# Visualizing the image of the entropy map when $$d = 3$$.

gi

## The inequalities defining the bipyramid Ccal

$E =[[ 1, 1, 1, 1], [ 1, 0, 1, 2], [-1, -1, -1, -2], [ 0, -1, 1, 0], [ 0, 0, -1, 0], [ 0, 1, -1, -1]]; ## The 6 inequalities defining the facets of the bipyramid Ccal # Eij means the facet of "i independent from j"$E12 =[[ 1,   1,   1,   1],
[-1,  -1,  -1,  -1],
[ 1,   0,   1,   2],
[-1,  -1,  -1,  -2],
[ 0,  -1,   1,   0],
[ 0,   0,  -1,   0],
[ 0,   1,  -1,  -1]];

$E13 =[[ 1, 1, 1, 1], [ 1, 0, 1, 2], [-1, -1, -1, -2], [ 0, -1, 1, 0], [ 0, 1, -1, 0], [ 0, 0, -1, 0], [ 0, 1, -1, -1]];$E23 =[[ 1,   1,   1,   1],
[ 1,   0,   1,   2],
[-1,   0,  -1,  -2],
[-1,  -1,  -1,  -2],
[ 0,  -1,   1,   0],
[ 0,   0,  -1,   0],
[ 0,   1,  -1,  -1]];

# Eij_k means the facet of "i independent from j given k"

$E12_3 =[[ 1, 1, 1, 1], [ 1, 0, 1, 2], [-1, -1, -1, -2], [ 0, -1, 1, 0], [ 0, 0, -1, 0], [ 0, 1, -1, -1], [ 0, -1, 1, 1]];$E13_2 =[[1, 1, 1, 1],
[1, 0, 1, 2],
[-1,-1,-1,-2],
[1, 1, 1, 2],
[0, -1, 1, 0],
[0, 0, -1, 0] ,
[0, 1, -1, -1]];

$E23_1 =[[ 1, 1, 1, 1], [ 1, 0, 1, 2], [-1, -1, -1, -2], [ 0, -1, 1, 0], [ 0, 0, -1, 0], [ 0, 0, 1, 0], [ 0, 1, -1, -1]]; #The three polyhedera that make the image of H in dimension 3.$C1 = [[1,1,1,1],[-1,-2,0,-1],[0,0,-1,0],[0,1,0,-1],[0,-1,0,1]];
$C2 = [[1,1,1,1],[0,-1,1,0],[0,0,-1,0],[1,2,0,1],[1,1,0,2],[-1,-1,0,-2]];$C3 = [[1,1,1,1],[0,0,-1,0],[-1,-1,-1,-2],[0,-1,0,1],[1,2,0,1],[-1,-2,0,-1]];

$P = new Polytope(INEQUALITIES=>$E); $P12 = new Polytope(INEQUALITIES=>$E12); $P13 = new Polytope(INEQUALITIES=>$E13); $P23 = new Polytope(INEQUALITIES=>$E23); $P12_3 = new Polytope(INEQUALITIES=>$E12_3); $P13_2 = new Polytope(INEQUALITIES=>$E13_2); $P23_1 = new Polytope(INEQUALITIES=>$E23_1);

$Ccal1 = new Polytope(INEQUALITIES=>$C1);  $Ccal2 = new Polytope(INEQUALITIES=>$C2);  $Ccal3 = new Polytope(INEQUALITIES=>$C3);

compose($P->VISUAL(FacetTransparency=>0.2),$P12->VISUAL FacetColor=>"red",FacetTransparency=>0.7), $P13->VISUAL(FacetColor=>"blue",FacetTransparency=>0.7),$P23->VISUAL(FacetColor=>"green",FacetTransparency=>0.7), $P12_3->VISUAL(FacetColor=>"orange"),$P13_2->VISUAL(FacetColor=>"yellow"),$P23_1->VISUAL(FacetColor=>"black"),); compose($P->VISUAL(FacetTransparency=>0.2), $P12->VISUAL(FacetColor=>"red", FacetTransparency=>0.7),$P13->VISUAL(FacetColor=>"blue",FacetTransparency=>0.7), $P23->VISUAL(FacetColor=>"green", FacetTransparency=>0.7),$P12_3->VISUAL(FacetColor=>"orange"), $P13_2->VISUAL(FacetColor=>"yellow"),$P23_1->VISUAL(FacetColor=>"black"), $Ccal1->VISUAL,$Ccal2->VISUAL,$Ccal3->VISUAL); compose($P->VISUAL(FacetTransparency=>0.2),$Ccal1->VISUAL(FacetColor=>"green"),$Ccal2->VISUAL(FacetColor=>"green"),\$Ccal3->VISUAL(FacetColor=>"green"));