Editing Glide reflection

{{short description|Geometric transformation combining reflection and translation}}
[[File:Glide reflection.svg|right|thumb|A glide reflection is the composition of a reflection across a line and a translation parallel to the line.]]
[[File:Krok_6.svg|thumb|This footprint trail has glide-reflection symmetry. Applying the glide reflection maps each left footprint into a right footprint and vice versa.]]

In [[geometry]], a '''glide reflection''' or '''transflection''' is a [[geometric transformation]] that consists of a [[reflection (mathematics)|reflection]] across a [[hyperplane]] and a [[translation (geometry)|translation]] ("glide") in a direction [[parallel (geometry)|parallel]] to that hyperplane, combined into a single transformation. Because the distances between [[point (geometry)|points]] are not changed under glide reflection, it is a [[motion (geometry)|motion]] or [[isometry]]. When the context is the two-dimensional [[Euclidean plane]], the hyperplane of reflection is a [[straight line]] called the ''glide line'' or ''glide axis''. When the context is [[three-dimensional space]], the hyperplane of reflection is a plane called the ''glide plane''. The [[displacement vector]] of the translation is called the ''glide vector''.

When some geometrical object or configuration appears unchanged by a transformation, it is said to have [[symmetry (geometry)|symmetry]], and the transformation is called a [[symmetry operation]]. ''Glide-reflection symmetry'' is seen in [[frieze group]]s (patterns which repeat in one dimension, often used in decorative borders), [[wallpaper group]]s (regular [[tessellation]]s of the plane), and [[space group]]s (which describe e.g. [[crystal]] symmetries). Objects with glide-reflection symmetry are in general not [[reflection symmetry|symmetrical under reflection]] alone, but two applications of the same glide reflection result in a double translation, so objects with glide-reflection symmetry always also have a simple [[translational symmetry]].

When a reflection is composed with a translation in a direction perpendicular to the hyperplane of reflection, the composition of the two transformations is a reflection in a parallel hyperplane. However, when a reflection is composed with a translation in any other direction, the composition of the two transformations is a glide reflection, which can be uniquely described as a reflection in a parallel hyperplane composed with a translation in a direction parallel to the hyperplane.

A single glide is represented as [[frieze group]] p11g. A glide reflection can be seen as a limiting [[rotoreflection]], where the rotation becomes a translation.  It can also be given a [[Schoenflies notation]] as S<sub>2∞</sub>, [[Coxeter notation]] as [∞<sup>+</sup>,2<sup>+</sup>], and [[orbifold notation]] as ∞×.

== 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]].

== Wallpaper groups ==
Glide-reflection symmetry with respect to two parallel lines with the same translation implies that there is also translational symmetry in the direction perpendicular to these lines, with a translation distance which is twice the distance between glide reflection lines. This corresponds to [[wallpaper group]] pg; with additional symmetry it occurs also in pmg, pgg and p4g.

If there are also true reflection lines in the same direction then they are evenly spaced between the glide reflection lines. A glide reflection line parallel to a true reflection line already implies this situation. This corresponds to wallpaper group cm. The translational symmetry is given by oblique translation vectors from one point on a true reflection line to two points on the next, supporting a [[rhombus]] with the true reflection line as one of the diagonals. With additional symmetry it occurs also in cmm, p3m1, p31m, p4m and p6m.

In the [[Euclidean plane]] 3 of 17 [[wallpaper group]]s require glide reflection generators. p2gg has orthogonal glide reflections and 2-fold rotations. cm has parallel mirrors and glides, and pg has parallel glides. (Glide reflections are shown below as dashed lines)
{| class=wikitable style="text-align: center
|+ Wallpaper group lattice domains, and fundamental domains (yellow)
|-
! Crystallographic name
| pgg
| cm
| pg
|-
! Conway name
| 22×
| *×
| ××
|-
! Diagram
|[[File:Wallpaper group diagram pgg.svg|240px]]
|[[File:Wallpaper group diagram cm rotated.svg|240px]]
|[[File:Wallpaper group diagram pg.svg|240px]]
|-
! Example
|[[File:SymBlend pgg.svg|240px]]
|[[File:SymBlend cm.svg|240px]]
|[[File:SymBlend pg.svg|240px]]
|}

