Here's the "simpler" form that might work with the rhombic dodecahedron:
points = {{0, 0, 0}, {12, 0, 0}, {0, 0, 12}, {0, 12, 0}, {3, 3, -3}, {3, -3, 3}, {-3, 3, 3}, {9, 9, 3}, {9, 3, 9}, {3, 9, 9}, {8, 8, 8}}/6;
ConvexHullMesh[points]
This could work with the following dodecahedron:
Select[Tuples[Range[-2,2],{3}], MemberQ[{{1,1,1},{0,0,2}},Sort[Abs[#]]]&]
