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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.
For each of the types D 1, D 2, and D 4 the distinction between the 3, 4, and 2 wallpaper groups, respectively, is determined by the translation vector associated with each reflection in the group: since isometries are in the same coset regardless of translational components, a reflection and a glide reflection with the same mirror are in the ...
p2mm: TRHVG (translation, 180° rotation, horizontal line reflection, vertical line reflection, and glide reflection) Formally, a frieze group is a class of infinite discrete symmetry groups of patterns on a strip (infinitely wide rectangle), hence a class of groups of isometries of the plane, or of a strip.
[2] [23] The composition of two glide reflections results in a translation symmetry with twice the translation vector. The symmetry group comprising glide reflections and associated translations is the frieze group p11g, and is isomorphic with the infinite cyclic group Z.
In abstract algebra, a cyclic group or monogenous group is a group, denoted C n ... With one generator, p1 has translations and p11g has glide reflections.
It has additional glide reflections whose axes are not reflection axes; rotations of order two (180°) are centred at the intersection of the glide reflection axes. All rotation centres lie on reflection axes. This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes.
Glide reflection. Glide reflections, denoted by G c,v,w, where c is a point in the plane, v is a unit vector in R 2, and w is non-null a vector perpendicular to v are a combination of a reflection in the line described by c and v, followed by a translation along w. That is, ,, =,, or in other words,
reflection with respect to a plane, and a translation in that plane, a rotation about an axis perpendicular to the plane, or a reflection with respect to a perpendicular plane; glide reflection with respect to a plane, and a translation in that plane; inversion in a point and any isometry keeping the point fixed