Drawing a Truncated Octahedron

Question

I'm trying to draw a truncated octahedron in MATLAB. This is also known as a permutahedron so my strategy is to link up all the vertices via adjacent transpositions of permutations in $S_4$. What I would like is this:

Here's the code:

%Size of S_n
n=4;
%Identity permutation
p=1:n;
%Matrix of Permutations as rows
Pmat=perms(p);
%Total number of perms
Nsize=factorial(n);

%Adjacency matrix=1 when two perms are linked by an adjacent transposition
Padj=zeros(Nsize,Nsize);
for i=1:(Nsize-1)
    for j=(i+1):Nsize
        p1=Pmat(i,:);
        p2=Pmat(j,:);
        for k=1:(n-1);
            p1temp=p1; 
            p1temp(k)=p1(k+1);
            p1temp(k+1)=p1(k);
            if p1temp==p2
                Padj(i,j)=1;
            end
        end
    end
end

%Make it symmetric 
Padj=(Padj+Padj');

if n==4

    figure(1)
    %first vertex Pmat(1,:) has coords [d,0,0,0]
    d=-1;


    %Specify rotation matrix to get coords of other points (want to kill
    %last entry to make all coords [a,b,c,0]

    R=0.5*[-3/d,-1/d,1/d,3/d; 1/d -3/d 3/d -1/d; 1, 3, 3, 1; 1,1,1,1];

    %Center permutahedron at origin
    Pmat_shift=Pmat-5/2;

    %Calculate all other coords
    Pmat_coord=zeros(Nsize,n);
    for i=1:Nsize
        Pmat_coord(i,:)=(R*(Pmat_shift(i,:)'))';
    end

    %Plot all points
    hold on
    for i=1:Nsize
        scatter3(Pmat_coord(i,1),Pmat_coord(i,2),Pmat_coord(i,3),'o')
        text(Pmat_coord(i,1),Pmat_coord(i,2),Pmat_coord(i,3),['  ',num2str(Pmat(i,:))])
    end

    %draw all edges
    for i=1:Nsize
        for j=(i+1):Nsize
            if Padj(i,j)==1        
                plot3(Pmat_coord([i,j],1),Pmat_coord([i,j],2),Pmat_coord([i,j],3));
            end
        end
    end
    hold off

end
axis square;

Here's the result:

So something is very wrong. The edges connect the right pairs of vertices but something is still off. I think what's going on here is that I need to look at the inverse permutation of each vertex to get things right but, I'm not sure why this is necessary since I thought that by definition, the truncated octahedron is the projection from 4 dimensions into 3 dimensions of all coordinate permutations of (1,2,3,4), subsequently normalized. What's going here?

Answer 1

Please take a look at this tutorial posted at this link.

http://blogs.mathworks.com/graphics/2016/01/29/tiling-hexagons-and-other-permutohedra/

I think it does exactly what you want but incorporates graph theoretical concepts for a cleaner presentation.