File:Permutohedron of S3; cube permutations.png

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Summary

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Description
the original cube position
permutohedron in cube

Permutohedron of the symmetric group S3 showing the six permutations of the cube that leave the main diagonal fixed

See v:Full octahedral group

the same in two dimensions


 
This image was created with POV-Ray.
(The labels are added with Inkscape.)
Date
Source Own work
Author
Watchduck
You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".
Other versions
All 48 permutations assigned to vertices of a truncated cuboctahedron, containing this hexagon in the bottom left corner.
Cayley graphs with vertices in the same positions

Licensing

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I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International, 3.0 Unported, 2.5 Generic, 2.0 Generic and 1.0 Generic license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
You may select the license of your choice.


POV-Ray source

[edit]
#version 3.6;
global_settings { assumed_gamma 1.0 }
#default{ finish{ ambient 0.1 diffuse 0.9 conserve_energy}}

#include "colors.inc"
#include "math.inc"


///////////////////////////// camera and light

#declare Camera_Position = <12, 12, -50>;
camera{
    location Camera_Position
    right    x*image_width/image_height
    angle    7.5
    look_at  <-0.05, -0.14, 0.1>
}


light_source{ <-400, 500, -300> color White*0.9 shadowless}

light_source{ Camera_Position  color rgb<0.9,0.9,1>*0.1 shadowless}
sky_sphere{ pigment{ White } }


///////////////////////////// JF solid

#declare LightColor = rgb<0.6, 0.55, 0.55>;
#declare DarkColor = rgb<0.005, 0.004, 0.004>;
#declare CubeColor = rgb<0.25, 0.25, 0.25>;


#declare BigCyl = cylinder{ <0, 0, 0>, <0.9, 0, 0>, 1.5 }
#declare SmallCyl = cylinder{ <-0.1, 0, 0>, <1, 0, 0>, 0.6 }
#declare Ring = difference{
    object{BigCyl}
    object{SmallCyl}
}
#declare CutBox = box{
    <-10,-10,-10> <10,10,10>
    translate 10*y
}
#declare HalfRing = difference{
    object{Ring}
    object{CutBox}
}
#declare CutBox = object{
    CutBox
    rotate -45*x
    pigment{color DarkColor}
}
#declare LetterJTail = difference{
    object{HalfRing}
    object{CutBox}
}


#declare LetterStem = box{
    <0, 0, 0.6>
    <0.9, 2.4, 1.5>
}


#declare LetterFBar1 = box{
    <0, 1.5, 0.6>
    <-3, 2.4, 1.5>
}
#declare CutBox = box{
    <-10,-10,-10> <10,10,10>
    translate <-12.6, 0, 0>
    pigment{color DarkColor}
}
#declare LetterFBar1 = difference{
    object{LetterFBar1}
    CutBox
}


#declare LetterFBar2 = object{
    LetterFBar1
    translate <0, -1.5, 0>
}
#declare CutBox = object{
    CutBox
    translate <1.3, 0, 0>
}
#declare LetterFBar2 = difference{
    object{LetterFBar2}
    CutBox
}


#declare LetterJFSolid = union{
    object{LetterJTail}
    object{LetterStem}
    object{LetterFBar1}
    object{LetterFBar2}
    pigment{color LightColor}
}


///////////////////////////// JF outline

#declare R = 0.03;

#declare PointFBar1a = <0.9, 2.4, 0.6>;
#declare PointFBar1b = PointFBar1a + 0.9*z;
#declare PointFBar1c = PointFBar1a - 3.5*x;
#declare PointFBar1d = PointFBar1b - 3.5*x;

#declare PointFBar2c = PointFBar1c - 0.9*y;
#declare PointFBar2d = PointFBar2c + 0.9*z;
#declare PointFBar2a = PointFBar2c + 2.6*x;
#declare PointFBar2b = PointFBar2a + 0.9*z;

