Science Fair Project Encyclopedia

Science Fair Project Encyclopedia Contents Page

Categories: Geometry

Chirality (mathematics)

Contents

1 Geometry

1.1 Chirality in three dimensions
1.2 Chirality in two dimensions

2 Knot theory

3 Related topics

Geometry

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly can't be mapped to its mirror images by rotations and translations alone. Such objects then come in two forms, called enantiomorphs. The word chirality is derived from the greek χειρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the greek εναντιος (enantios) 'opposite' and μορφη (morphe) 'form'. A non-chiral figure is also called achiral.

A figure is achiral if and only if its symmetry group contains at least one indirect (orientation-reversing) isometry. (If you need further explanation of the concept: the orientation-reversing isometries of three-dimensional space which have a fixed point are just the improper rotations.)

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. The helix and Möbius strip also exhibit chirality.

Chirality in three dimensions

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 $F$ is a plane $P$ , such that $F$ is invariant under the mapping $(x,y,z)\mapsto(x,y,-z)$ , when $P$ is chosen to be the $x$ - $y$ -plane of the coordinate system. A center of symmetry of a figure $F$ is a point $C$ , such that $F$ is invariant under the mapping $(x,y,z)\mapsto(-x,-y,-z)$ , when $C$ 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

$F_0=\left\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,-2,-1)\right\}$

which is invariant under the orientation reversing isometry $(x,y,z)\mapsto(-y,x,-z)$ and thus achiral, but it has neither plane nor center of symmetry. The figure

$F_1=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}$

also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry.

Chirality in two dimensions

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 $F$ is a line $L$ , such that $F$ is invariant under the mapping $(x,y)\mapsto(x,-y)$ , when $L$ is chosen to be the $x$ -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.

Knot theory

A knot is called achiral if it can be continuously deformed into its own mirror image, otherwise it is called chiral. More precisely, a knot $K$ is achiral if and only if there exists an orientation-reversing homeomorphism of R³ mapping $K$ to itself. For example the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral.

All Science Fair Projects