Converting non-periodic tilings with Tile(1, 1) into tilings with three types of pentagons, I

TL;DR

This paper demonstrates how non-periodic tilings with Tile(1, 1) can be transformed into non-periodic tilings with three types of pentagons, revealing two distinct patterns without subdividing rhombuses.

Contribution

It introduces a method to convert non-periodic tilings with Tile(1, 1) into pentagon tilings, identifying two resulting patterns while avoiding subdivision of rhombuses.

Findings

Two patterns of pentagon tilings identified

Conversion preserves non-periodicity

Rhombus subdivision is not performed

Abstract

Non-periodic tilings with Tile(1, 1) using the substitution method, as presented by Smith et al. in [2] and [3], can be converted into non-periodic tilings with three types of pentagons. When arbitrary replacements are excluded, the resulting non-periodic tilings with three types of pentagons exhibit two patterns. Note that, during the conversion process in this manuscript, the rhombus is not subdivided into smaller similar rhombuses.