Glide reflection

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Glide reflection

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a glide reflection is a type of isometry

Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
of the Euclidean plane: the combination of a reflection

Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
in a line and a translation

Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
along that line. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector.

The combination of a reflection in a line and a translation in a perpendicular direction is a reflection in a parallel line.

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Encyclopedia

In geometry

Geometry

Isometry

Reflection (mathematics)

Translation (geometry)

Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
.

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

to (x + 1, −y).

It fixes a system of parallel lines.

The isometry group

Isometry group

In mathematics, the isometry group of a metric space is the Set of all isometry from the metric space onto itself, with the function composition as group operation....
generated by just a glide reflection is an infinite cyclic group

Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
.

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

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
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

Frieze group

A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetry in the pattern....
nr. 2.

Example pattern with this symmetry group: +++ + +++ + + + +++ +++ +++ + + + + +++ + Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about an axis perpendicular to the line of reflection. It is isomorphic to a semi-direct product of Z and C₂.

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

Reflection symmetry

The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kite s and the isosceles trapezoids.For each line or plane of reflection, the symmetry group is isomorphic with Cs , one of the three types of order two , hence algebraically C2....
and translational symmetry

Translational symmetry

In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation....
.

In 3D the glide reflection is called a glide plane
Glide plane

In crystallography, a glide plane is symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged....
. It is a reflection in a plane combined with a translation parallel to the plane.

See also: congruence (geometry)

Congruence (geometry)

In geometry, two sets of point are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translation s, rotations and reflection s....
, similarity (mathematics), wallpaper group

Wallpaper group

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry in the pattern. Such patterns occur frequently in architecture and decorative art....
, frieze group

Frieze group

A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetry in the pattern....
.

External links

at cut-the-knot
Cut-the-knot

Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....