In the case of symmetry groups in the plane, additional parameters are the direction of the translation vector, and, for the frieze groups with horizontal line reflection, glide reflection, or 180° rotation (groups 3–7), the position of the reflection axis or rotation point in the direction perpendicular to the translation vector. The actual symmetry groups within a frieze group are characterized by the smallest translation distance, and, for the frieze groups with vertical line reflection or 180° rotation (groups 2, 5, 6, and 7), by a shift parameter locating the reflection axis or point of rotation. Many authors present the frieze groups in a different order. There are seven frieze groups, listed in the summary table. A symmetry group of a frieze group necessarily contains translations and may contain glide reflections, reflections along the long axis of the strip, reflections along the narrow axis of the strip, and 180° rotations. p2mm: TRHVG (translation, 180° rotation, horizontal line reflection, vertical line reflection, and glide reflection)įormally, a frieze group is a class of infinite discrete symmetry groups of patterns on a strip (infinitely wide rectangle), hence a class of groups of isometries of the plane, or of a strip.p2mg: TRVG (translation, 180° rotation, vertical line reflection, and glide reflection).p11g: TG (translation and glide reflection).p11m: THG (translation, horizontal line reflection, and glide reflection).p1m1: TV (translation and vertical line reflection).p1: T (translation only, in the horizontal direction).They are related to the more complex wallpaper groups, which classify patterns that are repetitive in two directions, and crystallographic groups, which classify patterns that are repetitive in three directions. The set of symmetries of a frieze pattern is called a frieze group.įrieze groups are two-dimensional line groups, having repetition in only one direction. Frieze patterns can be classified into seven types according to their symmetries. Such patterns occur frequently in architecture and decorative art. In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Type of symmetry group Examples of frieze patterns
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