Symmetric

Symmetric describes a property of an object, relation, or system that remains unchanged under a set of transformations or exchanges. In mathematics and related fields, symmetry denotes invariance under a group of operations; the collection of such operations is called a symmetry group. The concept appears in geometry, algebra, and the natural sciences, and it often carries the sense of balance or proportion.

In geometry, an object is symmetric if there exists isometries that map it onto itself. Common types include reflectional (mirror) symmetry, rotational symmetry, and translational symmetry. A circle has rotational and reflectional symmetry about any diameter; a regular polygon has discrete rotational symmetry and mirror symmetries. Bilateral symmetry is a form seen in many organisms, with a single plane dividing the body into mirror-image halves.

In algebra, the term appears in several forms. A relation R is symmetric if aRb implies bRa. A matrix is called symmetric if it equals its transpose, A = A^T; symmetric matrices have real eigenvalues and are orthogonally diagonalizable. The notation of symmetric bilinear forms and the role of the symmetric group Sn in permutation symmetry are common in higher mathematics.

Beyond pure math, symmetry concepts appear in set theory with symmetric difference, A Δ B, the elements in exactly one of A or B. In cryptography, symmetric-key algorithms use the same key for encryption and decryption, contrasting with public-key (asymmetric) systems. In aesthetics and design, symmetry is used to convey balance and harmony.