Wallpaper Groups Research Articles

Deformed Lattice Detection in Real-World Images Using Mean-Shift Belief Propagation

We propose a novel and robust computational framework for automatic detection of deformed 2D wallpaper patterns in real-world images. The theory of 2D crystallographic groups provides a sound and natural correspondence between the underlying lattice of a deformed wallpaper pattern and a degree-4 graphical model. We start the discovery process with unsupervised clustering of interest points and voting for consistent lattice unit proposals. The proposed lattice basis vectors and pattern element contribute to the pairwise compatibility and joint compatibility (observation model) functions in a Markov Random Field (MRF). Thus, we formulate the 2D lattice detection as a spatial, multitarget tracking problem, solved within an MRF framework using a novel and efficient Mean-Shift Belief Propagation (MSBP) method. Iterative detection and growth of the deformed lattice are interleaved with regularized thin-plate spline (TPS) warping, which rectifies the current deformed lattice into a regular one to ensure stability of the MRF model in the next round of lattice recovery. We provide quantitative comparisons of our proposed method with existing algorithms on a diverse set of 261 real-world photos to demonstrate significant advances in accuracy and speed over the state of the art in automatic discovery of regularity in real images.

Hyperbolization of Euclidean Ornaments

In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that are discussed in this article. In particular, it is discussed which hyperbolic symmetry groups are suitable generalizations of Euclidean wallpaper groups. Furthermore it is shown how one can take advantage of methods from discrete differential geometry in order to perform the conformal deformation of the fundamental region. Finally it is demonstrated how a reverse pixel lookup strategy can be used to obtain hyperbolic images with optimal resolution.

Planar wallpaper group metamaterials for novel terahertz applications

We present novel metamaterial structures based upon various planar wallpaper groups, in both hexagonal and square unit cells. An investigation of metamaterials consisting of one, two, and three unique sub-lattices with resonant frequencies in the terahertz (THz) was performed. We describe the theory, perform simulations, and conduct experiments to characterize these multiple element metamaterials. A method for using these new structures as a means for bio/chemical hazard detection, as well as electromagnetic signature control is proposed.

Islamic geometrical patterns indexing and classification using discrete symmetry groups

In this article, we propose a general computational model for the extraction of symmetry features of Islamic geometrical patterns' (IGP) images. We describe IGP images using the discrete symmetry groups theory. Our model contains the three following steps. (1) By noting that these patterns fall into three major categories, we begin our indexation process by classifying every pattern into one of these categories. The first pattern category describes all the patterns generated by translation along one direction. Every pattern of this category can be classified into one of the seven Frieze groups. The second type of pattern contains translational symmetries in two independent directions. Patterns of this category can be classified into one of the seventeen Wallpaper groups. The last type, called rosettes, describes patterns which begin at a central point and grow radially outward. We use rosette symmetry groups to classify patterns of this latter category. (2) For every pattern, we extract the symmetry features, namely, the symmetry group and the fundamental region, which is a representative region in the image from which the whole image can be regenerated. But for rosette groups, we can also compute the number of folds. (3) Finally, we describe the fundamental region by a simple color histogram and build the feature vector which is a combination of the symmetry feature (defined in the second step) and histogram information. Experiments show promising results for either IGP images' classification or indexing. Efforts for the subsequent task of classifying Islamic geometrical patterns' images can be significantly reduced.

Multidimensional Manhattan Street Networks

We formally define the $n$-dimensional Manhattan street network $M_n$—a special case of an $n$-regular digraph—and we study some of its structural properties. In particular, we show that $M_n$ is a Cayley digraph, which can be seen as a subgroup of the $n$-dimensional version of the wallpaper group $pgg$. These results induce a useful new representation of $M_n$, which can be applied to design a local (shortest-path) routing algorithm and to study some other metric properties, such as the diameter. We also show that the $n$-dimensional Manhattan street networks are Hamiltonian and, in the standard case (that is, in dimension two), we give sufficient conditions for a $2$-dimensional Manhattan street network to be decomposable into two arc-disjoint Hamiltonian cycles.

