Randkreise

Randkreise, literally “edge circles” in German, is a term used in geometry to describe the circles that lie on the exterior boundary of a finite arrangement of mutually tangent circles. In circle packing and related constructions, Randkreise are the peripheral circles that form or touch the outer boundary of the configuration, distinguishing them from the interior circles that are fully surrounded by others.

In a typical circle packing, a Randkreis is a circle that is tangent to at least two other circles and participates in defining the outer boundary of the packing. When an enclosing circle bounds the entire configuration, that enclosing circle is often counted among the Randkreise as part of the boundary structure. The study of Randkreise focuses on how these outer circles influence the shape, density, and geometry of the packing, as well as on the relations between the radii of boundary circles and the inner arrangement.

Randkreise appear in several classical topics, including Apollonian gaskets, where families of tangent circles generate intricate boundary patterns, and in computational geometry and graphics, where boundary circles help in algorithmically extending packings or tilings. They also relate to Descartes’ circle theorem and other results that describe how tangent circles constrain one another.

See also: circle packing, Apollonian gasket, tangent circles, Descartes’ theorem, boundary circle. Etymology: Randkreise combines Rand (“edge” or “margin”) and Kreise (“circles”) in German.