#declare PointFBar3a = PointFBar2a - 0.6*y;
#declare PointFBar3b = PointFBar3a + 0.9*z;
#declare PointFBar3c = PointFBar3a - 1.3*x;
#declare PointFBar3d = PointFBar3b - 1.3*x;

#declare PointFBar4a = PointFBar3a - 0.9*y;
#declare PointFBar4b = PointFBar4a + 0.9*z;
#declare PointFBar4c = PointFBar4a - 1.3*x;
#declare PointFBar4d = PointFBar4b - 1.3*x;

#declare PointJFaceA = PointFBar4a + 0.9*x;
#declare PointJFaceB = PointFBar4b + 0.9*x;

#declare PointJEndA = <0.9, -sqrt(1.125), -sqrt(1.125)>;
#declare PointJEndB = <0.9, -sqrt(0.18), -sqrt(0.18)>;
#declare PointJEndC = PointJEndA - 0.9*x;
#declare PointJEndD = PointJEndB - 0.9*x;


#declare SmallTorus = object{
    torus {0.6, R}
    rotate 90*z
}
#declare BigTorus = object{
    torus {1.5, R}
    rotate 90*z
}
#declare Tori1 = union{
    object{SmallTorus}
    object{BigTorus}
}
#declare Tori2 = object{
    Tori1
    translate 0.9*x
}
#declare Tori = union{
    object{Tori1}
    object{Tori2}
}

#declare CutBox = box{
    <-10,-10,-10> <10,10,10>
    translate 10*y
}
#declare Tori = difference{
    object{Tori}
    object{CutBox}
}
#declare CutBox = object{
    CutBox
    rotate -45*x
}
#declare Tori = difference{
    object{Tori}
    object{CutBox}
}


#declare LetterJFOutlines = union{

    sphere{PointFBar1a, R}
    sphere{PointFBar1b, R}
    sphere{PointFBar1c, R}
    sphere{PointFBar1d, R}

    sphere{PointFBar2a, R}
    sphere{PointFBar2b, R}
    sphere{PointFBar2c, R}
    sphere{PointFBar2d, R}

    sphere{PointFBar3a, R}
    sphere{PointFBar3b, R}
    sphere{PointFBar3c, R}
    sphere{PointFBar3d, R}

    sphere{PointFBar4a, R}
    sphere{PointFBar4b, R}
    sphere{PointFBar4c, R}
    sphere{PointFBar4d, R}

    sphere{PointJEndA, R}
    sphere{PointJEndB, R}
    sphere{PointJEndC, R}
    sphere{PointJEndD, R}

    cylinder{PointFBar1a, PointFBar1b, R}
    cylinder{PointFBar1b, PointFBar1d, R}
    cylinder{PointFBar1d, PointFBar1c, R}
    cylinder{PointFBar1c, PointFBar1a, R}

    cylinder{PointFBar2a, PointFBar2b, R}
    cylinder{PointFBar2b, PointFBar2d, R}
    cylinder{PointFBar2d, PointFBar2c, R}
    cylinder{PointFBar2c, PointFBar2a, R}

    cylinder{PointFBar3a, PointFBar3b, R}
    cylinder{PointFBar3b, PointFBar3d, R}
    cylinder{PointFBar3d, PointFBar3c, R}
    cylinder{PointFBar3c, PointFBar3a, R}

    cylinder{PointFBar4a, PointFBar4b, R}
    cylinder{PointFBar4b, PointFBar4d, R}
    cylinder{PointFBar4d, PointFBar4c, R}
    cylinder{PointFBar4c, PointFBar4a, R}

    cylinder{PointFBar1c, PointFBar2c, R}
    cylinder{PointFBar1d, PointFBar2d, R}

    cylinder{PointFBar2a, PointFBar3a, R}
    cylinder{PointFBar2b, PointFBar3b, R}

    cylinder{PointFBar3c, PointFBar4c, R}
    cylinder{PointFBar3d, PointFBar4d, R}

    cylinder{PointFBar1a, PointJFaceA, R}
    cylinder{PointFBar1b, PointJFaceB, R}

    cylinder{PointJEndA, PointJEndB, R}
    cylinder{PointJEndB, PointJEndD, R}
    cylinder{PointJEndD, PointJEndC, R}
    cylinder{PointJEndC, PointJEndA, R}

