Inflation rule for Gummelt coverings with decorated decagons and its implication for quasi-unit-cell models

The equivalence between quasi-unit-cell models and Penrose-tile models on the level of decorations is proved using inflation rules for Gummelt coverings with decorated decagons. Owing to overlaps, Gummelt arrangement of decorated decagons gives rise to nine different (context-dependent) decagon decorations in the covering. The inflation rules for decagons for each of nine types are presented and it is shown that inflations from differently typed decagons always produce different decorations of inflated decagons. However, if the original decagon region is divided into 'equivalent' rhombus Penrose tiles, typed-decagon arrangements in the tiles (of the same shape) become identical for the fourfold inflated decagons. This implies that a decagonal quasi-unit-cell model can be reinterpreted as a Penrose-tile model with fourfold deflated supertiles.