Glide reflection

In geometry, a glide reflection or transflection is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation.

Because the distances between points are not changed under glide reflection, it is a motion or isometry.

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.

This corresponds to wallpaper group pg; with additional symmetry it occurs also in pmg, pgg and p4g.

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.

In the Euclidean plane 3 of 17 wallpaper groups require glide reflection generators.

Additionally, a centered lattice can cause a glide plane to exist in two directions at the same time.

[2] 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.

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

[4] In Conway's Game of Life, 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.