Is it a pyritohedron? On Mon, Sep 9, 2013 at 10:11 AM, James Buddenhagen <jbuddenh@gmail.com> wrote:

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.

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