Glide plane

In geometry and crystallography, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged.

Glide planes are noted by a, b or c, depending on which axis the glide is along. 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.

In geometry, a glide plane operation is a type of isometry of the Euclidean space: the combination of a reflection in a plane and a translation 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 generated by just a glide plane operation 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 plane operation, so the even powers of the glide plane operation form a translation group.

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