GlideReflexionen

GlideReflexionen, commonly referred to as glide reflections in English, denote a class of plane isometries formed by composing a reflection across a straight line with a translation along that same line. This transformation is orientation-reversing and preserves distances, but it has no fixed points unless the translation is trivial, in which case it reduces to a reflection. GlideReflexionen occur as a standard type of symmetry in Euclidean plane geometry and crystallography.

Formal definition: Let l be a line in the plane and t be a nonzero vector parallel to l. A glideReflexion g is defined by g = T_t ∘ R_l, where R_l is reflection across l and T_t is translation by t. Equivalently, g(p) is the reflection of p across l followed by translating along l by t. The square of a glideReflexion satisfies g^2 = T_{2t}, a translation along l by twice the step. If t = 0, g reduces to the pure reflection R_l.

Geometric properties and classification: GlideReflexionen are orientation-reversing isometries without fixed points (unless the translation is zero). They can be described by the reflecting line l and the translation direction along l. In the context of plane symmetry groups, glideReflexionen combine with translations and rotations to produce a variety of repeating patterns and are a key feature in several wallpaper groups.

Applications and occurrences: GlideReflexionen appear in tiling and pattern design, where motifs drift along a direction, and in crystallography to describe layer symmetries. In mathematics, they help classify plane isometries and analyze symmetry groups of patterns, such as certain wallpaper groups.