Editing Tetrahedron (section)

==Irregular tetrahedra==
{| class=wikitable width=440 align=right
|[[File:Tetrahedral subgroup tree.png|240px]]<br>Tetrahedral symmetry subgroup relations
|[[File:Tetrahedron symmetry tree.png|200px]]<br>Tetrahedral symmetries shown in tetrahedral diagrams
|}Tetrahedra which do not have four equilateral faces are categorized and named by the symmetries they do possess.

If all three pairs of opposite edges of a tetrahedron are [[perpendicular]], then it is called an ''[[orthocentric tetrahedron]]''. When only one pair of opposite edges are perpendicular, it is called a '''''semi-orthocentric tetrahedron'''''.
In a ''[[trirectangular tetrahedron]]'' the three face angles at ''one'' vertex are [[right angle]]s, as at the corner of a cube.

An '''isodynamic tetrahedron''' is one in which the [[cevian]]s that join the vertices to the [[Incircle and excircles of a triangle|incenters]] of the opposite faces are [[Concurrent lines|concurrent]].

An '''isogonic tetrahedron''' has concurrent cevians that join the vertices to the points of contact of the opposite faces with the [[inscribed sphere]] of the tetrahedron.
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===Disphenoid===
{{Main|Disphenoid}}
[[File:Oblate tetrahedrille cell.png|thumb|A space-filling tetrahedral disphenoid inside a cube. Two edges have [[dihedral angle]]s of 90°, and four edges have dihedral angles of 60°.]]
A [[disphenoid]] is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute.  The regular tetrahedron is a special case of a disphenoid. Other names for the same shape include bisphenoid, isosceles tetrahedron and equifacial tetrahedron.
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===Orthoschemes===
[[File:Triangulated cube.svg|class=skin-invert-image|thumb|400px|A cube dissected into six characteristic orthoschemes.]]
A '''3-orthoscheme''' is a tetrahedron where all four faces are [[Triangle#By internal angles|right triangles]]. A 3-orthoscheme is not a disphenoid, because its opposite edges are not of equal length. It is not possible to construct a [[Disphenoid#Special cases and generalizations|disphenoid]] with right triangle or obtuse triangle faces.

An [[Schläfli orthoscheme|orthoscheme]] is an irregular [[simplex]] that is the [[convex hull]] of a [[Tree (graph theory)|tree]] in which all edges are mutually perpendicular. In a 3-dimensional orthoscheme, the tree consists of three perpendicular edges connecting all four vertices in a linear path that makes two right-angled turns. The 3-orthoscheme is a tetrahedron having two right angles at each of two vertices, so another name for it is '''''birectangular tetrahedron'''''. It is also called a '''''quadrirectangular''''' tetrahedron because it contains four right angles.<ref>{{Cite journal | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1–3 | pages=59–71 | doi=10.1016/0898-1221(89)90148-X | doi-access=free }}</ref>

Coxeter also calls quadrirectangular tetrahedra "characteristic tetrahedra", because of their integral relationship to the regular polytopes and their symmetry groups.{{Sfn|Coxeter|1973|pp=71-72|loc=§4.7 Characteristic tetrahedra}} For example, the special case of a 3-orthoscheme with equal-length perpendicular edges is [[Polyhedron#Ancient|characteristic of the cube]], which means that the cube can be subdivided into instances of this orthoscheme. If its three perpendicular edges are of unit length, its remaining edges are two of length {{radic|2}} and one of length {{radic|3}}, so all its edges are edges or diagonals of the cube. The cube {{CDD|node_1|4|node|3|node}} can be dissected into six such 3-orthoschemes {{CDD|node|4|node|3|node}} four different ways, with all six surrounding the same {{radic|3}} cube diagonal. The cube can also be dissected into 48 ''smaller'' instances of this same characteristic 3-orthoscheme (just one way, by all of its symmetry planes at once). The characteristic tetrahedron of the cube is an example of a [[Heronian tetrahedron]].

Every regular polytope, including the regular tetrahedron, has its [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]. There is a 3-orthoscheme, which is the "characteristic tetrahedron of the regular tetrahedron". The regular tetrahedron {{Coxeter–Dynkin diagram|node_1|3|node|3|node}} is subdivided into 24 instances of its characteristic tetrahedron {{Coxeter–Dynkin diagram|node|3|node|3|node}} by its planes of symmetry. The 24 characteristic tetrahedra of the regular tetrahedron occur in two mirror-image forms, 12 of each.

