Editing Parallelepiped (section)

==Special cases by symmetry==
{| class=wikitable width=440 align=center
|[[File:Full octahedral group; subgroups Hasse diagram; inversion.svg|350px]]<BR>Octahedral symmetry subgroup relations with [[point reflection|inversion center]]
|[[File:Special_cases_of_parallelepiped.svg|300px]]<BR>Special cases of the parallelepiped
|}
{| class=wikitable width=900
!Form
![[Cube]]
![[Square cuboid]]
![[Trigonal trapezohedron]]
![[Rectangular cuboid]]
!Right rhombic [[Prism (geometry)|prism]]
!Right parallelogrammic [[Prism (geometry)|prism]]
!Oblique rhombic [[Prism (geometry)|prism]]
|- align=center
!Constraints
|<math>a=b=c</math><BR><math>\alpha=\beta=\gamma=90^\circ</math>
|<math>a=b</math><BR><math>\alpha=\beta=\gamma=90^\circ</math>
|<math>a=b=c</math><BR><math>\alpha=\beta=\gamma</math>
|&nbsp;<BR><math>\alpha=\beta=\gamma=90^\circ</math>
|<math>a=b</math><BR><math>\alpha=\beta=90^\circ</math>
|&nbsp;<BR><math>\alpha=\beta=90^\circ</math>
|<math>a=b</math><BR><math>\alpha=\beta</math>
|- align=center
![[List of finite spherical symmetry groups|Symmetry]]
|[[Octahedral symmetry|O<sub>h</sub>]]<br/>order 48
|D<sub>4h</sub><br/>order 16
|D<sub>3d</sub><br/>order 12
|colspan="2"|D<sub>2h</sub><br/>order 8
|colspan="2"|[[Cyclic symmetries|C<sub>2h</sub>]]<br/>order 4
|- align=center
!Image
|[[File:Cubic.svg|80px]]
|[[File:Tetragonal.svg|60px]]
|[[File:Rhombohedral.svg|80px]]
|[[File:Orthorhombic.svg|60px]]
|[[File:Rhombic prism.svg|60px]]
|[[File:Monoclinic2.svg|60px]]
||[[File:Clinorhombic prism.svg|80px]]
|- align=center
!Faces
|6 squares
|2 squares,<BR>4 rectangles
|6 rhombi
|6 rectangles
|4 rectangles,<BR>2 rhombi
|4 rectangles,<BR>2 parallelograms
|2 rhombi,<BR>4 parallelograms
|}
*The parallelepiped with O<sub>h</sub> symmetry is known as a '''[[cube]]''', which has six congruent square faces.
*The parallelepiped with D<sub>4h</sub> symmetry is known as a '''[[square cuboid]]''', which has two square faces and four congruent rectangular faces.
*The parallelepiped with D<sub>3d</sub> symmetry is known as a '''[[trigonal trapezohedron]]''', which has six congruent [[rhombus|rhombic]] faces (also called an '''isohedral rhombohedron''').
*For parallelepipeds with D<sub>2h</sub> symmetry, there are two cases:
**'''[[Rectangular cuboid]]''': it has six rectangular faces (also called a '''rectangular parallelepiped''', or sometimes simply a ''cuboid'').
**'''Right rhombic prism''': it has two rhombic faces and four congruent rectangular faces.
**:Note: the fully rhombic special case, with two rhombic faces and four congruent square faces <math>(a=b=c)</math>, has the same name, and the same symmetry group (D<sub>2h</sub> , order 8).
*For parallelepipeds with C<sub>2h</sub> symmetry, there are two cases:
**'''Right parallelogrammic prism''': it has four rectangular faces and two parallelogrammic faces.
**'''Oblique rhombic prism''': it has two rhombic faces, while of the other faces, two adjacent ones are equal and the other two also (the two pairs are each other's mirror image).