halfrotations

A half-rotation, or half-turn, is a rotation by 180 degrees about a fixed point in the plane. In the plane this operation is equivalent to central symmetry with respect to that point. Geometrically, every point p is mapped to a point p' such that the fixed point, or center, lies at the midpoint of the segment pp'. Only the center is fixed by the transformation.

In three-dimensional space, a 180-degree rotation can be performed about an axis. This half-turn leaves every point on the axis fixed and pairs up points off the axis. Like its planar counterpart, such a transformation is an isometry of order two, meaning applying it twice yields the identity.

Algebraically and in terms of symmetry, a half-turn is an element of order two in the rotation group. In crystallography and the study of tessellations, half-turns correspond to 2-fold rotational symmetry and are often denoted as C2 in wallpaper groups. They play a key role in describing the repetitive structure of many geometric patterns.

In complex analysis, a half-turn about a point z0 is the map z -> 2z0 - z, which mirrors the point z across z0. This viewpoint highlights the connection between half-turns and point reflections.

Applications and appearances of half-turns include geometric constructions, computer graphics, art, and the analysis of symmetric patterns. They provide a simple yet fundamental example of an isometry with order two and are closely related to notions of central symmetry and 2-fold symmetry in various mathematical contexts.