Editing Trapezohedron (section)

==Symmetry==
The [[symmetry group]] of an {{mvar|n}}-gonal trapezohedron is {{math|1=D{{sub|''n''d}} = D{{sub|''n''v}}}}, of order {{math|4''n''}}, except in the case of {{math|1=''n'' = 3}}: a cube has the larger symmetry group {{math|O<sub>d</sub>}} of order {{math|1=48 = 4×(4×3)}}, which has four versions of {{math|D<sub>3d</sub>}} as subgroups.

The [[Point groups in three dimensions#Rotation groups|rotation group]] of an {{mvar|n}}-trapezohedron is {{math|D<sub>''n''</sub>}}, of order {{math|2''n''}}, except in the case of {{math|1=''n'' = 3}}: a cube has the larger rotation group {{math|O}} of order {{math|1=24 = 4×(2×3)}}, which has four versions of {{math|D<sub>3</sub>}} as subgroups.

Note: Every {{mvar|n}}-trapezohedron with a [[Regular skew polygon|regular zig-zag skew]] {{math|'''2'''''n''}}-gon base and {{math|2''n''}} congruent non-twisted kite faces has the same (dihedral) symmetry group as the [[Dual uniform polyhedron|dual-uniform]] {{mvar|n}}-trapezohedron, for {{math|''n'' ≥ 4}}.

One degree of freedom within symmetry from {{math|D<sub>''n''d</sub>}} (order {{math|4''n''}}) to {{math|D<sub>''n''</sub>}} (order {{math|2''n''}}) changes the congruent kites into congruent quadrilaterals with three edge lengths, called ''twisted kites'', and the {{mvar|n}}-trapezohedron is called a ''twisted trapezohedron''. (In the limit, one edge of each quadrilateral goes to zero length, and the {{mvar|n}}-trapezohedron becomes an {{mvar|n}}-[[bipyramid]].)

If the kites surrounding the two peaks are not twisted but are of two different shapes, the {{mvar|n}}-trapezohedron can only have {{math|C<sub>''n''v</sub>}} (cyclic with vertical mirrors) symmetry, order {{math|2''n''}}, and is called an ''unequal'' or ''asymmetric trapezohedron''. Its dual is an ''unequal {{mvar|n}}-[[antiprism]]'', with the top and bottom {{mvar|n}}-gons of different radii.

If the kites are twisted and are of two different shapes, the {{mvar|n}}-trapezohedron can only have {{math|C<sub>''n''</sub>}} (cyclic) symmetry, order {{mvar|n}}, and is called an ''unequal twisted trapezohedron''.

{| class=wikitable
|+ style="text-align:center;"|Example: variations with hexagonal trapezohedra (''n'' = 6)
!Trapezohedron type
!colspan=2|Twisted trapezohedron
!Unequal trapezohedron
!Unequal twisted trapezohedron
|- style="text-align:center;"
![[List of finite spherical symmetry groups|Symmetry group]]
|colspan=2|D<sub>6</sub>, (662), [6,2]<sup>+</sup>
|C<sub>6v</sub>, (*66), [6]
|C<sub>6</sub>, (66), [6]<sup>+</sup>
|- style="text-align:center;"
!Polyhedron image
|[[File:Twisted_hexagonal_trapezohedron.png|160px]]
|[[File:Twisted_hexagonal_trapezohedron2.png|160px]]
|[[File:Unequal_hexagonal_trapezohedron.png|160px]]
|[[File:Unequal_twisted_hexagonal_trapezohedron.png|160px]]
|- style="text-align:center;"
!Net
|[[File:Twisted_hexagonal_trapezohedron net.png|160px]]
|[[File:Twisted_hexagonal_trapezohedron2 net.png|160px]]
|[[File:Unequal_hexagonal_trapezohedron net.png|160px]]
|[[File:Unequal_twisted_hexagonal_trapezohedron net.png|160px]]
|}