The mystery of perpendicular fivefold axes and the fourth dimension in intermetallic structures

The structures of eight related known intermetallic structure types are the impetus to this paper: Li21Si5, Mg44Rh7, Zn-13(Fe,N)(2), Mg6Pd, Na6Tl, Zn91I11, Li13Na29Ba19, and Al69Ta39. All belong to the F (4) over bar 3m space group, have roughly 400 atoms in their cubic unit cells, are built up at least partially from the gamma-brass structure, and exhibit pseudo-tenfold symmetric diffraction patterns. These pseudo-tenfold axes lie in the < 110 > directions, and thus present a paradox. The < 110 > set is comprised of three pairs of perpendicular directions. Yet no 3D point group contains a single pair of perpendicular fivefold axes (by Friedel's Law, a fivefold axis leads to a tenfold diffraction pattern). The current work seeks to resolve this paradox. Its resolution is based on the largest of all 4D Platonic solids, the 600-cell. We first review the 600-cell, building an intuition discussing 41) polyhedroids (4D polytopes). We then show that the positions of common atoms in the F (4) over bar 3m structures lie close to the positions of vertices in a 3D projection of the 600-cell. For this purpose, we develop a projection method that we call intermediate projection. The introduction of the 600-cell resolves the above paradox. This 4D Platonic solid contains numerous orthogonal fivefold rotations. The six fivefold directions that are best preserved after projection prove to lie along the < 110 > directions of the F (4) over bar 3m structures. Finally, this paper shows that at certain ideal projected cluster sizes related to one another by the golden mean (tau = (1+ root 5)/2), constructive interference leading to tenfold diffraction patterns is optimized. It is these optimal values that predominate in actual F (4) over bar 3m structures. Explicit comparison of experimental cluster sizes and theoretically derived cluster sizes shows a clear correspondence, both for isolated and crystalline pairs of projected 600-cells.