Taking the 32 vertex 6 Demicube with an even number of -1 elements and projecting with 3 of the 4 rows of the H4 folding matrix gives an dodecahedron hull with 12 vertices and 30 edges on the hull out of 240 edges of 6D length Sqrt[2]).

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While taking the 32 vertex 6 Demicube with odd number of -1 elements and projecting with 3 of the 4 rows of the H4 folding matrix gives the icosahedron with 20 vertices and 30 edges on the hull out of 240 edges of 6D length Sqrt[2]).