Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone.
A chiral object and its mirror image are said to be enantiomorphs.
The word chirality is derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'.
Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule.
Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes.
In contrast thin gloves may not be considered chiral if you can wear them inside-out.
[1] The J-, L-, S- and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space.
A figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry.
A general definition of chirality based on group theory exists.
The resulting chirality definition works in spacetime.
[3][4] In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry.
For that reason, a triangle is achiral if it is equilateral or isosceles, and is chiral if it is scalene.
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 axis of symmetry.
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[5] is achiral.
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
A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it is called a chiral knot.
This means that as physical necklaces on a table the left and right ones can be rotated into their mirror image while remaining on the table. The one in the middle, however, would have to be picked up and turned in three dimensions.
