Table of contents |
2 Basic descriptions 3 Examples 4 Orbifold notation 5 Related Topics 6 External links |
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).
In fact each isometry is one of the following four types
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.
The seventeen wallpaper patterns can be described as follows:
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.
There are 230 three-dimensional crystallographic groups called space groups.
There are 4783 four-dimensional symmetry groups.
http://www.xahlee.org/Wallpaper_dir/c0_WallPaper.html The Discontinuous Groups of Rotation and Translation in the PlanePlane crystallographic groups or wallpaper groups
All isometries in the plane are bijections and are obtained from combinations of:
translations by a = (a1,a2): ta(u) = u+a where + is vector addition
rotations about the origin: ρθ(u1,u2) = (u1cosθ−u2sinθ,u1sinθ+u2cosθ)
reflections about the x axis: r(u1,u2) = (u1,−u2)
There are only 17 wallpaper group classes, because we only consider discrete motions. This means we do not allow arbitrarily small rotations or translations.Basic descriptions
All of the above patterns are translated across the plane and therefore the translation of the pattern has been left out of the description. Examples
The following are examples of the 17 wallpaper plane classesOrbifold notation
To be addedRelated Topics
External links