Narrow Results

Not much is known about the topological structure of a connected self-similar tile whose interior is disconnected, and even less is understood if the interior consists of infinitely many components. We introduce a technique to show that for a large class of self-similar tiles in ℝ², the closure of each component of the interior is homeomorphic to a disk. This allows us to prove such a result for the Eisenstein set, the fundamental domain of a well-known quadratic canonical number system, and some other well-known fractal tiles.

In this paper, we present an idea of creating fractals by using the geometric arc as the basic element. This approach of generating fractals, through the tuning of just three parameters, gives a universal way to obtain many novel fractals including the classic ones. Although this arc-fractal system shares similar features with the well-known Lindenmayer system, such as the same set of invariant points and the ability to tile the space, they do have different properties. One of which is the generation of pseudo-random number, which is not available in the Lindenmayer system. Furthermore, by assuming that coastline formation is based purely on the processes of erosion and deposition, the arc-fractal system can also serve as a dynamical model of coastal morphology, with each level of its construction corresponds to the time evolution of the shape of the coastal features. Remarkably, our results indicate that the arc-fractal system can provide an explanation on the origin of fractality in real coastline.

The object under study is a particular closed and simple curve on the square lattice ℤ² related with the Fibonacci sequence F_n. It belongs to a class of curves whose length is 4F_3n+1, and whose interiors tile the plane by translation. The limit object, when conveniently normalized, is a fractal line for which we compute first the fractal dimension, and then give a complexity measure.

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.