The Gummelt decagon as a 'quasi unit cell'

Steinhardt, Jeong, Saitoh, Tanaka, Abe & Tsai [Nature (London) (1998), 396, 55-57] have demonstrated that the structure of decagonal Al-Ni-Co can be built from overlapping clusters of a single type. The structure arises from a decoration of the decagons of a Gummelt covering. The unit (essentially a decagonal prism) was called by Steinhardt et al. a 'quasi unit cell'. In this work, a classification scheme is proposed for 'G patterns' - quasiperiodic patterns obtained by decorating a decagonal quasi unit cell. The classification makes use of the fact that G patterns can also be derived from decoration of a tiling. The tiles are analogues, for decagonal quasiperiodic patterns, of the 'asymmetric units' of a periodic pattern; they provide a simple mode of description and classification of the 'Gummelt-type structures'. Four existing models for decagonal phases are considered from this viewpoint