);
}
%Drawing all the relative lines
\foreach \n/\m  in {
17/29,2/1,26/14,13/25,27/15,5/53,33/45,47/35,55/7,21/23,38/50,
36/48,28/16,8/56,11/59,41/42,54/6,34/46}{
\draw[rel] (\n)--(\m);
}
%Drawing front  vertices.
\foreach \n in {
27,35,21,38,23,36,48,56,41,59,42,54,6,14,1,2,15,33,45,53,5,13,25}{
\node[Verf node] at (\n) {};
}
% %use to identify all the nodes in drawing lines process
%     \foreach \n in {1,...,60}{
%       \node[circle,fill=blue!20] at (\n) {\Large \n};
%       }

}
\newcommand{\rotateRPY}[4][0/0/0]% point to be saved to \savedxyz, roll, pitch, yaw
{   \pgfmathsetmacro{\rollangle}{#2}
\pgfmathsetmacro{\pitchangle}{#3}
\pgfmathsetmacro{\yawangle}{#4}

% to what vector is the x unit vector transformed, and which 2D vector is this?
\pgfmathsetmacro{\newxx}{cos(\yawangle)*cos(\pitchangle)}% a
\pgfmathsetmacro{\newxy}{sin(\yawangle)*cos(\pitchangle)}% d
\pgfmathsetmacro{\newxz}{-sin(\pitchangle)}% g
\path (\newxx,\newxy,\newxz);
\pgfgetlastxy{\nxx}{\nxy};

% to what vector is the y unit vector transformed, and which 2D vector is this?
\pgfmathsetmacro{\newyx}{cos(\yawangle)*sin(\pitchangle)*sin(\rollangle)-sin(\yawangle)*cos(\rollangle)}% b
\pgfmathsetmacro{\newyy}{sin(\yawangle)*sin(\pitchangle)*sin(\rollangle)+ cos(\yawangle)*cos(\rollangle)}% e
\pgfmathsetmacro{\newyz}{cos(\pitchangle)*sin(\rollangle)}% h
\path (\newyx,\newyy,\newyz);
\pgfgetlastxy{\nyx}{\nyy};

% to what vector is the z unit vector transformed, and which 2D vector is this?
\pgfmathsetmacro{\newzx}{cos(\yawangle)*sin(\pitchangle)*cos(\rollangle)+ sin(\yawangle)*sin(\rollangle)}
\pgfmathsetmacro{\newzy}{sin(\yawangle)*sin(\pitchangle)*cos(\rollangle)-cos(\yawangle)*sin(\rollangle)}
\pgfmathsetmacro{\newzz}{cos(\pitchangle)*cos(\rollangle)}
\path (\newzx,\newzy,\newzz);
\pgfgetlastxy{\nzx}{\nzy};

% transform the point given by #1
\foreach \x/\y/\z in {#1}
{   \pgfmathsetmacro{\transformedx}{\x*\newxx+\y*\newyx+\z*\newzx}
\pgfmathsetmacro{\transformedy}{\x*\newxy+\y*\newyy+\z*\newzy}
\pgfmathsetmacro{\transformedz}{\x*\newxz+\y*\newyz+\z*\newzz}
\xdef\savedx{\transformedx}
\xdef\savedy{\transformedy}
\xdef\savedz{\transformedz}
}
}

\tikzset{RPY/.style={x={(\nxx,\nxy)},y={(\nyx,\nyy)},z={(\nzx,\nzy)}}}

\begin{document}
\begin{tikzpicture}[% Sets all the styles
x={(-0.86in, -0.5in)}, y = {(0.86in, -0.5in)}, z = {(0, 1in)},
scale = 1,
Verf node/.style = {circle,fill = WIRE!50, draw,ultra thick, minimum size = 1.5cm},
Vertb node/.style = {circle,fill = WIRE!25, draw=black!50,thick, minimum size = 1cm},
gro/.style = {line width=7pt, red,>=triangle 60,->,shorten >= 1cm, shorten <= 1cm},
grob/.style = {line width=7pt,red!25,>=triangle 60,->,shorten >= .8cm, shorten <= .8cm},
rel/.style = {line width=7pt,WIRE,shorten >= 1cm, shorten <= 1cm },
relb/.style = { line width=7pt,WIRE!30,shorten >= .8cm, shorten <= .8cm }
]

\rotateRPY{30}{10}{10} % 30-10-10
\begin{scope}[RPY]
\somedrawing
\end{scope}
\end{tikzpicture}
\end{document}