Motif-based defect detection for patterned fabric

This paper proposes a generalized motif-based method for detecting defects in 16 out of 17 wallpaper groups in 2D patterned texture. It assumes that most patterned texture can be decomposed into lattices and their constituents—motifs. It then utilizes the symmetry property of motifs to calculate the energy of moving subtraction and its variance among different motifs. By learning the distribution of these values over a number of defect-free patterns, boundary conditions for discerning defective and defect-free patterns can be determined. This paper presents the theoretical foundation of the method, and defines the relations between motifs and lattice, from which a new concept called energy of moving subtraction is derived using norm metric measurement between a collection of circular shift matrices of motif and itself. It has been shown in this paper that the energy of moving subtraction amplifies the defect information of the defective motif. Together with its variance, an energy-variance space is further defined where decision boundaries are drawn for classifying defective and defect-free motifs. As the 16 wallpaper groups of patterned fabric can be transformed into three major groups, the proposed method is evaluated over these three major groups, from which 160 defect-free lattices samples are used for defining the decision boundaries, with 140 defect-free and 113 defective samples used for testing. An overall detection success rate of 93.32% is achieved for the proposed method. No other generalized approach can achieve this success rate has been reported before, and hence this result outperforms all other previously published approaches.

The Multidimensional Manhattan Network

The n-dimensional Manhattan network Mn—a special case of n-regular digraph—is formally defined and some of its structural properties are studied. In particular, it is shown that Mn is a Cayley digraph, which can be seen as a subgroup of the n-dim version of the wallpaper group pgg. These results induce a useful new presentation of Mn, which can be applied to design a (shortest-path) local routing algorithm and to study some other metric properties. Also it is shown that the n-dim Manhattan networks are Hamiltonian and, in the standard case (that is, dimension two), they can be decomposed in two arc-disjoint Hamiltonian cycles. Finally, some results on the connectivity and distance-related parameters of Mn, such as the distribution of the node distances and the diameter are presented.

Symmetries of projected wallpaper patterns

In this paper we study periodic functions of one and two variables that are invariant under a subgroup of the Euclidean group. Starting with a function defined on the plane we obtain a function of one variable by two methods: we project the values of the function on a strip into its edge, by integrating along the width; and we restrict the function to a line. If the functions had been obtained by solving a partial differential equation equivariant under the Euclidean group, how do their symmetries compare to those of solutions of equations formulated directly in one dimension? Some of the symmetries of projected and of restricted functions can be obtained knowing the symmetries of the original functions only. There are also some extra symmetries arising for special widths of the strip and for some special positions of the line used for restriction. We obtain a general description of the two types of symmetries and discuss how they arise in the wallpaper groups (crystalographic groups of the plane). We show that the projections and restrictions of solutions of p.d.e.s in the plane may have symmetry groups larger than those of solutions of problems formulated in one dimension.

Making Quilts without Sewing: Investigating Planar Symmetries in Southern Quilts

An activity for “making quilts without sewing” that enables high school students to develop their understanding of planar symmetries and wallpaper patterns. This activity incorporates the culture and traditions of quilting into the study of geometry. From the same block, students can make quilts with different patterns by using various combinations of transformations, in addition to the possible combinations of transformations that might have been used to create a quilt or wallpaper pattern. Students may also reflect upon the cultural activity of quilt makers and the art of quilt making.

Crystallography of two-dimensional photonic lattices formed by holography of three noncoplanar beams

A systematically theoretical study on how to form two-dimensional photonic lattices with various plane groups by three elliptically polarized beams is presented. It is shown that nine plane groups can be formed in the photonic lattices by use of an intuitionistic intensity pattern-superposition method; however, we demonstrate that the other eight plane groups cannot be constructed. A phase shift associated with the interference intensity and the elliptic polarization is derived, and a relevant formula for interference intensity is deduced. The phase shift can be used to obtain the lower symmetries in some wallpaper groups such as p1, pm, cm, and p3m1 without introducing additional undesired symmetries. This analysis may lay the foundation for the study of space groups in holographic three-dimensional photonic crystals and multidimensional photonic quasicrystals.

Optical manifestations of planar chirality.

We report that planar chiral structures affect the polarization state of light in a manner similar to three-dimensional chiral (optical active) media. In experiments with artificial metal-on-silicon chiral planar gratings of 442 wallpaper group symmetry, containing millions of chiral elements per square centimeter, we observed rotation of the polarization azimuth in excess of 30 degrees of light diffracted from it. The rotation was found to change its sign for two enantiomeric forms of the media and to have components associated with both the structural arrangement and the chirality of individual structural elements.

