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Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group.

Symmetry (Mathematics)

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The reprint is devoted to the phenomena associated with exact or approximate axial symmetry in different areas of technical physics and mechanical engineering science. How can the symmetry of the problem be used most efficiently for its analysis? Why is the symmetry broken or why is it still approximately retained? These and other questions are discussed based on systems from different fields of engineering.

Symmetry --- History. --- Aesthetics --- Proportion

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The present reprint provides some theoretical results (and their applications) in the fields of mathematical analysis and functional analysis, in which the concept of symmetry plays an essential role. More specifically, various problems are investigated in areas, such as: optimization problems, polynomial approximation on unbounded subsets, moment problems, variational inequalities, evolutionary problems, dynamical systems, generalized convexity, partial differential equations, and special spaces of self-adjoint operators. With various examples and applications to complement and substantiate the mathematical developments, the present reprint is a valuable guide for researchers, engineers, and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science, and economics.

Symmetry. --- Symmetry (Art) --- Form (Aesthetics) --- Proportion (Art) --- Aesthetics --- Proportion

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Symmetry is found everywhere, to a greater or lesser degree. It is essential in all sciences and also as the basic substrate in all arts. The same happens with Complex Networks, which are the majority of all those that interest for advanced knowledge, that is, for research. That is why these two themes have been unified here, whose intersection we are dealing with: in its first part, of fundamentals, and in its second part of applications, which are multiple in the world today.

fuzzy symmetry --- near symmetry --- graph theory --- fuzzy optimization --- measure theory --- complex networks --- combinatorial and computational group theory --- theoretical computer science --- complex systems --- symmetry measures --- similarity --- symmetry as a new and very important bridge between sciences and humanities --- discrete applied mathematics --- information theory --- stability --- fuzzy measure theory --- complexity theory --- symmetry and asymmetry --- chirality --- entropy measures --- mathematical analysis --- combinatorial optimization --- preferential attachment --- clustering

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Symmetry has a central role in the study of vision. The concept of symmetry has an ancient origin in considerations of visual appearance; in modern times, abstracted and formalized into Group Theory, it has found spectacular applications, far beyond the visible; but its importance for vision persists in many ways including: • As a non-accidental feature of an image that cues affordances, 3D structure or the semantic categories of object present. • As a redundant aspect of an image which many be exploited for simplicity and compactness of encoding. • As a salient feature that draws attention, and evokes distinctive brain responses. • As a constraint on priors on the distribution of structures to be found in the natural world. • As an aesthetic principle. • As a design principle for vision systems. • The original idea for a Special Issue came from a symposium at the European Conference in Visual Perception, in 2015, on the topic of brain responses to visual symmetry, but we have now extended the scope. This Special Issue is devoted to provide a shared place for cutting edge studies on how and why symmetry is processed and exploited by biological and artificial visual systems.

image processing --- vision --- visual perception --- visual preference --- 3D shape --- perceptual grouping --- Symmetry