Glide reflection

Example of a glide reflection

In geometry, a glide reflection is a type of opposite isometry of the Euclidean plane: the combination of a reflection in a line and a translation 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. However, a glide reflection cannot be reduced like that. Thus the effect of a reflection combined with any translation is a glide reflection, with as special case just a reflection. These are the two kinds of indirect isometries in 2D.

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

(x, y) → (x + 1, −y).

It fixes a system of parallel lines.

The isometry group generated by just a glide reflection is an infinite cyclic group.^[1]

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 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 a point on 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 and translational symmetry.

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 3D the glide reflection is called a glide plane. It is a reflection in a plane combined with a translation parallel to the plane.

Glide symmetry can be observed in nature among certain fossils of the Ediacara biota; the machaeridians; and certain palaeoscolecid worms.^[2]

Glide reflection in cellular automata

Glide reflection is common in Conway's Game of Life. The four most common spaceships: the glider, LWSS, MWSS and HWSS all move using glide reflection. Every two iterations these spaceships appear to have been glide reflected to their new positions. Here is the evolution of the lightweight spaceship:

This then carries on glide reflecting in a similar manner.

References

External links

• Glide Reflection at cut-the-knot