Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells

Abstract

It is shown that copies of seven elementary cells suffice to fill any region of Euclidean three-dimensional space. The seven elementary cells have four basic convex polyhedral shapes and three of them appear in two different sizes. The space filling is non-periodic, has a central point, and preserves the full icosahedral group.

Crystallography of quasi-crystals

“…Therefore, the symmetry group of the Penrose pattern is the holohedral space group dis- The crystallographic aspects of Penrose tiling have previously been studied by Mackay (1982), who determined the diffraction pattern by optical means. Kramer (Kramer, 1982(Kramer, , 1985Kramer & Ned, 1984) has given a crystallographic discussion of aperiodic tilings that is a generalization of that by de Bruijn.…”

On periodic and non-periodic space fillings ofEmobtained by projection

“…If the icosahedral rotations of the regular dodecahedron are interpreted as permutations of the twelve faces, we obtain the explicit form of this embedding. The enumeration of the dodecahedral faces is taken from Kramer (1982) and leads to Table 1. The representation D of S(12) in ~_-12 is given by the standard permutation matrices.…”

“…For a different association of the icosahedral group to non-periodic space filling of E 3 we refer to Kramer (1982).…”

Tiling Groupoids and Bratteli Diagrams

“…In particular, it was very convenient to represent the structure of these materials by various examples of tilings, such as the Penrose tiling or its 3-dimensional analogs [22,27,32]. It was shown subsequently [24] that such structures are liable to describe precisely the icosahedral phase of quasicrystalline alloys such as AlCuFe.…”