International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 610611
Section 4.6.3.3.2.2. Diffraction symmetry ^{a}Laboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, WolfgangPauliStrasse 10, CH8093 Zurich, Switzerland 
The diffraction symmetry of decagonal phases can be described by the Laue groups or . The set of all vectors H forms a Fourier module of rank 5 in physical space which can be decomposed into two submodules . corresponds to a module of rank 4 in a 2D subspace, corresponds to a module of rank 1 in a 1D subspace. Consequently, the first submodule can be considered as a projection from a 4D reciprocal lattice, , while the second submodule is of the form of a regular 1D reciprocal lattice, . The diffraction pattern of the Penrose tiling decorated with equal point scatterers on its vertices is shown in Fig. 4.6.3.17. All Bragg reflections within are depicted. Without intensitytruncation limit, the diffraction pattern would be densely filled with discrete Bragg reflections. To illustrate their spatial and intensity distribution, an enlarged section of Fig. 4.6.3.17 is shown in Fig. 4.6.3.18. This picture shows all Bragg reflections within . The projected 4D reciprocallattice unit cell is drawn and several reflections are indexed. All reflections are arranged along lines in five symmetryequivalent orientations. The perpendicularspace diffraction patterns (Figs. 4.6.3.19 and 4.6.3.20) show a characteristic starlike distribution of the Bragg reflections. This is a consequence of the pentagonal shape of the atomic surfaces: the Fourier transform of a pentagon has a starlike distribution of strong Fourier coefficients.

Schematic diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_{r} = 4.04 Å). All reflections are shown within and . 

The perpendicularspace diffraction pattern of the Penrose tiling (edge length of the Penrose unit rhombs a_{r} = 4.04 Å). All reflections are shown within and . 
The 5D decagonal space groups that may be of relevance for the description of decagonal phases are listed in Table 4.6.3.1. These space groups are a subset of all 5D decagonal space groups fulfilling the condition that the 5D point groups they are associated with are isomorphous to the 3D point groups describing the diffraction symmetry. Their structures are comparable to 3D hexagonal groups. Hence, only primitive lattices exist. The orientation of the symmetry elements in the 5D space is defined by the isomorphism of the 3D and 5D point groups. However, the action of the tenfold rotation is different in the subspaces and : a rotation of in is correlated with a rotation of in . The reflection and inversion operations are equivalent in both subspaces.
