punktsymmetrie

Punktsymmetrie, also known as point symmetry or central symmetry, is a geometric property of an object or a shape. An object possesses punktsymmetrie if it can be rotated by 180 degrees around a central point, and the object remains unchanged. This central point is referred to as the center of symmetry.

Imagine a point P located at the center of symmetry. For every point X on the object, there must be a corresponding point X' on the object such that P is the midpoint of the line segment XX'. In other words, X and X' are equidistant from P and lie on opposite sides of P along the same straight line.

Examples of objects exhibiting punktsymmetrie include a circle, where any point on the circumference has a diametrically opposite point also on the circumference. A square has punktsymmetrie, with its center being the point of symmetry. An equilateral triangle, however, does not possess punktsymmetrie. In three dimensions, a sphere or a cube exhibits punktsymmetrie. The concept is fundamental in various fields, including geometry, crystallography, and physics, where it describes the arrangement of atoms or molecules. The mathematical notation for a point reflection through a point c is often given by the transformation R_c(x) = 2c - x.