a glide reflection

A glide reflection is exactly what footprints in the sand do: step, then step on the other side a little further along, then back, alternating left and right as you walk. Each footprint is the mirror image of the last, slid forward. The motion combines two simpler ones — a reflection across a line, plus a translation along that very same line — done together. Neither alone gives footprints; you need the flip and the slide, in the same direction as the mirror.

Precisely, a glide reflection is the composition of a reflection across a line m with a translation by a vector parallel to m (the glide vector). The order does not matter: reflecting then sliding gives the same result as sliding then reflecting, because the slide runs along the mirror. Because it contains a reflection, a glide reflection reverses orientation — it is an opposite isometry. Its remarkable feature is that, unlike a plain reflection, it has no fixed points at all (the slide carries every point, even those on m, off their starting spot). Doing a glide reflection twice gives a pure translation by twice the glide vector, with no flipping left over.

Why does it deserve its own name? Because it is the fourth and final kind of plane isometry, and the easiest to forget. The classification theorem says every plane isometry is a translation, a rotation, a reflection, or a glide reflection — and glide reflection is precisely the opposite isometry with no fixed point, the one that is neither a pure reflection nor a pure slide. It also generates important repeating patterns: a frieze of alternating footprints has glide symmetry that no combination of a plain reflection and a plain translation can capture on its own.