Abstract. This paper is the last of a series of three articles presenting a classification of Vornoi and Delone tilings determined by point sets Σ(Ω) ('quasicrystals'), built by the standard projection of the root lattice of type A₄ to a two-dimensional plane spanned by the roots of the Coxeter group H₂ (dihedral group of order 10). The acceptance window Ω for Σ(Ω) in the present paper is a regular decagon of any radius 0 < r < ∞. There are 14 distinct VT sets of Voronoi tiles and 6 sets DT of Delone tiles, up to a uniform scaling by the factor and . The number of Voronoi tiles in different quasicrystal tilings varies between 3 and 12. Similarly, the number of Delone tiles is varying between 4 and 6. There are 7 VT sets of the 'generic' type and 7 of the 'singular' type. The latter occur for seven precise values of the radius of the acceptance window. Quasicrystals with acceptance windows with radii in between these values have constant VT sets, only the relative densities and arrangement of the tiles in the tilings change. Similarly, we distinguish singular and generic sets DT of Delone tiles.

PACS numbers: 02.20.Bb, 61.44.Br

Print publication: Issue 9 (4 March 2005)
Received 8 July 2004, in final form 4 November 2004
Published 16 February 2005