Fractal Tilings Based on Successive Adjacent Substitution Rule

A fractal tiling or -tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. -tilings have complicated structures and strong visual appeal. However, so far, the discovered -tilings are very limited since constructing such -tilings needs special talent. Based on the idea of hierarchically subdividing adjacent tiles, this paper presents a general method to generate -tilings. Penrose tilings are utilized as illustrators to show how to achieve it in detail. This method can be extended to treat a large number of tilings that can be constructed by substitution rule (such as chair and sphinx tilings and Amman tilings). Thus, the proposed method can be used to create a great many of -tilings.