Puolikierrosta

Puolikierrosta, or a half-turn, is a geometric transformation that rotates every point of a plane around a fixed center by 180 degrees. The center remains fixed, and for any point P the image P' lies on the line PC such that the center C is the midpoint of PP'. This makes puolikierrosta equivalent to a point reflection about the center and to a rotation by π radians. Consequently, the transformation is an isometry of order two: applying it twice returns the original figure.

Properties of puolikierrosta include the preservation of distances and areas, and the mapping of lines to lines (for lines not passing through the center, the image is a parallel line; lines through the center map to themselves). A figure that coincides with its image under a half-turn has twofold rotational symmetry about the center. The operation is orientation-preserving, as it corresponds to a rotation rather than a reflection.

Relation to other concepts: puolikierrosta is the Finnish term for a half-turn and is often described as central symmetry with center C. In symmetry terminology it is referred to as a C2 rotation in the plane. It is one of the basic plane symmetries alongside identity and other rotational or reflective symmetries.

Examples and applications: Shapes such as squares and rectangles exhibit puolikierrosta about their center, meaning rotating them by 180 degrees yields the same figure. In art and tiling, twofold rotational symmetry is common. In computer graphics and image processing, 180-degree rotation is used for data augmentation and pattern analysis. In crystallography and mathematics, it serves as a fundamental symmetry operation and a simple model of twofold symmetry.