A while back the fact that no regular dodecahedron can be placed in R^3 with all vertices having integer coordinates came up here. Here, just for fun, is a near miss: The faces are congruent planar non-regular pentagons each with face angles: [108.0000197, 107.9999984, 107.9999918, 107.9999918, 107.9999984], as compared to a regular pentagon with all face angles 108 degrees. This irregular dodecahedron has vertices with integer x,y,z coordinates: 2550409 2550409 2550409 4126648 0 1576240 1576240 4126648 0 2550409 2550409 -2550409 4126648 0 -1576240 0 1576240 4126648 0 -1576240 4126648 2550409 -2550409 2550409 -1576240 4126648 0 -2550409 2550409 2550409 0 1576240 -4126648 0 -1576240 -4126648 2550409 -2550409 -2550409 -2550409 2550409 -2550409 -2550409 -2550409 2550409 -4126648 0 1576240 -2550409 -2550409 -2550409 -1576240 -4126648 0 1576240 -4126648 0 -4126648 0 -1576240 and is the convex hull of those points.