the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a frieze group generated by a single glide reflection. Chirality in three dimensions In three dimensions, every figure that possesses a mirror plane of symmetry S1, an inversion center of symmetry S2, or a higher improper rotation (rotoreflection) Sn axis of symmetry is achiral. (A plane of symmetry of a figure is a plane , such that is invariant under the mapping , when is chosen to be the --plane of the coordinate system. A center of symmetry of a figure is a point , such that is invariant under the mapping , when is chosen to be the origin of the coordinate system.) Note, however, that there are achiral figures lacking both plane and center of symmetry. An example is the figure which is invariant under the orientation reversing isometry and thus achiral, but it has neither plane nor center of symmetry. The figure also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry. Achiral figures can have a center axis. Knot theory A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it is called a chiral knot. For example, the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral. See also Chiral polytope Chirality (physics) Chirality (chemistry) Asymmetry Skewness Vertex algebra References Further reading External links Symmetry, Chirality, Symmetry Measures and Chirality Measures: General Definitions Chiral Polyhedra by Eric W. Weisstein, The Wolfram Demonstrations Project. Chiral manifold at the Manifold Atlas. Knot theory Polyhedra Chirality Topology