• Very nice! +1 Would you perhaps consider not redefining \phi but using some other name for the macro of the angle since otherwise this code may clash with some basic uses of \phi in formulae? BTW, how did you find out which vertices are in the back? Did you do it by hand or did you use some automatic method? In the latter case, it would be great if you could describe it since there are many other Cayley diagrams that could then be drawn analogously. – marmot May 7 '18 at 0:28
• @marmot, you're right, it can cause problems, I'll change it to \r; As for finding which vertices go back, I found them manually, there are some lines of code that list the vertices, the vertices from 1 to 20 define at least 3 faces, rotating in 3D can be confirmed according to the displacement which goes in front or behind (using TikzEdt semi WYSIWYG editor), I chose a face that is completely oriented forward and draw the red arrows in a clockwise direction, since this can be completed all step by step, visually. – J Leon V. May 7 '18 at 1:20
• Maybe not even \r but something more complicated such that it does not interfere with other definitions. And maybe do the definition inside the \somedrawing group, possibly with \pgfmathsetmacro, such that it becomes local, and thus has less chance of interfering with other macros. – marmot May 7 '18 at 1:28
• in terms of automating it, I think the trick is to strategically list the vertices, I do not master the topic but there is a type of formula or notation for that topology in this link, another problem is to draw the arrows in 3D, and finally define faces with certain transparency so that the posterior vertices are noticed, in addition to a perspective rendering ... – J Leon V. May 7 '18 at 1:36
• I see. I was wondering if you found the vector pointing to the viewer and computed the projection of the elements on that vector. (I was playing a bit with that before I switched to asymptote.) – marmot May 7 '18 at 1:45

Here is a proposal. I started with these examples but then I switched gear and got the vertices, edges and faces from Mathematica because that allowed me to draw the arrows on isolated surfaces only. You need to compile with -shell-escape.

\documentclass[border=3.14mm]{standalone}
\usepackage{asypictureB}
\begin{document}
\begin{asypicture}{name=CayleyA5}
import solids;
import three;

triple vertices[];
vertices[0]=(-0.16245984811645317, -2.118033988749895, 1.2759762125280598);
vertices[1]=(-0.16245984811645317, 2.118033988749895, 1.2759762125280598);
vertices[2]=(0.16245984811645317, -2.118033988749895, -1.27597621252806);
vertices[3]=(0.16245984811645317, 2.118033988749895, -1.27597621252806);
vertices[4]=(-0.2628655560595668, -0.8090169943749473, -2.327438436766327);
vertices[5]=(-0.2628655560595668, -2.4270509831248424, -0.42532540417601994);
vertices[6]=(-0.2628655560595668, 0.8090169943749475, -2.327438436766327);
vertices[7]=(-0.2628655560595668, 2.4270509831248424, -0.42532540417601994);
vertices[8]=(0.2628655560595668, -0.8090169943749473, 2.327438436766327);
vertices[9]=(0.2628655560595668, -2.4270509831248424, 0.42532540417601994);
vertices[10]=(0.2628655560595668, 0.8090169943749475, 2.327438436766327);
vertices[11]=(0.2628655560595668, 2.4270509831248424, 0.42532540417601994);
vertices[12]=(0.6881909602355868, -0.5, -2.327438436766327);
vertices[13]=(0.6881909602355868, 0.5, -2.327438436766327);
vertices[14]=(1.2139220723547204, -2.118033988749895, 0.42532540417601994);
vertices[15]=(1.2139220723547204, 2.118033988749895, 0.42532540417601994);
vertices[16]=(-2.0645728807067605, -0.5, 1.2759762125280598);
vertices[17]=(-2.0645728807067605, 0.5, 1.2759762125280598);
vertices[18]=(-1.3763819204711736, -1., 1.8017073246471935);
vertices[19]=(-1.3763819204711736, 1., 1.8017073246471935);
vertices[20]=(-1.3763819204711736, -1.6180339887498947, -1.27597621252806);
vertices[21]=(-1.3763819204711736, 1.618033988749895, -1.27597621252806);
vertices[22]=(-0.6881909602355868, -0.5, 2.327438436766327);
vertices[23]=(-0.6881909602355868, 0.5, 2.327438436766327);
vertices[24]=(1.3763819204711736, -1., -1.8017073246471935);
vertices[25]=(1.3763819204711736, 1., -1.8017073246471935);
vertices[26]=(1.3763819204711736, -1.6180339887498947, 1.2759762125280598);
vertices[27]=(1.3763819204711736, 1.618033988749895, 1.2759762125280598);
vertices[28]=(-1.7013016167040798, 0., -1.8017073246471935);
vertices[29]=(1.7013016167040798, 0., 1.8017073246471935);
vertices[30]=(-1.2139220723547204, -2.118033988749895, -0.42532540417601994);
vertices[31]=(-1.2139220723547204, 2.118033988749895, -0.42532540417601994);
vertices[32]=(-1.9641671727636467, -0.8090169943749473, -1.27597621252806);
vertices[33]=(-1.9641671727636467, 0.8090169943749475, -1.27597621252806);
vertices[34]=(2.0645728807067605, -0.5, -1.27597621252806);
vertices[35]=(2.0645728807067605, 0.5, -1.27597621252806);
vertices[36]=(2.2270327288232132, -1., -0.42532540417601994);
vertices[37]=(2.2270327288232132, 1., -0.42532540417601994);
vertices[38]=(2.3894925769396664, -0.5, 0.42532540417601994);
vertices[39]=(2.3894925769396664, 0.5, 0.42532540417601994);
vertices[40]=(-1.1135163644116066, -1.8090169943749475, 1.2759762125280598);
vertices[41]=(-1.1135163644116066, 1.8090169943749475, 1.2759762125280598);
vertices[42]=(1.1135163644116066, -1.8090169943749475, -1.27597621252806);
vertices[43]=(1.1135163644116066, 1.8090169943749475, -1.27597621252806);
vertices[44]=(-2.3894925769396664, -0.5, -0.42532540417601994);
vertices[45]=(-2.3894925769396664, 0.5, -0.42532540417601994);
vertices[46]=(-1.6392474765307403, -1.8090169943749475, 0.42532540417601994);
vertices[47]=(-1.6392474765307403, 1.8090169943749475, 0.42532540417601994);
vertices[48]=(1.6392474765307403, -1.8090169943749475, -0.42532540417601994);
vertices[49]=(1.6392474765307403, 1.8090169943749475, -0.42532540417601994);
vertices[50]=(1.9641671727636467, -0.8090169943749473, 1.2759762125280598);
vertices[51]=(1.9641671727636467, 0.8090169943749475, 1.2759762125280598);
vertices[52]=(0.85065080835204, 0., 2.327438436766327);
vertices[53]=(-2.2270327288232137, -1., 0.42532540417601994);
vertices[54]=(-2.2270327288232137, 1., 0.42532540417601994);
vertices[55]=(-0.8506508083520399, 0., -2.327438436766327);
vertices[56]=(-0.5257311121191336, -1.6180339887498947, -1.8017073246471935);
vertices[57]=(-0.5257311121191336, 1.618033988749895, -1.8017073246471935);
vertices[58]=(0.5257311121191336, -1.6180339887498947, 1.8017073246471935);
vertices[59]=(0.5257311121191336, 1.618033988749895, 1.8017073246471935);

