A higher dimensional description of the structure of β-Mn

The structure of β-Mn crystallizes in space group P4 ₁32. The pseudo 8-fold nature of the 4 ₁ axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually perpendicular 8-fold axes cannot generate P4 ₁32 on projection to 3-d space and how this may instead be accomplished from a six-dimensional model. A procedure for generating the actual structure of β-Mn lifted to six-dimensional space is given.