Examples of Parallelograms

A parallelogram is a four-sided figure in which opposite sides are parallel. Each side of a parallelogram is also equal in length to its opposite side. The area of a parallelogram is equal to the length of the base times the shortest distance to the opposite side.

Symmetry

Symmetry, orderly, mutually corresponding arrangement of various parts of a body, producing a proportionate, balanced form. The principle of symmetry is of great importance in the fields of biology, mathematics, and mineralogy.

In biology, the regular distribution of various parts of an animal's body on two opposite sides of a linear axis, or a median plane, is known as bilateral symmetry. The proportional arrangement of similar parts of a body around a central axis, as in the case of jellyfish or starfish, is known as radial symmetry. The bodies of protozoans, such as those of the order Radiolaria, which have a round form about a central point or nucleus, are said to have a spherical symmetry.

In geometry, symmetry is a feature of certain plane and solid shapes. So-called symmetry operations are those mathematical transformations that produce a figure identical to the original or a mirror image of the original figure. Symmetry operations are defined with respect to a given point (center of symmetry), line (axis of symmetry), and plane (plane of symmetry).

In mineralogy, laws of symmetry apply to the angular structure of crystals. All classes of crystal are divided into six systems that are based on the length of their axes and other details of symmetry. See Crystal; Metallography.

In physics, a system is said to exhibit symmetry if it remains unchanged in the course of operations such as mirror reversal, reversal in the direction of time, and space-time translation. Many physical systems obey such symmetries, to which the conservation laws of physics are also related. This relationship has come to be of particular importance in particle physics, where certain symmetries called internal symmetries are observed. Such symmetries exist in the mathematical “space” of that realm and underlie the conservation of such quantities as charge, parity, baryon and lepton number, and total strangeness, even as certain particles are substituted for one another. In current theoretical physics, however, such symmetries are now known to be only approximate. Except for baryon and lepton number, that is, they are violated in their physical manifestations. When internal symmetries do not operate the same way but instead can be different at each point in space-time, they are called gauge symmetries. Theorists currently hope to reduce all such symmetries to gauge symmetries in their effort to develop a grand unification theory that can incorporate all of the fundamental interactions of matter (see Unified Field Theory). See also Elementary Particles.

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