geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a glide reflection or transflection is a

geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function wh ...

that consists of a reflection across a

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

and a

translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

("glide") in a direction parallel to that hyperplane, combined into a single transformation. Because the distances between

points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...

are not changed under glide reflection, it is a

motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

. When the context is the two-dimensional

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

, the hyperplane of reflection is a straight line called the ''glide line'' or ''glide axis''. When the context is

three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...

, the hyperplane of reflection is a plane called the ''glide plane''. The

displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along ...

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 Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

, and the transformation is called a symmetry operation. ''Glide-reflection symmetry'' is seen in frieze groups (patterns which repeat in one dimension, often used in decorative borders),

wallpaper group A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...

s (regular

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

s of the plane), and

space group In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that ...

s (which describe e.g.

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...

symmetries). Objects with glide-reflection symmetry are in general not 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 In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation. Analogously, an operato ...

. 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 In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection ( ...

, where the rotation becomes a translation. It can also be given a Schoenflies notation as S_2∞, Coxeter notation as ��⁺,2⁺ and

orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Horton Conway, John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curv ...

as ∞×.

Frieze groups

In the Euclidean plane, reflections and glide reflections are the only two kinds of indirect (orientation-reversing)

isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

. 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 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 In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element ...

generated by just a glide reflection is an infinite

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

. 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 In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

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: 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''₂. Example pattern with this symmetry group: 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 In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...

and

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

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 In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...

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

3 of 17

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)

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 In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector In mathematics, a unit vector i ...

. 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. 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

generated by just a glide reflection is an infinite

. 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

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

and

Examples and applications

Glide symmetry can be observed in nature among certain fossils of the

Ediacara biota The Ediacaran (; formerly Vendian) biota is a taxonomic period classification that consists of all life forms that were present on Earth during the Ediacaran Period (). These were enigmatic tubular and frond-shaped, mostly sessile, organis ...

; the machaeridians; and certain

palaeoscolecid The palaeoscolecids are a group of extinct ecdysozoan worms resembling armoured priapulids. They are known from the Lower Cambrian to the lower Ludfordian (late Silurian); they are mainly found as disarticulated sclerites, but are also preserved ...

worms. It can also be seen in many extant groups of sea pens. In

Conway's Game of Life The Game of Life, also known as Conway's Game of Life or simply Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is a zero-player game, meaning that its evolution is determined by its initial ...

, a commonly occurring pattern called the 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. 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 In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...

square wave Square wave may refer to: *Square wave (waveform) A square wave is a non-sinusoidal waveform, non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same ...

s, and triangle waves.

Notes

References

External links

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 Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...

Euclidean symmetries Transformation (function) Crystallography