A new method to analyse mosaics based on Symmetry Group theory applied to Islamic Geometric Patterns

[EN] This article presents a new method for analysing mosaics based on the mathematical principles of Symmetry Groups. This method has been developed to get the understanding present in patterns by extracting the objects that form them, their lattice, and the Wallpaper Group. The main novelty of this method resides in the creation of a higher level of knowledge based on objects, which makes it possible to classify the objects, to extract their main features (Point Group, principal axes, etc.), and the relationships between them. In order to validate the method, several tests were carried out on a set of Islamic Geometric Patterns from different sources, for which the Wallpaper Group has been successfully obtained in 85% of the cases. This method can be applied to any kind of pattern that presents a Wallpaper Group. Possible applications of this computational method include pattern classification, cataloguing of ceramic coatings, creating databases of decorative patterns, creating pattern designs, pattern comparison between different cultures, tile cataloguing, and so on.The authors wish to thank the Patronato de la Alhambra y Generalife (Granada, Spain) and the Patronato del Real Alcázar de Sevilla (Seville, Spain) for their valuable collaboration in this research work.Albert Gil, FE.; Gomis Martí, JM.; Blasco, J.; Valiente González, JM.; Aleixos Borrás, MN. (2015). A new method to analyse mosaics based on Symmetry Group theory applied to Islamic Geometric Patterns. Computer Vision and Image Understanding. 130:54-70. doi:10.1016/j.cviu.2014.09.002S547013

Pembangkitan pola menggunakan konsep grup kertas dinding

Pola berulang dan simetris berdimensi dua sering dijumpai dalam kehidupan sehari-hari, seperti pada motif kain, paving jalan, barisan tapak kaki seseorang di pasir. Pola berulang dan simetris tersebut terbentuk dari transformasi, yang terdiri dari translasi, rotasi, refleksi, dan refleksi-glide. Dalam matematika, pola berulang dan simetris pada bidang datar, yang terbentuk dari transformasi, termasuk dalam grup simetri bidang datar. Grup simetri bidang datar ini sering disebut juga sebagai grup kristalografi bidang datar atau grup kertas dinding (wallpaper group). Terdapat 17 grup kertas dinding yang dikenal. Tujuan dari penelitian ini adalah membangkitkan berbagai macam pola dari suatu pola dasar dengan menerapkan konsep grup kertas dinding. Penelitian dilakukan dengan (1) menentukan dan membuat suatu pola dasar yang digunakan untuk membentuk pola lain, (2) membangkitkan 17 pola yang dibentuk dari 17 grup kertas dinding, dan (3) mengidentifikasi pola gambar yang diperoleh dari internet termasuk ke dalam salah satu grup kertas dinding. Pola dasar yang digunakan dalam penelitian ini adalah spiral. Hasil yang didapat berupa 17 motif spiral yang berbeda dan juga empat motif yang diambil dari internet yang dapat dikenali sebagai salah satu pola dari 17 pola grup kertas dinding.Two-dimensional repetitive and symmetrical patterns are found in everyday life, such as in cloth motifs, decorative art, road pavement design, and floor tiles. The repetitive and symmetrical patterns are formed by geometric transformation, which comprises translation, rotation, reflection, and glide reflection. In mathematics, the study of repetitive patterns on a plane, formed by geometric transformations, belongs to the plane symmetry group. This plane symmetry group is often referred to as plane crystallographic group or wallpaper group that comprises 17 known patterns. The purpose of this research is to generate various patterns from a basic pattern by applying the concept from the   wallpaper group. Furthermore, this research also identifies four cloth motifs and categorizes them to the wallpaper group. This research was conducted by: (1) determining and creating a basic pattern that will be used to form other patterns, (2) generating 17 patterns from the basic pattern based on the patterns in the wallpaper group, (3) identifying some existing motifs downloaded from the Internet and categorize them to the wallpaper group. The basic pattern used in this research is a spiral pattern. The results obtained from this research are the 17 different spiral motifs as well as the identification of four motifs obtained from the Internet to one of the 17 patterns of wallpaper group