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

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.