geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

and

crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...

, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a

translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

parallel with that plane, may leave the

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, macro ...

unchanged. Glide planes are noted by ''a'', ''b'' or ''c'', depending on which axis the glide is along. 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, for example) does not necessaril ...

. 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.

Formal treatment

, a glide plane operation is a type of

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'' ...

of the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

: the combination of a reflection in a plane and a

in that plane. 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 plane and a translation in a perpendicular direction is a reflection in a parallel plane. However, a glide plane operation with a nonzero translation vector in the plane cannot be reduced like that. Thus the effect of a reflection combined with ''any'' translation is a glide plane operation in the general sense, with as special case just a reflection. The glide plane operation in the strict sense and the pure reflection are two of the four kinds of indirect isometries in 3D. The

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

generated by just a glide plane operation is an infinite

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

. Combining two equal glide plane operations gives a pure translation with a translation vector that is twice that of the glide plane operation, so the even powers of the glide plane operation form a translation group. In the case of glide plane 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 plane operation, and hence the group generated by it. For any symmetry group containing glide plane symmetry, the translation vector of any glide plane operation 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 In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...

. See also lattice.

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 Crystallography Symmetry