Abstract
Abstract
The α AlMnSi crystal structure consists of two kinds of polyhedra: Mackay icosahedra (MI) and double Mackay icosahedra (DMI). An MI and a DMI both have icosahedral symmetry and consist of 54 and 156 atoms respectively. They are similar to each other, but a DMI is 1•618 times bigger than an MI. If these polyhedra connect with each other along the two-, three- and fivefold axes, an AlMnSi quasicrystal can be built up. We term the MI and the DMI the first-order icosahedron and the first-order DMI respectively. The second-order icosahedron consists of 12 first-order DMIs and 20 first-order icosahedra. The second-order DMI consists of 31 first-order DMIs and 32 first-order icosahedra. The second-order icosahedron and DMI are 2•618 times bigger than the first-order ones. In general, the Nth-order icosahedron consists of 12 (N−1)th-order DMIs and 20 (N−1)th-order icosahedra; the Nth-order DMI, 31 (N−1)th-order DMIs and 32 (N−1)th-order icosahedra. The Nth-order icosahedron and DMI are 2•618 times bigger than the (N−1)th ones. This can go on continuously to fill expanding space.