Gleichartigkeit

Gleichartigkeit is a concept in geometry describing similarity of figures or objects up to scaling. In Euclidean geometry, two plane figures are Gleichartig (similar) if there exists a similarity transformation that maps one onto the other. Equivalently, their corresponding angles are equal and their corresponding sides are in a constant ratio. The notion also extends to three-dimensional objects and higher dimensions, where a uniform scale factor applies in all directions.

Formally, two figures F and F' are Gleichartig if there is a positive real number k and an isometry I (a composition of rotation, reflection, and translation) such that F' = k · I(F). The linear measurements are multiplied by k, areas by k², and volumes by k³. This leads to invariant properties such as angle equality and proportionality of sides.

Key consequences include the Satz von Ähnlichkeit: if two objects have equal corresponding angles, they are Gleichartig; conversely, similar figures have equal corresponding angles. In practice, similarity is characterized by proportional corresponding sides and equal angles, independent of overall size.

Examples include a triangle with sides 3-4-5 being Gleichartig with a 6-8-10 triangle (scaling by factor 2). Circles are Gleichartig to each other regardless of radius, since all circles are similar under scaling.

Relation to other concepts: Gleichartigkeit differs from congruence, which requires identical size and shape (scale factor 1). Related notions include Ähnlichkeit, Skalierung, and geometric transformations used to describe and analyze similarity.