In all cases we multiply the position vector by an orthogonal matrix and add a vector; if the determinant is 1 we have a rotation, a translation, or the identity, and if it is & minus; 1 we have a glide reflection or a reflection.
42.
""'Charnia " "'is a genus of frond-like Ediacaran lifeforms with segmented, leaf-like ridges branching alternately to the right and left from a zig-zag medial suture ( thus exhibiting glide reflection, or opposite isometry ).
43.
In the case of symmetry groups in the plane, additional parameters are the direction of the translation vector, and, for the frieze groups with horizontal line reflection, glide reflection, or 180?rotation ( groups 3 7 ), the position of the reflection axis or rotation point in the direction perpendicular to the translation vector.
44.
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 same coset.
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