Motion Sickness Symptoms and Perception of Self Motion from Exposure to Different Wallpaper Patterns

Vection refers to the compelling visual illusion of self-motion experienced in virtual environments by stationary individuals when viewing moving visual surrounds. Early research with these perceptions generally employed a vertically striped drum of about 1.5 meters in diameter. The illusion of self-motion or vection generated in these optokinetic drums is particularly useful for studying the effects of optic flow imagery in virtual environments because the subject determines the precise time at which the vection occurs and the nauseogenic experience is stopped; thus problems of motion sickness, nausea, and vomiting can be monitored and/or minimized. This vection, presumably related to the reflexive optokinetic nystagmus (OKN), experimentally presents a non-invasive neural visual response which may be safely used for the investigation of possible motion sickness symptoms ranging from the so called Sopite syndrome to the nausea and vomiting which may be experienced in some virtual environments. Further, to be able to examine the effects of color, texture, and differing scene content, we constructed a vection drum with easily reconfigurable displays. Therefore, for stimulus material, instead of the conventional black and white stripes, we employed differing patterns of household wallpaper. Results showed that, while equal amounts of vection were produced with the wallpaper, one pattern appeared to produce less sickness.

Tessellations with symmetries of the wallpaper groups and the modular group in the hyperbolic 3-space from dynamics

Automatic generation of colored patterns in the hyperbolic 3-space is considered from a dynamical system’s point of view. The symmetries of various two-dimensional cross-sections of a tessellation are related to the 17 wallpaper groups and the modular group. The computer images generated in this paper reveal the exotic structures of the tessellations in the hyperbolic 3-space. A color scheme is described, which reflects the rate of convergence of various orbits and, at the same time, enhances the artistic appeal of a generated image. This method can be used to create a great variety of aesthetic symmetrical patterns.

Tort-Decó: a “turtle geometry”-based package for drawing periodic designs

We have developed a package called Tort-Decó. The aim of this project was to construct a package for drawing periodic tilings by computer and an AI aid for wallpaper group and minimal generator regions recognition. The three topics covered are cyclic and dihedral groups, freezes, and wallpaper groups. There are several packages allowing users to apply movements to a subset of the plane. Some of them are specifically designed to draw periodic designs (RepTyles, Tesselmania!, Kali …). However, we know of no exhaustive approach based upon the use of “Turtle Graphics”. This approach evolved from [1] (where the motifs were fixed) and makes it possible to describe and store the periodic designs in an extremely short form. Moreover, the strong hierarchy of the consecutive steps in our approach makes the change of group clear. Finally, the classification expert system for wallpaper groups included is a novelty. It is well known that wallpaper group and minimal generator regions recognition is an algorithmic problem. Nevertheless, we have found that our students and nonmathematicians (like geologists) have serious difficulties in using these algorithms. Therefore, we decided to construct an AI aid to guide users in their search (in a similar way to expert systems designed to repair car and locomotive breakdowns).

Un promeneur analytique: Smirnoff VN. Préface de M. Moscovici. Paris: Calmann-Lévy, coll. « Essai Psychanalyse ; 1998. 324 p

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.

Indications à suivre dans le traitement moral de la folie: Leuret F. (1846). Préface de P. Morel. Paris: L'Harmattan, coll. « Trouvailles et retrouvailles ; 1998. 114 p

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.

Spiral tilings with colour symmetry from dynamics

Computer generation of spiral tilings with colour symmetry is considered from a dynamical system's point of view. A relation is established between spiral symmetry and wallpaper symmetry. Wallpaper patterns can be created by means of equivariant mappings. Associated spiral patterns can then be obtained accordingly. Patterns with colour symmetry are particular striking. A few colour schemes are discussed for the creation of patterns with colour symmetry. This method can be used to create a great variety of artistic spiral patterns.

`Smaller and Smaller' from Dynamics

By studying the periodic drawings of the Dutch Artist M.C. Escher, we find that the simplest group of the seventeen wallpaper group, P1, can be `visualized' in the upper halfplane as a symmetrical tessellation with self-similar properties. Iterative mappings are constructed and an algorithm is presented to automatically generate coloured patterns with this symmetry. This method can be used to create a great variety of artistic symmetrical patterns.

Frieze groups

The study of frieze patterns is traditionally associated with group theory, a branch of abstract mathematics. In fact, frieze and wallpaper patterns appear almost universally in books on group theory because they are perfect illustrations of those ideas. Because the pattern is algorithmic, one can synthesize it and generate texture (such as when scan converting) or analyze it to find the texture coordinates (such as when ray tracing). The author takes a very informal and visual approach to the subject.

USING WALLPAPER GROUPS TO MOTIVATE GROUP THEORY

ABSTRACT The study of the wallpaper groups and some related topics can be very effective motivation for the concepts of abstract algebra. Several student projects are discussed along with the students' reactions.