Editing Chirality (mathematics) (section)

==Chirality in three dimensions==
[[File:Chiralität von Würfeln V.1.svg|thumb|Pair of chiral [[dice]] (enantiomorphs)]]
In three dimensions, every figure that possesses a [[mirror plane of symmetry]] ''S<sub>1</sub>'', an inversion center of symmetry ''S<sub>2</sub>'', or a higher [[improper rotation]] (rotoreflection) ''S<sub>n</sub>'' axis of symmetry<ref>{{cite web|title=2. Symmetry operations and symmetry elements|url=http://chemwiki.ucdavis.edu/Theoretical_Chemistry/Symmetry/Symmetry_operations_and_symmetry_elements|website=chemwiki.ucdavis.edu|date=3 March 2014 |access-date=25 March 2016}}</ref> is achiral. (A ''plane of symmetry'' of a figure <math>F</math> is a plane <math>P</math>, such that <math>F</math> is invariant under the mapping <math>(x,y,z)\mapsto(x,y,-z)</math>, when <math>P</math> is chosen to be the <math>x</math>-<math>y</math>-plane of the coordinate system. A ''center of symmetry'' of a figure <math>F</math> is a point <math>C</math>, such that <math>F</math> is invariant under the mapping <math>(x,y,z)\mapsto(-x,-y,-z)</math>, when <math>C</math> 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

:<math>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\}</math>

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

:<math>F_1=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}</math>

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

Achiral figures can have a [[Point groups in three dimensions#Center of symmetry|center axis]].