glidecovering

Glidecovering is a concept in tiling theory and geometric group theory that describes a way of covering a space with tiles generated from a single base tile by a symmetry group that includes glide reflections. A glide reflection is the combination of a reflection across a line and a translation along that line. In a glidecovering, the action of the glide-reflection group produces a tiling that fills the plane (or another space) without gaps or overlaps, while recording how the tiling is built from the base tile.

In formal terms, a glidecovering can be described as a tiling produced by a discrete group of isometries G that contains glide reflections. If a fundamental tile D is chosen, the tiling is obtained by applying elements of G to D, with the interiors of the resulting tiles being pairwise disjoint. The quotient space by G can be viewed as a basic domain or orbit space, and the original tiling serves as a geometric covering of that quotient.

A typical example is a strip or ribbon tiling formed by repeatedly reflecting a rectangle across one edge and translating along that edge, generating a pattern with glide symmetry. Such tilings illustrate how glide reflections contribute to the repetition and regularity characteristic of glidecoverings, while also highlighting relationships to broader symmetry concepts.

Glidecoverings are related to wallpaper groups that include glide reflections and are studied for their implications in symmetry, crystallography, and computational texture generation. See also glide reflection, tilings, wallpaper group, and covering space.