Editing Tetrahedron (section)

===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>
|-
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!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>
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!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|
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!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|
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!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>.