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

TL;DR

This paper establishes a formal connection between quasi-unit-cell models and Penrose-tile models using inflation rules for decorated Gummelt coverings with decagons, revealing how these models relate through inflation and decoration patterns.

Contribution

It introduces inflation rules for decorated decagons in Gummelt coverings and demonstrates their equivalence to Penrose-tile models through decoration and inflation analysis.

Findings

Inflation rules for nine types of decorated decagons are derived.

Different decagon types produce distinct decorations upon inflation.

Decagon arrangements become identical within tiles after fourfold inflation, linking quasi-unit-cell and Penrose models.

Abstract

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. Due 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 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 super-tiles.