Pyörähdysryhmään

Pyörähdysryhmään is a Finnish term that translates to "rotation group" in English. In mathematics, a rotation group is a group consisting of all rotations in a given Euclidean space about a fixed point. These rotations are typically represented by orthogonal matrices with a determinant of one.

For example, in two-dimensional Euclidean space (R^2), the rotation group consists of all rotations about the origin. These can be represented by 2x2 matrices of the form:

[ cos(theta) -sin(theta) ]

[ sin(theta) cos(theta) ]

where theta is the angle of rotation. The composition of two rotations corresponds to the multiplication of their respective matrices. The identity element of the group is the rotation by zero degrees, represented by the identity matrix.

In three-dimensional Euclidean space (R^3), the rotation group is also known as SO(3). It consists of all rotations about any axis passing through the origin. This group is more complex than its 2D counterpart and has applications in various fields, including physics, chemistry, and computer graphics. Understanding rotation groups is fundamental in studying symmetries and transformations in geometry and related disciplines. The concept extends to higher dimensions as well, forming Lie groups that play a crucial role in advanced mathematical and physical theories.