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A glide reflection is the composition of a reflection across a line and a translation parallel to the line. This footprint trail has glide-reflection symmetry. Applying the glide reflection maps each left footprint into a right footprint and vice versa.
A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell.
A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a ...
This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a parallel line. The identity isometry, defined by I(p) = p for all points p is a special case of a translation, and also a special case of a rotation. It is the ...
glide reflection with respect to a plane, and a translation in that plane inversion in a point and any isometry keeping the point fixed rotation by 180° about an axis and reflection in a plane through that axis
Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold rotational symmetry. Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis of reflection and glide reflection.
A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]
Fractals also exhibit a form of scale symmetry, where smaller portions of the fractal are similar in shape to larger portions. [11] Other symmetries include glide reflection symmetry (a reflection followed by a translation) and rotoreflection symmetry (a combination of a rotation and a reflection [12]).