Editing Glide reflection (section)

== Frieze groups ==

In the Euclidean plane, reflections and glide reflections are the only two kinds of indirect (orientation-reversing) [[Euclidean group#Overview of isometries in up to three dimensions|isometries]].

For example, there is an isometry consisting of the reflection on the ''x''-axis, followed by translation of one unit parallel to it. In coordinates, it takes

{{block indent | em = 1.5 | text = (''x'', ''y'') → (''x'' + 1, −''y'').}}

This isometry maps the ''x''-axis to itself; any other line which is parallel to the ''x''-axis gets reflected in the ''x''-axis, so this system of parallel lines is left invariant.

The [[isometry group]] generated by just a glide reflection is an infinite [[cyclic group]].<ref>{{cite book| title = Transformation Geometry: An Introduction to Symmetry|series=[[Undergraduate Texts in Mathematics]]|first=George E.|last=Martin |publisher=Springer|year=1982| isbn = 9780387906362| page=64 | url = https://books.google.com/books?id=KW4EwONsQJgC&pg=PA64}}.</ref>

Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group.

In the case of glide-reflection symmetry, the [[symmetry group]] of an object contains a glide reflection, and hence the group generated by it. If that is all it contains, this type is [[frieze group]] p11g.

Example pattern with this symmetry group:
[[File:Frieze example p11g.png]]

A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach.

Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a [[semi-direct product]] of '''Z''' and ''C''<sub>2</sub>.

Example pattern with this symmetry group:
[[File:Frieze example p2mg.png]]

For any symmetry group containing some glide-reflection symmetry, the translation vector of any glide reflection is one half of an element of the translation group. If the translation vector of a glide reflection is itself an element of the translation group, then the corresponding glide-reflection symmetry reduces to a combination of [[reflection symmetry]] and [[translational symmetry]].