Katz and Gratias Model

This bulk model of icosahedral Al–Pd–Mn is Katz and Gratias' simplest model of Al–Pd–Mn. The underlying model of atomic positions was developed for icosahedral Al–Cu–Fe in references [1,2]. It is the primitive tiling P decorated by Bergman (and therefore automatically Mackay) polyhedra. The atomic positions in the model are deterministic.

The deterministic atomic positions of this model are defined uniquely by the polyhedral windows (atomic surfaces) Wbc, Wn', and Wn in the coding space Eshown below [1,2]. Each window defines a quasilattice from a corresponding copy of an icosahedrally projected D6 module. These windows corresond to the windows Wa, Wq, and Wb respectively, for the Papadopolos and Kasner model. The model of atomic positions describes both i-Al–Pd–Mn and i-Al–Cu–Fe.


Atomic surfaces (windows) Wbc, Wn', and Wn from left to right. The edge lengths of the triacontahedra Wbc, and Wn areτ-1 and τ, respectively.


Appropriate normals to 2-fold, 5-fold and 3-fold planes in "Katz-Gratias AlPdMn.dat" are as follows:

(0, 0, 1) is 2-fold
τ, 0) is 5-fold
τ, τ-1) is a 3-fold

If you need instructions on how to determine atomic 2-fold, 3-fold, and 5-fold planes orthogonal to the given directions, click HERE.

The file available for downloading is in ASCII format. Specific information about the format can be found at the top of the file.



[1] A. Katz and D. Gratias J. Non-Cryst. Solids, 1993, 153&154, 187.
[2] A. Katz and D. Gratias in Proceedings of the 5th International Conference on Quasicrystals, Ed. by C. Janot and R. Mosseri, World Scientific, Singapore, 1995, 164.