Plane crystallographic groups or wallpaper groups

There are seventeen different types of wallpaper patterns. As opposed to the frieze patterns, these patterns cover the entire plane and can be extended infinitely in any direction on the plane. Discrete frieze patterns extend infinitely in only one direction, and not across the entire plane.

An isometry or rigid motion in the plane is a map m : R² → R² such that m is distance-preserving. i.e. d(u,v) = d(m(u),m(v)) where d((u₁,u₂),(v₁,v₂)) = sqrt((u₁−v₁)² + (u₂−v₂)²).
All isometries in the plane are bijections and are obtained from combinations of:
translations by a = (a₁,a₂): t_a(u) = u+a where + is vector addition
rotations about the origin: ρ_θ(u₁,u₂) = (u₁cosθ−u₂sinθ,u₁sinθ+u₂cosθ)
reflections about the x axis: r(u₁,u₂) = (u₁,−u₂)

In fact each isometry is one of the following four types

• translation (as above)
• rotation by θ about a: ρ_θ(u−a)+a
• reflection about any line l in R²: r_a,p where a is a vector in the direction of the line, p is a point on the line
• glide reflection about line l: g_a,p(u) = t_a(r_a,p(u))

There are only 17 wallpaper group classes, because we only consider discrete motions. This means we do not allow arbitrarily small rotations or translations.

To decide which of the 17 classes any particular plane periodic pattern belongs to, determine the rotational symmetry and whether or not the pattern has reflection symmetry or nontrivial glide-reflection symmetry. If necessary, the last step is to determine the locations of the centers of rotation.

Basic descriptions

The seventeen wallpaper patterns can be described as follows: (Grouped by possible shapes of the lattice cells)

• Parallelogram | Rectangle | Square
  • No rotations, no reflections, no glide-reflection (only translations) (p1)
  • Rotation of 180 degrees only (p2)

• Rhombus | Rectangle | Square
  • No rotation, at least one reflection, and a glide-reflection that is not on the reflection axis (cm)
  • Rotation of 180 degrees not on a reflection axis, reflections in two directions (cmm)

• Rectangle | Square
  • No rotation, at least one reflection, no glide-reflections that can be located outside of the reflection axis (pm)
  • Glide-reflections only (pg)
  • Rotation of 180 degrees, a glide-reflection (pgg)
  • Rotation of 180 degrees located on the reflection axis, reflections in two directions (pmm)
  • Rotation of 180 degrees, reflections only in one direction (pmg)

• Square
  • Rotation of 90 degrees only (p4)
  • Rotation of 90 degrees, reflections in four directions (p4m)
  • Rotation of 90 degrees, reflections in less than four directions (p4g)

• Triangle | 60/120 rhombus
  • Rotation of 120 degrees only (p3)
  • Rotation of 120 degrees, reflections with a threefold center located on the axis (p3m1)
  • Rotation of 120 degrees, reflections with a threefold center not located on the axis (p31m)

• Hexagon
  • Rotation of 60 degrees only (p6)
  • Rotation of 60 degrees, at least one reflection (p6m)

All of the above patterns are translated across the plane and therefore the translation of the pattern has been left out of the description.

Symmetry diagrams and examples

Translation cell	Example

A generating region of a periodic pattern is the smallest portion of the lattice unit whose images under the full symmetry group of the pattern cover the plane.

Orbifold notation

To be added

Software

There exist several software graphic tools that will let you create 2D patterns using wallpaper symmetry groups. Usually, you can edit the original tile and its copies in the entire pattern are updated automatically.

• Inkscape, a free vector editor, supports all 17 groups plus arbitrary scales, shifts, and rotates per row or per column, optionally randomized to a given degree.

• SymmetryWorks is a commercial plugin for Adobe Illustrator, supports all 17 groups.

• Arabeske is a free standalone tool, supports a subset of wallpaper groups.

Related topics

There are 230 three-dimensional crystallographic groups called space groups. There are 4783 four-dimensional symmetry groups.

External links

• Article "The Discontinuous Groups of Rotation and Translation in the Plane" by Xah Lee
• Article "The 17 plane symmetry groups" by David E. Joyce
• Introduction to Wallpaper Patterns by Chaim Goodman-Strauss and Heidi Burgiel
• Description by Silvio Levy