§1 Quasicrystals

§ 1.5. Theory predates discovery of icosahedral long-range order.

The point symmetry of the regular pentagon or the regular icosahedron is forbidden in a periodic crystal of three-dimensional space. Insisting on icosahedral point symmetry, we constructed in 1983 a quasiperiodic long-range repetition pattern with two rhombohedral building blocks and icosahedral point symmetry. Icosahedral point symmetry in diffraction was discovered in 1984 and interpreted as aperiodic long-range order in quasicrystals.

In 1983 I began to study aperiodic tilings obtained by projection. Harald Bohr [2], [3] pp. 111-125 already in 1925 had presented a geometric analysis of quasiperiodic functions. He considered a lattice in n-dimensional Euclidean space, cut by a planar section of dimension m<n, chosen irrational and so not parallel to any net plane of the lattice. Rational sections are excluded since they would again be periodic. Bohr showed that the restriction of a periodic function in n-space to its values on an irrational section would produce a quasiperiodic function. Starting from a \Lambda-periodic function f^{p}(x) defined on all x\in E^{n} with the property eq. 3, and choosing an irrational subspace E^{m}_{{\parallel}} and a fixed point c_{{\perp}} from the orthogonal complement E^{{(n-m)}}_{{\perp}}\in E^{n}, a quasiperiodic function f^{{qp}}(x_{{\parallel}}) is given as the restriction of the periodic function to its values on the irrational section,

\displaystyle x=x_{{\parallel}}+x_{{\perp}}, (6)
\displaystyle f^{{qp}}(x_{{\parallel}}):=f^{p}(x_{{\parallel}}+x_{{\perp}})|_{{x_{{\perp}}=c_{{\perp}}}}=f^{p}(x_{{\parallel}}+c_{{\perp}}).

An idea not discussed by H Bohr was that a non-crystallographic point group can guide the choice of the irrational section. Bohr's analysis, contrary to scaling, provides a Fourier analysis of quasiperiodic functions and assures that they have a pure point Fourier spectrum, see the next subsection. An embedding into a high-dimensional lattice was also introduced by Janner in 1976 [15] to describe what was called modulated crystals.

N G de Bruijn [5], [6] employed pentagonal point symmetry to construct the Penrose rhombus tiling by projection from a lattice in 5 dimensions. Together with my student Roberto Neri I then inquired how the point group of the icosahedron could be made compatible with an n-dimensional lattice. Our answer, technically obtained from the theory of induced representations, was an embedding of icosahedral symmetry into a 6-dimensional hypercubic lattice. This embedding is already minimal, that is, the standard representation of the icosahedral group cannot be embedded into a lattice in a space of dimension smaller than 6. Within 6-dimensional space E^{6}, there is a 3D linear subspace E^{3}, called the parallel space E^{3}=E^{3}_{{\parallel}}, such that the six basis vectors of the hypercubic lattice Z^{6}\in E^{6} project into the six 5-fold axes of the icosahedron. Clearly then this 3-dimensional subspace is transformed into itself under icosahedral rotations and provides the physical space for the tiling construction. Any element of the icosahedral group is presented by a permutation or change of sign of these six axes. Our work [21] (received on Dec 5, 1983) was published in 1984 before the discovery of icosahedral quasicrystals by Shechtman et al [55]. We constructed two icosahedral tiles in the shape of rhombohedra as projections of 3-dimensional boundaries of the 6-dimensional hypercube. The 3D projection of this hypercube is Kepler's triacontahedron [17]. The two rhombohedral tiles are shown in the left lower part of Fig. 10. The tiling was constructed from a hexagrid as described in detail in [26], [22]. In this way we built the first 3-dimensional icosahedral tiling model by projection. This tiling in our present terminology would be denoted as ({\cal T},Z^{6}).

A model of this icosahedral tiling with rhombohedral tiles we constructed in 1985, with substantial computational and technical help by L. Kramer, and in collaboration with H. Weitzel, TU Darmstadt. The model (see picture) was presented at the exhibition Symmetrie in Kunst, Natur und Wissenschaft, Mathildenhöhe Darmstadt 1986, see [48] Band 3 p. 91.

Figure 4. Model of a Quasicrystal constructed 1985 for the exhibition Symmetrie in Kunst, Natur und Wissenschaft, Mathildenhöhe Darmstadt 1986.

In 1984, Shechtman et al in [55] (received on Oct 9, 1984), reported the experimental discovery of material with sharp diffraction peaks of icosahedral point symmetry and interpreted it as evidence for long-range non-periodic order. Developments of the field of quasicrystals are described in [50], [30] and [13].