Type Synthesis of Long Symmetric Planar Shape-Morphing Mechanism Arrays

This paper describes a novel type synthesis methodology for creating planar shape-morphing mechanism chains of any length with specified mobility (degree-of-freedom). It is expected that this will be of use in designing complex shape-morphing objects. The methodology is based on using graph theory in conjunction with symmetry theory for borders (Frieze groups). It is shown that the methodology can produce chains of a variety of lengths and mobility, and these chains have symmetric topologies and that symmetric kinematics may be possible for some of the chains, which simplifies their analysis. Techniques for decreasing and increasing the mobility of the chains are discussed. Finally, a gallery of shape morphing chains is given to provide concrete illustrations of the diversity of designs generated using this methodology.

Refinements of Wilker–Huygens-Type Inequalities via Trigonometric Series

The study of even functions is important from the symmetry theory point of view because their graphs are symmetrical to the Oy axis; therefore, it is essential to analyse the properties of even functions for x greater than 0. Since the functions involved in Wilker–Huygens-type inequalities are even, in our approach, we use cosine polynomials expansion method in order to provide new refinements of the above-mentioned inequalities.

SYNTHESIS OF IMAGES-ORNAMENTS

The article developed the mathematical model for the synthesis of ornamental images and implemented the software editor of ornamental images, based on symmetry theory. The paper shows the fundamental role of symmetry. It is analyzed that the symmetry theory methods are used in physics, chemistry, biology, and engineering. It was found that symmetry is based on transformation and storage. In addition, the symmetrical system is based on a set of invariants that are built according to certain rules. It is shown that the symmetry of borders and the symmetry of mesh ornaments are used in ornaments. The synthesis of ornamental images is considered on the example of Ukrainian folk embroidery. The contribution of foreign and domestic scientists to the development of the symmetry theory and synthesis of images is analyzed. It is indicated that Ukrainian folk embroidery is the valuable property of the cultural and material heritage of people and an important source of research. It is analyzed that there are more than 100 types of different embroidery techniques. The role of famous Ukrainian artists in the popularization and organization of Ukrainian folk embroidery museums is presented. It is investigated that embroidery is built from separate motives or from ornaments. Ornaments consist of sub-ornaments. A sub-ornament is a pattern consisting of rhythmically ordered identical elements (built on one group transformation). Subornaments are divided into reports. The report is called the minimum for the area of the area that can cover the sub-ornament, using only transfers. The report, in turn, is divided into even smaller particles: motive or elementary picture. It is found that in embroidery ornaments there are 7 groups of stripe and 12 - plan. Mathematical models of images-ornaments synthesis for groups of a strip and plan groups are developed. Mathematical models are given for ideal ornaments. If offsets of axes or centers of symmetries, it is necessary to adjust the coefficients of transformation matrices displacement. Samples of embroidered ornaments of the corresponding plane and stripe groups are provided. Editor of image-ornaments has been developed, which allows the synthesis of complex ornamental images based on analytical formulas of elementary picture, sub-ornament, and ornament. Examples of real and synthesized samples of Ukrainian folk embroidery are provided. The scientific novelty of the work lies in the development of mathematical models of ornaments on the basis of symmetry groups on the strip and the plane. The practical value of the work lies in the development of an image editor-ornaments.

Local and global color symmetries of a symmetrical pattern

This study addresses the problem of arriving at transitive perfect colorings of a symmetrical pattern {\cal P} consisting of disjoint congruent symmetric motifs. The pattern {\cal P} has local symmetries that are not necessarily contained in its global symmetry group G. The usual approach in color symmetry theory is to arrive at perfect colorings of {\cal P} ignoring local symmetries and considering only elements of G. A framework is presented to systematically arrive at what Roth [Geom. Dedicata (1984), 17, 99–108] defined as a coordinated coloring of {\cal P}, a coloring that is perfect and transitive under G, satisfying the condition that the coloring of a given motif is also perfect and transitive under its symmetry group. Moreover, in the coloring of {\cal P}, the symmetry of {\cal P} that is both a global and local symmetry, effects the same permutation of the colors used to color {\cal P} and the corresponding motif, respectively.