Editing Tetrahedron (section)

===Isometries of irregular tetrahedra===
The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a [[Point groups in three dimensions|3-dimensional point group]] is formed. Two other isometries (C<sub>3</sub>, [3]<sup>+</sup>), and (S<sub>4</sub>, [2<sup>+</sup>,4<sup>+</sup>]) can exist if the face or edge marking are included. Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges.
{| class=wikitable style="text-align:center; width:800px;"
!colspan=4|Tetrahedron name
!rowspan=3|Edge<br>equivalence<br>diagram
!rowspan=3|Description
|-
!colspan=4|[[List of spherical symmetry groups|Symmetry]]
|-
![[Schönflies notation|Schön.]]
![[Coxeter notation|Cox.]]
![[Orbifold notation|Orb.]]
![[Symmetry order|Ord.]]
|-
!colspan=4|Regular tetrahedron
|rowspan=2|[[File:Regular tetrahedron diagram.png|60px]]
|rowspan=2 align=left|{{center|Four '''equilateral''' triangles}}It forms the symmetry group ''T''<sub>d</sub>, isomorphic to the [[symmetric group]], ''S''<sub>4</sub>. A regular tetrahedron has [[Coxeter diagram]] {{CDD|node_1|3|node|3|node}} and [[Schläfli symbol]] {3,3}.
|-
||''T''<sub>d</sub><br>''T''||[3,3]<br>[3,3]<sup>+</sup>||*332<br>332|| 24<br>12
|-
!colspan=4|Triangular pyramid
|rowspan=2|[[File:Isosceles trigonal pyramid diagram.png|60px]]
|rowspan=2 align=left|{{center|An '''equilateral''' triangle base and three equal '''isosceles''' triangle sides}}It gives 6 isometries, corresponding to the 6 isometries of the base. As permutations of the vertices, these 6 isometries are the identity 1, (123), (132), (12), (13) and (23), forming the symmetry group ''C''<sub>3v</sub>, isomorphic to the [[symmetric group]], ''S''<sub>3</sub>. A triangular pyramid has Schläfli symbol {3}∨(&nbsp;).
|-
||''C''<sub>3v</sub><br>C<sub>3</sub> ||[3]<br>[3]<sup>+</sup> || *33<br>33 ||6<br>3

|-
!colspan=4|Mirrored sphenoid
|rowspan=2|[[File:Sphenoid diagram.png|60px]]
|rowspan=2 align=left|{{center|Two equal '''scalene''' triangles with a common base edge}}This has two pairs of equal edges (1,3), (1,4) and (2,3), (2,4) and otherwise no edges equal. The only two isometries are 1 and the reflection (34), giving the group ''C''<sub>s</sub>, also isomorphic to the [[cyclic group]], '''Z'''<sub>2</sub>.
|-
|''C''<sub>s</sub><br>=''C''<sub>1h</sub><br>=''C''<sub>1v</sub>||[&nbsp;] ||*|| 2

|-
!colspan=4|Irregular tetrahedron<br>(No symmetry)
|rowspan=2|[[File:Scalene tetrahedron diagram.png|60px]]
|rowspan=2 align=left|{{center|Four unequal triangles}}

Its only isometry is the identity, and the symmetry group is the [[trivial group]]. An irregular tetrahedron has Schläfli symbol (&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;).
|-
|C<sub>1</sub>||[&nbsp;]<sup>+</sup>||1||1

|-
!colspan=6 |[[Disphenoid]]s (Four equal triangles)
|-
!colspan=4|[[Tetragonal disphenoid]]
|rowspan=2|[[File:Tetragonal disphenoid diagram.png|60px]]
|rowspan=2 align=left|{{center|Four equal '''isosceles''' triangles}}

It has 8 isometries. If edges (1,2) and (3,4) are of different length to the other 4 then the 8 isometries are the identity 1, reflections (12) and (34), and 180° rotations (12)(34), (13)(24), (14)(23) and improper 90° rotations (1234) and (1432) forming the symmetry group ''D''<sub>2d</sub>.  A tetragonal disphenoid has Coxeter diagram {{CDD|node_h|2x|node_h|4|node}} and Schläfli symbol s{2,4}.
|-
|''D''<sub>2d</sub><br>S<sub>4</sub> ||[2<sup>+</sup>,4]<br>[2<sup>+</sup>,4<sup>+</sup>] ||2*2<br>2×|| 8<br>4
|-
!colspan=4|[[Rhombic disphenoid]]
|rowspan=2|[[File:Rhombic disphenoid diagram.png|60px]]
|rowspan=2 align=left|{{center|Four equal '''scalene''' triangles}}

It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the [[Klein four-group]] ''V''<sub>4</sub> or '''Z'''<sub>2</sub><sup>2</sup>, present as the point group ''D''<sub>2</sub>.  A rhombic disphenoid has Coxeter diagram {{CDD|node_h|2x|node_h|2x|node_h}} and Schläfli symbol sr{2,2}.
|-
|''D''<sub>2</sub> ||[2,2]<sup>+</sup> ||222||4

|-
!colspan=6 |Generalized disphenoids (2 pairs of equal triangles)
|-
!colspan=4|[[Digonal disphenoid]]
|rowspan=2|[[File:Digonal disphenoid diagram2.png|80px]]<br>[[File:Digonal disphenoid diagram.png|80px]]
|rowspan=2 align=left|{{center|Two pairs of equal '''isosceles''' triangles}} This gives two opposite edges (1,2) and (3,4) that are perpendicular but different lengths, and then the 4 isometries are 1, reflections (12) and (34) and the 180° rotation (12)(34). The symmetry group is ''C''<sub>2v</sub>, isomorphic to the [[Klein four-group]] ''V''<sub>4</sub>. A digonal disphenoid has Schläfli symbol {&nbsp;}∨{&nbsp;}.
|-
|''C''<sub>2v</sub><br>''C''<sub>2</sub> ||[2]<br>[2]<sup>+</sup> ||*22<br>22||4<br>2

|-
!colspan=4|Phyllic disphenoid
|rowspan=2|[[File:Half-turn tetrahedron diagram.png|80px]]<br> [[File:Half-turn tetrahedron diagram2.png|80px]]
|rowspan=2 align=left|{{center|Two pairs of equal '''scalene''' or '''isosceles''' triangles}}

This has two pairs of equal edges (1,3), (2,4) and (1,4), (2,3) but otherwise no edges equal. The only two isometries are 1 and the rotation (12)(34), giving the group ''C''<sub>2</sub> isomorphic to the [[cyclic group]], '''Z'''<sub>2</sub>.
|-
|''C''<sub>2</sub> ||[2]<sup>+</sup> ||22 ||2
|}