Pöörlemisrühm

Pöörlemisrühm is a term used in mathematics, specifically within group theory, to describe the set of all rotations of a geometric object that leave it unchanged. These rotations are typically considered in three-dimensional Euclidean space. A rotation is defined by an axis and an angle of rotation. When we talk about the pöörlemisrühm of an object, we are referring to the collection of all possible rotational symmetries. This collection forms a mathematical group under the operation of composition of rotations.

For example, the pöörlemisrühm of a cube contains rotations by 0, 90, 180, and 270 degrees about axes passing through the centers of opposite faces. It also includes rotations about axes passing through opposite vertices and about axes passing through the midpoints of opposite edges. The identity rotation, which involves no change, is always part of any pöörlemisrühm. The order of the group, which is the number of distinct rotations it contains, is a significant property of the object's symmetry. Understanding the pöörlemisrühm provides deep insights into the geometric structure and symmetries of various objects, from simple shapes like spheres to more complex molecules.