Home > Chirality (mathematics)

A figure is achiral if and only if its symmetry group contains at least one indirect (orientation reversing) isometry; see improper rotation for a discussion in group theory terms, and handedness for informal explanation.

Many familiar objects are chiral - for instance, a right glove and left glove are enantiomorphic, and so are the S and Z tetrominoes of the popular video game Tetris. A helix also has chirality.

In three dimensions, every figure which possesses a plane of symmetry or a center 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.

In two dimensions, every figure which possesses a line of symmetry is achiral, and it can be shown that every bounded achiral figure must have a line of symmetry. (A line of symmetry of a figure is a line , such that is invariant under the mapping , when is chosen to be the -axis of the coordinate system.) Consider the following pattern:

This figure is chiral, as it is not identical to its mirror image:

But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no line of symmetry.

Related topics

• Helix
• Chirality (physics)
• Chirality (chemistry)