pts4= Point @@@ Permutations[Range[4]]

mat={{1,-1,0,0}, {0,1,-1,0}, {0,0,1,-1}};mat=Drop[Orthogonalize[{{1,1,1,1},{1,-1,0,0}, {0,1,-1,0}, {0,0,1,-1}}], 1]

pts3=pts4 /. Point[xs__] :> Point @@ (
mat.{xs}
)

l=Length[pts3];
Graphics3D[{EdgeForm[None],
Table[
Polygon[List @@@ pts3[[{i,j,k}]]],
{i,1,l-2}, {j, i+1,l-1}, {k, j+1, l}
]},
Boxed->False
]

Graphics3D[ConvexHullMesh[pts3/.Point->List], Boxed->False]

l=Length[pts3];
Rasterize[Graphics3D[
Table[
Polygon[List @@@ pts3[[{i,j,k}]]],
{i,1,l-2}, {j, i+1,l-1}, {k, j+1, l}
]
]]

PolyhedronData["Dodecahedron"]

bhex={{1,2,3}, {1,3,2},{2,3,1},{3,2,1},{3,1,2},{2,1,3}};
Graphics3D[Polygon[bhex]]

Graphics3D[
Join[
Table[
Polygon[Insert[#, 4, i]& /@ bhex],
{i,4}
],
Table[
Polygon[Insert[#, 1, i]& /@ (1+bhex)],
{i,4}
]
] /. {x1_, x2_, x3_, x4_} :> mat.{x1,x2,x3,x4}
]

Graphics3D[{Opacity[0.7],
Join[
Table[
Polygon[Insert[#, 4, i]& /@ bhex],
{i,4}
],
Table[
Polygon[Insert[#, 1, i]& /@ (1+bhex)],
{i,4}
],
{
Polygon[{{1,2,3,4}, {1,2,4,3}, {2,1,4,3}, {2,1,3,4}}],
Polygon[{{1,3,2,4}, {1,4,2,3}, {2,4,1,3}, {2,3,1,4}}],
Polygon[{{1,4,3,2}, {1,3,4,2}, {2,3,4,1}, {2,4,3,1}}],
Polygon[{{3,1,2,4}, {4,1,2,3}, {4,2,1,3}, {3,2,1,4}}],
Polygon[{{4,2,3,1}, {3,2,4,1}, {3,1,4,2}, {4,1,3,2}}],
Polygon[{{3,4,1,2}, {4,3,1,2}, {4,3,2,1}, {3,4,2,1}}]
}
] /. {x1_Integer, x2_Integer, x3_Integer, x4_Integer} :> mat.{x1,x2,x3,x4}
}
]