sümmeetriiaid

Sümmeetriiaid refers to symmetries, transformations that preserve the essential structure of a geometric object. In mathematics, the collection of all such transformations forms the symmetry group of the object, under the operation of composition. Symmetries can be discrete, as in polygons, or continuous, as in circles or spheres.

In the plane, a symmetry is an isometry that maps a figure to itself. Common plane symmetries include translations, rotations about a point, reflections across a line, and glide reflections. A regular n-gon has rotational symmetry of order n (rotating by 360/n leaves the figure unchanged) and reflection symmetries along lines through its center, giving a dihedral symmetry group of order 2n. A square, for example, has eight symmetries (the dihedral group D4), while a circle has infinitely many symmetries corresponding to any rotation around its center and any reflection across a line through the center.

In three dimensions, symmetries include rotations about an axis, reflections across a plane, and improper rotations (a rotation followed by reflection). The study of all possible symmetries of a shape or pattern leads to concepts such as space groups in crystallography and wallpaper groups in plane tilings, which classify periodic patterns by their symmetries.

Applications of symmetries span mathematics, physics, chemistry, computer science, and art. They aid in understanding shapes and patterns, simplifying complex problems through invariants, and analyzing molecular structures or crystal lattices. The concept of symmetry is central to both theoretical definitions and practical modeling across disciplines.