simetriye

Simetriye, or symmetry, is a property of an object or system that remains unchanged under a specified set of transformations. In mathematics and the sciences, symmetry is formalized as invariance under a group of operations, and it can be geometric or abstract.

Geometric symmetry includes reflections (mirror symmetry), rotations, translations, and glide reflections, as well as central symmetry about a point. Symmetries can be continuous, as in rotating a circle by any angle, or discrete, as in rotating a square by multiples of 90 degrees. In two dimensions, regular polygons have dihedral symmetries; in three dimensions, polyhedra and crystals have corresponding point and space groups.

Common examples: a circle possesses infinite rotational symmetry; a square has fourfold rotational symmetry and four mirror symmetries, forming the dihedral group D4; the hexagon has D6; the human face shows approximate bilateral symmetry.

In mathematics and physics, symmetries are described by groups. Continuous symmetries are modeled by Lie groups and are linked to conservation laws through Noether's theorem. Symmetry also governs selection rules in quantum mechanics and determines the spectral properties of molecules in chemistry.

Applications of symmetry span pattern design, art, architecture, crystallography, materials science, and computer graphics. Historically, the systematic study of symmetry emerged in the 19th century with the development of group theory and was deepened in the 20th century by connections between symmetries and physical laws.