== Space groups ==

Glide planes are noted in the [[Hermann–Mauguin notation]] by ''a'', ''b'' or ''c'', depending on which axis the glide is along. (The orientation of the plane is determined by the position of the symbol in the Hermann–Mauguin designation.) If the axis is not defined, then the glide plane may be noted by ''g''. When the glide plane is parallel to the screen, these planes may be indicated by a bent arrow in which the arrowhead indicates the direction of the glide. When the glide plane is perpendicular to the screen, these planes can be represented either by dashed lines when the glide is parallel to the plane of the screen or dotted lines when the glide is perpendicular to the plane of the screen. Additionally, a centered lattice can cause a glide plane to exist in two directions at the same time. This type of glide plane may be indicated by a bent arrow with an arrowhead on both sides when the glide plan is parallel to the plane of the screen or a dashed and double-dotted line when the glide plane is perpendicular to the plane of the screen. There is also the ''n'' glide, which is a glide along the half of a diagonal of a face, and the ''d'' glide, which is along a fourth of either a face or space diagonal of the [[unit cell]] . The latter is often called the diamond glide plane as it features in the diamond structure. The ''n'' glide plane may be indicated by diagonal arrow when it is parallel to the plane of the screen or a dashed-dotted line when the glide plane is perpendicular to the plane of the screen. A ''d'' glide plane may be indicated by a diagonal half-arrow if the glide plane is parallel to the plane of the screen or a dashed-dotted line with arrows if the glide plane is perpendicular to the plane of the screen. If a ''d'' glide plane is present in a crystal system, then that crystal must have a centered lattice.<ref>{{cite web |title=Glide Planes |url=http://img.chem.ucl.ac.uk/sgp/misc/glide.htm |website=Birkbeck College, University of London |accessdate=24 April 2019 |archive-date=21 July 2019 |archive-url=https://web.archive.org/web/20190721220633/http://img.chem.ucl.ac.uk/sgp/misc/glide.htm |url-status=live }}</ref>

In today's version of Hermann–Mauguin notation, the symbol ''e'' is used in cases where there are two possible ways of designating the glide direction because both are true. For example if a crystal has a base-centered [[Bravais lattice]] centered on the C face, then a glide of half a cell unit in the ''a'' direction gives the same result as a glide of half a cell unit in the ''b'' direction.

The [[isometry group]] generated by just a glide reflection is an infinite [[cyclic group]]. Combining two equal glide plane operations 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 the group generated by it. For any symmetry group containing a glide reflection, the glide vector is one half of an element of the translation group. If the translation vector of a glide plane operation is itself an element of the translation group, then the corresponding glide plane symmetry reduces to a combination of [[reflection symmetry]] and [[translational symmetry]].

== Examples and applications ==
Glide symmetry can be observed in nature among certain fossils of the [[Ediacara biota]]; the [[Machaeridia (annelid)|machaeridians]]; and certain [[palaeoscolecid]] worms.<ref>{{Cite journal | last1 = Waggoner | first1 = B. M.| title = Phylogenetic Hypotheses of the Relationships of Arthropods to Precambrian and Cambrian Problematic Fossil Taxa| jstor = 2413615| journal = Systematic Biology| volume = 45| issue = 2| pages = 190–222| year = 1996| doi = 10.2307/2413615| doi-access = free}}</ref>  It can also be seen in many extant groups of [[sea pen]]s.<ref>{{cite web|last1=Zubi|first1=Teresa|title=Octocorals (Stoloniferans, soft corals, sea fans, gorgonians, sea pens) - Starfish Photos - Achtstrahlige Korallen (Röhrenkorallen, Weichkorallen, Hornkoralllen, Seefedern, Fächerkorallen)|url=http://www.starfish.ch/c-invertebrates/octocorallia.html#Pennatulacea|website=starfish.ch|access-date=2016-09-08|date=2016-01-02|archive-date=2022-08-11|archive-url=https://web.archive.org/web/20220811220503/http://www.starfish.ch/c-invertebrates/octocorallia.html#Pennatulacea|url-status=live}}</ref>

In [[Conway's Game of Life]], a commonly occurring pattern called the [[Glider (Conway's Life)|glider]] is so named because it repeats its configuration of cells, shifted by a glide reflection, after two steps of the automaton. After four steps and two glide reflections, the pattern returns to its original orientation, shifted diagonally by one unit. Continuing in this way, it moves across the array of the game.<ref>{{cite conference
 | last = Wainwright | first = Robert T.
 | contribution = Life is universal!
 | doi = 10.1145/800290.811303
 | publisher = ACM Press
 | title = Proceedings of the 7th conference on Winter simulation - WSC '74
 | year = 1974| volume = 2
 | pages = 449–459
 | doi-access = free
 }}</ref>

In electrical engineering, if the graph of a periodic function also has glide symmetries, it is called "half-wave symmetric". This extra symmetry causes the fourier series of the function to only contain odd terms. Examples include the [[sine function]], [[square wave (waveform)|square wave]]s, and [[triangle wave]]s.

==See also==
* [[Screw axis]]
* [[Lattice (group)]]

==Notes==
{{Reflist}}

== References ==
* {{cite book | author=Walter Borchardt-Ott | title=Crystallography | url=https://archive.org/details/crystallography0000borc | url-access=registration | publisher=Springer-Verlag | year=1995 | isbn=3-540-59478-7}}

==External links==
* [http://www.cut-the-knot.org/Curriculum/Geometry/GlideReflection.shtml Glide Reflection] {{Webarchive|url=https://web.archive.org/web/20060404101708/http://www.cut-the-knot.org/Curriculum/Geometry/GlideReflection.shtml |date=2006-04-04 }} at [[cut-the-knot]]

[[Category:Euclidean symmetries]]
[[Category:Transformation (function)]]
[[Category:Crystallography]]