    object{Tori}

    pigment{color DarkColor}

}


///////////////////////////// cube

#declare CubeCoordinates = array[8]{
    <-2.25, -2.25, -2.25>,
    < 2.25, -2.25, -2.25>,
    <-2.25, -2.25,  2.25>,
    < 2.25, -2.25,  2.25>,
    <-2.25,  2.25, -2.25>,
    < 2.25,  2.25, -2.25>,
    <-2.25,  2.25,  2.25>,
    < 2.25,  2.25,  2.25>
}
#declare CubeEdgesAbstractCoordinates = array[12][2]{
    {0,1}, {2,3}, {4,5}, {6,7}, {0,2}, {1,3}, {4,6}, {5,7}, {0,4}, {1,5}, {2,6}, {3,7}
}
#declare Cube = union{
    #for(i, 0, 7)
        sphere{CubeCoordinates[i], 0.01}
    #end
    #for(i, 0, 11)
        cylinder{
            CubeCoordinates[CubeEdgesAbstractCoordinates[i][0]],
            CubeCoordinates[CubeEdgesAbstractCoordinates[i][1]],
            0.01
        }
    #end
    pigment{color CubeColor}
}


///////////////////////////// define complete cube compound

#declare LetterJF = union{
    object{LetterJFSolid}
    object{LetterJFOutlines}
    translate <0.85, -0.45, -0.25>
    rotate 90*y
}

#declare CubeLetterCompound = union{
    object{LetterJF}
    object{Cube}
    scale 0.2
}

///////////////////////////// macro to shorten hexagon edges

#macro edge(A, B)
    #local Dist = sqrt(2)*0.45;
    #local Dir = vnormalize(B-A);
    #local NewA = A + Dist*Dir;
    #local NewB = B - Dist*Dir;
    cylinder{NewA, NewB, R}
#end


///////////////////////////// hexagon

#declare R = 0.05;
#declare S = 4.5;

#declare Point0 = CubeCoordinates[4] + 0.5*S*z;
#declare Point1 = CubeCoordinates[4] + 0.5*S*x;
#declare Point2 = CubeCoordinates[2] + 0.5*S*y;
#declare Point3 = CubeCoordinates[2] + 0.5*S*x;
#declare Point4 = CubeCoordinates[1] + 0.5*S*y;
#declare Point5 = CubeCoordinates[1] + 0.5*S*z;

union{
    edge(Point0, Point1)
    edge(Point1, Point4)
    edge(Point4, Point5)
    edge(Point5, Point3)
    edge(Point3, Point2)
    edge(Point2, Point0)
    pigment{color rgb<0.01,0.01,0.01>}
}

///////////////////////////// place little cube compounds

object{
    CubeLetterCompound
    translate Point0
}

object{
    CubeLetterCompound
    scale -z rotate -90*y
    translate Point1
}

object{
    CubeLetterCompound
    scale -y rotate -90*x
    translate Point2
}

object{
    CubeLetterCompound
    rotate 90*x rotate 90*z
    translate Point3
}

object{
    CubeLetterCompound
    rotate -90*z rotate -90*x
    translate Point4
}

object{
    CubeLetterCompound
    scale -x rotate -90*z
    translate Point5
}

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Date/TimeThumbnailDimensionsUserComment
current13:11, 31 May 2017Thumbnail for version as of 13:11, 31 May 20174,632 × 4,807 (712 KB)Watchduck (talk | contribs){{Information |Description ={{en|1=upload forms suck}} |Source ={{own}} |Author =Watchduck |Date = |Permission = |other_versions = }}

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