Editing Octahedron (section)

=== Related figures ===
[[File:Compound of two tetrahedra.png|right|thumb|upright=0.8|The octahedron represents the central intersection of two tetrahedra]]
The interior of the [[polyhedral compound|compound]] of two dual [[tetrahedra]] is an octahedron, and this compound&mdash;called the [[stella octangula]]&mdash;is its first and only [[stellation]]. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. [[Rectification (geometry)|rectifying]] the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the [[cuboctahedron]] and [[icosidodecahedron]] relate to the other Platonic solids.

One can also divide the edges of an octahedron in the ratio of the [[golden ratio|golden mean]] to define the vertices of a [[regular icosahedron]]. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. Five octahedra define any given icosahedron in this fashion, and together they define a ''regular compound''. A regular icosahedron produced this way is called a ''snub octahedron''.{{r|kappraff}}

{{anchor|Trigonal antiprism}}The regular octahedron can be considered as the [[antiprism]], a [[Prism (geometry)|prism]] like polyhedron in which lateral faces are replaced by alternating equilateral triangles. It is also called ''trigonal antiprism''.{{sfnp|O'Keeffe|Hyde|2020|p=[https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA141 141]}} Therefore, it has the property of [[Quasiregular polyhedron|quasiregular]], a polyhedron in which two different polygonal faces are alternating and meet at a vertex.{{r|maekawa}}

[[Tetrahedral-octahedral honeycomb|Octahedra and tetrahedra]] can be alternated to form a vertex, edge, and face-uniform [[tessellation of space]]. This and the regular tessellation of [[cube]]s are the only such [[uniform honeycomb]]s in 3-dimensional space.

The uniform [[tetrahemihexahedron]] is a [[tetrahedral symmetry]] [[faceting]] of the regular octahedron, sharing [[edge arrangement|edge]] and [[vertex arrangement]]. It has four of the triangular faces, and 3 central squares.

A regular octahedron is a [[n-ball|3-ball]] in the [[Taxicab geometry|Manhattan ({{math|''ℓ''}}{{sub|1}}) metric]].