Rotaatioryhmä

Rotaatioryhmä is a term used in mathematics, specifically in group theory, to describe the group of all rotational symmetries of a given object. This object can be a geometric figure in two or three dimensions, or a more abstract mathematical structure. The elements of a rotaatioryhmä are the rotations themselves, and the group operation is the composition of these rotations, meaning performing one rotation after another.

For a two-dimensional object, like a regular polygon, the rotaatioryhmä consists of rotations around the center of the polygon that map the polygon onto itself. For example, a square has a rotaatioryhmä of order 4, with rotations by 0, 90, 180, and 270 degrees. A equilateral triangle has a rotaatioryhmä of order 3, with rotations by 0, 120, and 240 degrees. In general, a regular n-sided polygon has a rotaatioryhmä of order n.

In three dimensions, the concept extends to rotations around axes that pass through the object. For instance, a cube has a rotaatioryhmä of order 24, which includes rotations about axes passing through the centers of opposite faces, through opposite edges, and through opposite vertices. The structure of a rotaatioryhmä reveals important symmetries of the object it represents, and these groups are fundamental in various branches of mathematics, physics, and chemistry, particularly in the study of crystals and molecular structures. The set of rotations in a rotaatioryhmä forms a mathematical group because it is closed under composition, contains an identity element (no rotation), and each rotation has an inverse rotation.