Friezelpatronen

Friezelpatronen, commonly called frieze patronen in Dutch, describe frieze patterns in mathematics. A frieze pattern is a repeating arrangement of motifs along a strip that extends infinitely in one direction. They are one-dimensional analogues of two-dimensional wallpaper patterns and are used to study symmetry and tiling in a simple setting.

Symmetry is central to frieze patterns. The allowed symmetries include translations along the strip, reflections across a line parallel or perpendicular to the strip, and glide reflections (a reflection combined with a translation). Based on which symmetries occur, seven distinct frieze types are classified. Each type describes a different balance of repetition and reflection in the motif sequence.

Construction and examples: start with a basic motif arranged along the axis and repeat it along the length with a fixed step. If the pattern has a mirror line, the motif on one side mirrors the other. If glide symmetry is present, the pattern repeats after a combined reflection and shift. These patterns are used in design and crystallography to model linear repetitive structures.

History and significance: The study of frieze patterns was developed in the 20th century, notably by H. S. M. Coxeter and collaborators, leading to the standard seven-frieze-group classification. They provide a tractable entry point into the broader theory of symmetry groups.

See also: Frieze groups, Wallpaper groups, Pattern (mathematics).