Abstract
In the literature, we can find various methods for generating artistic patterns. One of the methods is the orbit trap method. In this paper, we propose various modifications of a variant of the orbit trap method that generates patterns with wallpaper symmetry. The first modification relies on replacing the Picard iteration (used in the original method) with the S-iteration known from the fixed point theory. Moreover, we extend the parameters in the S-iteration from scalar to vector ones. In the second modification, we replace the Euclidean metric used in the orbit traps with other metrics. Finally, we propose three new orbit traps. The presented examples show that using the proposed method, we are able to obtain a great variety of interesting patterns. Moreover, we show that a proper selection of the orbit traps and the mapping used by the method can lead to patterns that possess a local fractal structure.
Highlights
A pattern is a generic term for any type of repeated, often regular, arrangement [1]
Based on the results presented in [15] and the orbit trap method [16], Lu et al in [17] introduced a method that is the main subject of this paper, namely, the orbit trap method that generates patterns with wallpaper symmetry
We introduce modifications of the orbit trap method presented by Lu et al in [17]
Summary
A pattern is a generic term for any type of repeated, often regular, arrangement [1]. A motif can be repeated and arranged in many ways to create different types of patterns. Motifs can be arranged in many ways to create a regular pattern. We can find various methods for generating patterns that use mathematical equations. In the generation of patterns, various types of non-Euclidean geometries are used Mathematical objects such as whirls [9] or spirals [10,11] are commonly used in obtaining artistic patterns. We introduce modifications of the orbit trap method presented by Lu et al in [17]. We present examples showing that we can obtain a local fractal structure in the generated pattern when using an appropriate mapping and orbit traps.
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