Editing Octahedron (section)

==Related polyhedra==
A regular octahedron can be augmented into a [[tetrahedron]] by adding 4 tetrahedra on alternated faces. Adding tetrahedra to all 8 faces creates the [[stellated octahedron]].
{| class=wikitable
|[[File:Triangulated tetrahedron.png|120px]]
|[[File:Compound of two tetrahedra.png|120px]]
|-
![[tetrahedron]]
![[stellated octahedron]]
|}

The octahedron is one of a family of uniform polyhedra related to the cube.

{{Octahedral truncations}}

It is also one of the simplest examples of a [[hypersimplex]], a polytope formed by certain intersections of a [[hypercube]] with a [[hyperplane]].

The octahedron is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,''n''}, continuing into the [[Hyperbolic space|hyperbolic plane]].
{{Triangular regular tiling}}

===Tetratetrahedron===
The regular octahedron can also be considered a ''[[rectification (geometry)|rectified]] tetrahedron'' – and can be called a ''tetratetrahedron''. This can be shown by a 2-color face model. With this coloring, the octahedron has [[tetrahedral symmetry]].

Compare this truncation sequence between a tetrahedron and its dual:
{{Tetrahedron family}} <!-- This template shows too many figures. It needs replacing with the simple set described in the text -->

The above shapes may also be realized as slices orthogonal to the long diagonal of a [[tesseract]]. If this diagonal is oriented vertically with a height of 1, then the first five slices above occur at heights ''r'', {{sfrac|3|8}}, {{sfrac|1|2}}, {{sfrac|5|8}}, and ''s'', where ''r'' is any number in the range {{nowrap|0 < ''r'' ≤ {{sfrac|1|4}}}}, and ''s'' is any number in the range {{nowrap|{{sfrac|3|4}} ≤ ''s'' < 1}}.

The octahedron as a ''tetratetrahedron'' exists in a sequence of symmetries of quasiregular polyhedra and tilings with [[vertex configuration]]s (3.''n'')<sup>2</sup>, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With [[orbifold notation]] symmetry of *''n''32 all of these tilings are [[Wythoff construction]]s within a [[fundamental domain]] of symmetry, with generator points at the right angle corner of the domain.<ref>{{cite book |last=Coxeter |first=H.S.M. |author-link=Harold Scott MacDonald Coxeter |title-link=Regular Polytopes (book) |title=Regular Polytopes |edition=Third |date=1973 |publisher=Dover |isbn=0-486-61480-8 |at=Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction}}</ref><ref>{{citation |last=Huson |first=Daniel H. |title= Two Dimensional Symmetry Mutation |date=September 1998 |url=https://www.researchgate.net/publication/2422380}}</ref>
{{Quasiregular3 small table}}

===Trigonal antiprism===
As a trigonal [[antiprism]], the octahedron is related to the hexagonal dihedral symmetry family.
{{Hexagonal dihedral truncations}}

{{UniformAntiprisms}}

===Other related polyhedra===
Truncation of two opposite vertices results in a [[square bifrustum]].

The octahedron can be generated as the case of a 3D [[superellipsoid]] with all exponent values set to 1.