int edge[][] ={{1, 10}, {1, 41}, {1, 59}, {2,
12}, {2, 42}, {2, 60}, {3, 6}, {3,
43}, {3, 57}, {4, 8}, {4, 44}, {4,
58}, {5, 13}, {5, 56}, {5, 57}, {6,
10}, {6, 31}, {7, 14}, {7, 56}, {7,
58}, {8, 12}, {8, 32}, {9, 23}, {9,
53}, {9, 59}, {10, 15}, {11, 24},
{11, 53}, {11, 60}, {12, 16}, {13,
14}, {13, 25}, {14, 26}, {15, 27},
{15, 49}, {16, 28}, {16, 50}, {17,
18}, {17, 19}, {17, 54}, {18, 20},
{18, 55}, {19, 23}, {19, 41}, {20,
24}, {20, 42}, {21, 31}, {21, 33},
{21, 57}, {22, 32}, {22, 34}, {22,
58}, {23, 24}, {25, 35}, {25, 43},
{26, 36}, {26, 44}, {27, 51}, {27,
59}, {28, 52}, {28, 60}, {29, 33},
{29, 34}, {29, 56}, {30, 51}, {30,
52}, {30, 53}, {31, 47}, {32, 48},
{33, 45}, {34, 46}, {35, 36}, {35,
37}, {36, 38}, {37, 39}, {37, 49},
{38, 40}, {38, 50}, {39, 40}, {39,
51}, {40, 52}, {41, 47}, {42, 48},
{43, 49}, {44, 50}, {45, 46}, {45,
54}, {46, 55}, {47, 54}, {48, 55}};

int isolatedfaces[][] = {{53, 11, 24, 23, 9},
{51, 39, 40, 52, 30},
{60, 28, 16, 12, 2},
{20, 42, 48, 55, 18},
{19, 17, 54, 47, 41},
{1, 10, 15, 27, 59},
{36, 26, 44, 50, 38},
{4, 58, 22, 32, 8}, {34, 29, 33,
45, 46}, {21, 57, 3, 6, 31},
{37, 49, 43, 25, 35},
{13, 5, 56, 7, 14}};

// comment the following line for OpenGl
settings.render=5;

settings.tex="pdflatex";
settings.outformat="pdf"; // for opacity

size(10cm);

currentprojection=perspective(7,6,4); //if you want perspectivic look
//currentprojection=orthographic(1,1,0.5); //if you want othographic look
currentlight=(1,1,2);
// currentlight=nolight;

for(int i=0;i<60;++i)
draw(shift(vertices[i])*scale3(0.1)*unitsphere,blue+opacity(.7));

for(int i=0;i<90;++i)
draw(vertices[edge[i][0]-1] -- vertices[edge[i][1]-1]);

for(int i=0;i<isolatedfaces.length;++i)
{