Narrow Results

A fractal tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. In this paper, we investigate the boundary dimension of $s_n$- and $u_n$-tilings. We first derive an explicit recursion formula for the boundary edges of $s_n$- and $u_n$-tilings. Then we present an analytical expression for their fractal boundary dimensions using matrix methods. Results indicate that, as $n$ increases, the boundaries of $s_n$- and $u_n$-tilings will degenerate into general Euclidean curves. The method proposed in this paper can be extended to compute the boundary dimensions of other kinds of fractal tilings.

A fractal tiling or $f$-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of $f$-tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.