The 7 frieze groups: translations, reflections, and glide reflections
Borders on buildings, fences, and ribbons repeat in one direction. However wild the motif, every such pattern has one of just seven kinds of symmetry.
Compare the seven rows. Each starts from the same lopsided motif, so every difference you see comes from the symmetries applied.
Every repeating border pattern belongs to exactly one of these seven frieze groups, classified by their symmetry operations: translations, reflections, glide reflections, and 180° rotations.
A frieze pattern is a pattern that repeats in one direction -- think of decorative borders on buildings, fences, or ribbons. Despite the infinite variety of possible motifs, there are exactly 7 distinct frieze groups, classified by which combinations of translation, rotation, reflection, and glide reflection they possess.
This result is a beautiful theorem in crystallography: no matter how complex the repeating border pattern, its symmetry must fall into one of precisely seven categories.
All seven frieze groups displayed simultaneously using the same asymmetric motif. Each row shows how a different combination of symmetries transforms the motif. Notice how the same base shape produces dramatically different patterns depending on which symmetries are applied.
Key insight: The 7 groups form a hierarchy. Every frieze pattern has translation symmetry (p1). Adding horizontal reflection gives p11m, adding a 180° rotation gives p2, and so on. The most symmetric group, p2mm, has all four types of symmetry.
Draw a motif in the dashed box, then pick a frieze group. The overlaid mirror lines, glide axes, and rotation centers show how each copy relates to the original.
Draw your own motif and select a frieze group to see your design tiled along a strip. The symmetry elements -- mirror lines, glide axes, and rotation centers -- are overlaid so you can see exactly how each copy relates to the original.
Key insight: A glide reflection is a reflection followed by a translation along the reflection axis. Like a translation it has no fixed points, yet it is not equal to any single translation, rotation, or reflection — it is a genuinely new, fourth kind of plane isometry. Glide reflections are responsible for the "footprint" pattern: left, right, left, right...
Name the group for each pattern, then press Next Pattern. Ask in turn: vertical mirrors? Horizontal mirror? 180° rotation? Glide reflection?
Test your understanding by identifying which of the 7 frieze groups each pattern belongs to. Use the decision tree: Does it have vertical reflection? Horizontal reflection? 180° rotation? Glide reflection? These yes/no questions uniquely determine the group.
Key insight: The classification algorithm is a binary decision tree. First check for vertical mirrors, then horizontal mirrors, then 180° rotations, then glide reflections. At most 3 questions suffice to identify any frieze group uniquely.
Spot a mistake or have an idea?