Continuous rotation from cubic to icosahedral order

Al-Mn alloys display the competition of condensed matter phases with periodic cubic and non-periodic icosahedral order respectively. Both types of order are connected by a single continuous rotation M(β) in the hypercubic lattice of 6-space projected to 3-space. The rotation M(β) results from Schur's lemma [Schur (1905). Sitzungsber. Preuss. Akad. Wiss. pp. 406-432] applied to the cubic and icosahedral point groups. For 0 ≤ β < π) it preserves tetrahedral symmetry. For β^c = 0° one finds cubic symmetry, for βⁱ = 13.28° icosahedral symmetry. Implications for the Al-Mn structure are presented.