Editing Chirality (mathematics) (section)

==Chirality and symmetry group==
A figure is achiral<!--sic!--> if and only if its [[symmetry group]] contains at least one ''[[orientation-reversing]]'' isometry. (In Euclidean geometry any [[isometry]] can be written as <math>v\mapsto Av+b</math> with an [[orthogonal matrix]] <math>A</math> and a vector <math>b</math>. The [[determinant]] of <math>A</math> is either 1 or &minus;1 then. If it is &minus;1 the isometry is orientation-reversing, otherwise it is orientation-preserving.

A general definition of chirality based on group theory exists.<ref>{{cite journal | author = Petitjean, M. | title = Chirality in metric spaces. In memoriam Michel Deza | journal = Optimization Letters | year=2020 | volume=14 | issue=2 | pages=329–338 | doi = 10.1007/s11590-017-1189-7 | doi-access=free }}</ref> It does not refer to any orientation concept: an [[isometry]] is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime.<ref>{{cite journal | author = Petitjean, M. | title = Chirality in geometric algebra | journal=Mathematics | year=2021 | volume=9 | issue=13 | at=1521 | no-pp=yes | doi = 10.3390/math9131521 | doi-access=free }}</ref><ref>{{cite arXiv |last=Petitjean |first=M. |date=2022 |title=Chirality in affine spaces and in spacetime |eprint=2203.04066 |class=math-ph }}</ref>