Editing Antiprism (section)

=== Right antiprism ===
For an antiprism with [[Regular polygon|regular {{mvar|n}}-gon]] bases, one usually considers the case where these two copies are twisted by an angle of {{math|{{sfrac|180|''n''}}}} degrees. The axis of a regular polygon is the line [[perpendicular]] to the polygon plane and lying in the polygon centre.

For an antiprism with [[Congruence (geometry)|congruent]] regular {{mvar|n}}-gon bases, twisted by an angle of {{math|{{sfrac|180|''n''}}}} degrees, more regularity is obtained if the bases have the same axis: are ''[[coaxial]]''; i.e. (for non-[[Coplanarity|coplanar]] bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a '''right antiprism''', and its {{math|2''n''}} side faces are [[isosceles triangle]]s.<ref name=oh>{{cite book
 | last1 = Alsina | first1 = Claudi
 | last2 = Nelsen | first2 = Roger B.
 | year = 2015
 | title = A Mathematical Space Odyssey: Solid Geometry in the 21st Century
 | publisher = [[Mathematical Association of America]]
 | url = https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA87
 | page = 87
 | isbn = 978-1-61444-216-5
 | volume = 50
}}</ref>

The [[point groups in three dimensions|symmetry group]] of a right {{mvar|n}}-antiprism is {{math|1=D{{sub|''n''d}}}} of order {{math|4''n''}} known as an [[antiprismatic symmetry]], because it could be obtained by rotation of the bottom half of a prism by <math> \pi/n </math> in relation to the top half. A concave polyhedron created in this way would have this symmetry group, hence prefix "anti" before "prismatic".<ref>{{cite book
 | last1 = Flusser | first1 = J.
 | last2 = Suk | first2 = T.
 | last3 = Zitofa | first3 = B.
 | year = 2017
 | title = 2D and 3D Image Analysis by Moments
 | publisher = [[John Wiley & Sons]]
 | isbn = 978-1-119-03935-8
 | url = https://books.google.com/books?id=jwKLDQAAQBAJ&pg=PA126
 | page = 126
}}</ref> There are two exceptions having groups different than {{math|D<sub>''n''d</sub>}}:
*{{math|1=''n'' = 2}}: the regular [[tetrahedron]], which has the larger symmetry group {{math|T<sub>d</sub>}} of order [[List of spherical symmetry groups#Polyhedral symmetry|{{math|24}}]], which has three versions of {{math|D<sub>2d</sub>}} as subgroups;
*{{math|1=''n'' = 3}}: the regular [[octahedron]], which has the larger symmetry group {{math|O<sub>h</sub>}} of order {{math|48}}, which has four versions of {{math|D<sub>3d</sub>}} as subgroups.<ref>{{cite book
 | last1 = O'Keeffe | first1 = Michael
 | last2 = Hyde | first2 = Bruce G.
 | title = Crystal Structures: Patterns and Symmetry
 | year = 2020
 | url = https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA140
 | page = 140
 | publisher = [[Dover Publications]]
 | isbn = 978-0-486-83654-6
}}</ref>
If a right 2- or 3-antiprism is not uniform, then its symmetry group is {{math|D<sub>2d</sub>}} or {{math|D<sub>3d</sub>}} as usual.<br />
The symmetry group contains [[Inversion in a point|inversion]] [[if and only if]] {{mvar|n}} is odd.

The [[Rotation group SO(3)|rotation group]] is {{math|D<sub>''n''</sub>}} of order {{math|2''n''}}, except in the cases of:
*{{math|1=''n'' = 2}}: the regular tetrahedron, which has the larger rotation group {{math|T}} of order {{math|12}}, which has only one subgroup {{math|D<sub>2</sub>}};
*{{math|1=''n'' = 3}}: the regular octahedron, which has the larger rotation group {{math|O}} of order {{math|24}}, which has four versions of {{math|D<sub>3</sub>}} as subgroups.
If a right 2- or 3-antiprism is not uniform, then its rotation group is {{math|D<sub>2</sub>}} or {{math|D<sub>3</sub>}} as usual.<br />
The right {{mvar|n}}-antiprisms have congruent regular {{mvar|n}}-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform {{mvar|n}}-antiprism, for {{math|''n'' ≥ 4}}.