{| class="wikitable floatright"
!colspan=6|Characteristics of the regular tetrahedron{{Sfn|Coxeter|1973|pp=292–293|loc=Table I(i); "Tetrahedron, 𝛼<sub>3</sub>"}}
|-
!align=right|
!align=center|edge
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral
|-
!align=right|𝒍
|align=center|<small><math>2</math></small>
|align=center|<small>109°28′16″</small>
|align=center|<small><math>\pi - 2\kappa</math></small>
|align=center|<small>70°31′44″</small>
|align=center|<small><math>\pi - 2\psi</math></small>
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{4}{3}} \approx 1.155</math></small>
|align=center|<small>70°31′44″</small>
|align=center|<small><math>2\kappa</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|<small><math>1</math></small>
|align=center|<small>54°44′8″</small>
|align=center|<small><math>\tfrac{\pi}{2} - \kappa</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small>
|align=center|<small>54°44′8″</small>
|align=center|<small><math>\tfrac{\pi}{2} - \kappa</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
|
|
|
|
|
|-
!align=right|<small><math>_0R/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{3}{2}} \approx 1.225</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_1R/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_2R/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
|
|
|
|
|
|-
!align=right|<small><math>\kappa</math></small>
|align=center|
|align=center|<small>35°15′52″</small>
|align=center|<small><math>\tfrac{\text{arc sec }3}{2}</math></small>
|align=center|
|align=center|
|}
If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths <small><math>\sqrt{\tfrac{4}{3}}</math></small>, <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (edges that are the ''characteristic radii'' of the regular tetrahedron). The 3-edge path along orthogonal edges of the orthoscheme is <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, first from a tetrahedron vertex to an tetrahedron edge center, then turning 90° to an tetrahedron face center, then turning 90° to the tetrahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a [[30-60-90 triangle|60-90-30 triangle]] which is one-sixth of a tetrahedron face. The three faces interior to the tetrahedron are: a right triangle with edges <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, a right triangle with edges <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, and a right triangle with edges <small><math>\sqrt{\tfrac{4}{3}}</math></small>, <small><math>\sqrt{\tfrac{3}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>.

===Space-filling tetrahedra===
A '''space-filling tetrahedron''' packs with directly congruent or enantiomorphous ([[mirror image]]) copies of itself to tile space.{{Sfn|Coxeter|1973|pp=33–34|loc=§3.1 Congruent transformations}} The cube can be dissected into six 3-orthoschemes, three left-handed and three right-handed (one of each at each cube face), and cubes can fill space, so the characteristic 3-orthoscheme of the cube is a space-filling tetrahedron in this sense. (The characteristic orthoscheme of the cube is one of the [[Hill tetrahedra]], a family of space-filling tetrahedra. All space-filling tetrahedra are [[scissors-congruent]] to a cube.)

A disphenoid can be a space-filling tetrahedron in the directly congruent sense, as in the [[disphenoid tetrahedral honeycomb]]. Regular tetrahedra, however, cannot fill space by themselves (moreover, it is not scissors-congruent to any other polyhedra which can fill the space, see [[Hilbert's third problem]]). The [[tetrahedral-octahedral honeycomb]] fills space with alternating regular tetrahedron cells and regular [[octahedron]] cells in a ratio of 2:1.

===Fundamental domains===
[[File:Coxeter-Dynkin 3-space groups.png|480px|thumb|For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra. They can be seen as points on and within a cube.]]An irregular tetrahedron which is the [[fundamental domain]]{{Sfn|Coxeter|1973|p=63|loc=§4.3 Rotation groups in two dimensions; notion of a ''fundamental region''}} of a [[Coxeter group|symmetry group]] is an example of a [[Goursat tetrahedron]]. The Goursat tetrahedra generate all the regular polyhedra (and many other uniform polyhedra) by mirror reflections, a process referred to as [[Wythoff construction|Wythoff's kaleidoscopic construction]].

For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a [[kaleidoscope]]. Unlike a cylindrical kaleidoscope, Wythoff's mirrors are located at three faces of a Goursat tetrahedron such that all three mirrors intersect at a single point. (The [[Coxeter-Dynkin diagram]] of the generated polyhedron contains three ''nodes'' representing the three mirrors. The dihedral angle between each pair of mirrors is encoded in the diagram, as well as the location of a single ''generating point'' which is multiplied by mirror reflections into the vertices of the polyhedron.)

Among the Goursat tetrahedra which generate 3-dimensional [[Honeycomb (geometry)|honeycombs]] we can recognize an orthoscheme (the characteristic tetrahedron of the cube), a double orthoscheme (the characteristic tetrahedron of the cube face-bonded to its mirror image), and the space-filling disphenoid illustrated [[#Disphenoid|above]].{{Sfn|Coxeter|1973|pp=71–72|loc=§4.7 Characteristic tetrahedra}} The disphenoid is the double orthoscheme face-bonded to its mirror image (a quadruple orthoscheme). Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can be [[Dissection into orthoschemes|dissected into characteristic tetrahedra of the cube]].
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===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
|}