Abstract
A fractal tiling or f‐tiling is a tiling which possesses self‐similarity and the boundary of which is a fractal. f‐tilings have complicated structures and strong visual appeal. However, so far, the discovered f‐tilings are very limited since constructing such f‐tilings needs special talent. Based on the idea of hierarchically subdividing adjacent tiles, this paper presents a general method to generate f‐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 f‐tilings.
Highlights
IntroductionThe investigation of tilings is one of the most ancient parts of mathematics
In many ways, the investigation of tilings is one of the most ancient parts of mathematics
Except for (i + 1)th-layer tiles, denote by ∑i+1 tiles as the other substituted tiles contained in ∑i tiles, i = 4, 5, 6
Summary
The investigation of tilings is one of the most ancient parts of mathematics. Arts based on f -tilings look more visually pleasing (see two designs demonstrated in Figure 2) [15] Due to their aesthetic attraction, f -tilings have attracted much attention. Does there exist a general method to create f -tilings Well, we believe it is a quite attractive question. The subject of Penrose tilings has attracted much attention since Roger Penrose discovered them [19] and has been investigated extensively [20]. It is a simple but remarkable tiling used in demonstrating aperiodicity.
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