Editing Octahedron (section)

=== Characteristic orthoscheme ===
Like all regular convex polytopes, the octahedron can be [[Dissection into orthoschemes|dissected]] into an integral number of disjoint [[orthoscheme]]s, all of the same shape characteristic of the polytope. A polytope's [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] is a [[fundamental domain|fundamental]] property because the polytope is generated by reflections in the [[Facet (geometry)|facets]] of its orthoscheme. The orthoscheme occurs in two [[chiral]] forms which are mirror images of each other. The characteristic orthoscheme of a regular polyhedron is a [[Tetrahedron#Orthoschemes|quadrirectangular irregular tetrahedron]].

The faces of the octahedron's characteristic tetrahedron lie in the octahedron's mirror planes of [[Symmetry (geometry)|symmetry]]. The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess, among its mirror planes, some that do not pass through any of its faces. The octahedron's [[Coxeter group|symmetry group]] is denoted [[Octahedral symmetry|B<sub>3</sub>]]. The octahedron and its [[dual polytope]], the [[cube]], have the same symmetry group but different characteristic tetrahedra. 

The '''characteristic tetrahedron of the regular octahedron''' can be found by a canonical dissection{{Sfn|Coxeter|1973|p=130|loc=§7.6 The symmetry group of the general regular polytope|ps=; "simplicial subdivision".}} of the regular octahedron {{Coxeter–Dynkin diagram|node_1|3|node|4|node}} which subdivides it into 48 of these characteristic orthoschemes {{Coxeter–Dynkin diagram|node|3|node|4|node}} surrounding the octahedron's center. Three left-handed orthoschemes and three right-handed orthoschemes meet in each of the octahedron's eight faces, the six orthoschemes collectively forming a [[Tetrahedron#Trirectangular tetrahedron|trirectangular tetrahedron]]: a triangular pyramid with the octahedron face as its equilateral base, and its cube-cornered apex at the center of the octahedron.{{Sfn|Coxeter|1973|pp=70-71|loc=Characteristic tetrahedra; Fig. 4.7A}}

{| class="wikitable floatright"
!colspan=6|Characteristics of the regular octahedron{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i); "Octahedron, 𝛽<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>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|align=center|<small>109°28{{prime}}</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>54°44{{prime}}8{{pprime}}</small>
|align=center|<small><math>\tfrac{\pi}{2} - \kappa</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</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>45°</small>
|align=center|<small><math>\tfrac{\pi}{4}</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>35°15{{prime}}52{{pprime}}</small>
|align=center|<small><math>\kappa</math></small>
|align=center|<small>45°</small>
|align=center|<small><math>\tfrac{\pi}{4}</math></small>
|-
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!align=right|<small><math>_0R/l</math></small>
|align=center|<small><math>\sqrt{2} \approx 1.414</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_1R/l</math></small>
|align=center|<small><math>1</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_2R/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</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{{prime}}52{{pprime}}</small>
|align=center|<small><math>\tfrac{\text{arc sec }3}{2}</math></small>
|align=center|
|align=center|
|}
If the octahedron 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{2}</math></small>, <small><math>1</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small> (edges that are the ''characteristic radii'' of the octahedron). 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{2}{3}}</math></small>, first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to the octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a [[30-60-90 triangle|90-60-30 triangle]] which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a [[45-45-90 triangle|45-90-45 triangle]] with edges <small><math>1</math></small>, <small><math>\sqrt{2}</math></small>, <small><math>1</math></small>, a right triangle with edges <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>1</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, and a right triangle with edges <small><math>\sqrt{\tfrac{4}{3}}</math></small>, <small><math>\sqrt{2}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>.