Structure Factor for Generalized Penrose Tiling

M. Chodyń1, P. Kuczera1,2 and J. Wolny1

1Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Mickiewicz Avenue 30, 30-059 Krakow, Poland
2Laboratory of Crystallography, ETH Zurich, Wolfgang-Pauli-Strasse 10, Zurich, CH-8093, Switzerland

The Generalized Penrose Tiling (GPT)[1,2] can be considered a promising alternative for Penrose Tiling (PT) as a model for decagonal quasicrystal refinement procedure, particularly in the statistical approach (also called the Average Unit Cell (AUC) approach) [3]. The statistical method using PT has been successfully applied to the structure optimization of various decagonal phases [4]. The application of the AUC concept to the GPT will be presented.

In the higher dimensional (nD) approach, PT is obtained by projecting a 5D hypercubic lattice through a window consisting of four pentagons, called the atomic surfaces (ASs), in the perpendicular space. The vertices of these pentagons together with two additional points form a rhombicosahedron. The GPT is created by projecting the 5D hypercubic lattice through a window consisting of five polygons, generated by shifting the ASs along the body diagonal of the rhombicosahedron. Three of the previously pentagonal ASs will become decagon, one will remain pentagonal and one more pentagon will be created (for PT it is a single point). The structure of GPT will depend on the shift parameter, its building units are still thick and thin rhombuses, but the matching rules and the tiling changes. Diffraction pattern of GPT have peaks in the same positions as regular PT, however their intensities are different.

Binary decagonal quasicrystal structure with arbitrary decoration for a given shift value was simulated. Its diffraction pattern was calculated using AUC method[5,6]. Generated diffraction pattern were used as "experimental data set" in structure refinement algorithm made to test the refining of shift parameter.


1.       K. N. Ishihara, A. Yamamoto, Acta Cryst. (1988), A44, 508-516

2.       Pavlovitch, M. Kleman, J Phys a-Math Gen. (1987), 20, 687-702

3.       J. Wolny, Phil. Mag. (1998), 77, 395-414

4.       P. Kuczera, J. Wolny, W. Steurer, Acta Cryst. (2012), B68, 578–589 

5.       B. Kozakowski, J. Wolny, Acta Cryst (2010), A66, 489-498

6.       M. Chodyn, P. Kuczera, J. Wolny, Acta Cryst. (2015), A71, 161–168