Symmetry

Symmetry () in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art, and music. The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry. Symmetry (geometry) A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry (line or mirror symmetry) if there is a line (or in 3D a plane) going through it which divides it into two pieces that are mirror images of each other. An object has rotational symmetry if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape. An object has translational symmetry if it can be translated (moving every point of the object by the same distance) without changing its overall shape. An object has helical symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis.

Mean value theorems for collections of lattices with a prescribed group of symmetries

Mean value theorems for collections of lattices with a prescribed group of symmetries