symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. In art and design, it is often used in a somewhat loose sense, to mean a kind of balance in which the corresponding parts are not necessarily alike but only similar. A symmetrical design should produce a pleasing effect; if there is too close a correspondence, the effect may be monotonous. Ancient Greek architecture is particularly distinguished for its symmetry. In modern art, the Dutch artist M. C. Escher achieved a number of striking effects in his works exploring mathematical symmetry. A mathematical operation, or transformation, that results in the same figure as the original figure (or its mirror image) is called a symmetry operation. Such operations include reflection, rotation, double reflection, and translation. The set of all operations on a given figure that leave the figure unchanged constitutes the symmetry group for that figure. The symmetry groups of three-dimensional figures are of special interest because of their application in fields such as crystallography (see crystal). In general, a symmetry operation on a figure is defined with respect to a given point (center of symmetry), line (axis of symmetry), or plane (plane of symmetry). In biology, symmetry is studied in the correspondences between different parts of a given organism, as between the left and right halves of the human body or between the various segments of a starfish (see symmetry, biological). In physics, basic symmetries in nature underlie the various conservation laws. For example, the symmetry of space and time with respect to translation and rotation means that a given experiment should yield the same results regardless of where it is performed, what direction the equipment is pointing in, or when it is performed. These three symmetries can be shown to imply the laws of conservation of linear momentum, angular momentum, and energy, respectively.

See G. E. Martin, Transformation Geometry (1987); B. Bunch, Reality's Mirror (1989); M. C. Escher, Escher on Escher (tr. 1989).

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