Editing Octahedron (section)

== Other types of octahedra ==
[[File:Gyrobifastigium.png|thumb|upright=0.9|A regular faced convex polyhedron, the [[gyrobifastigium]].]]
An octahedron can be any polyhedron with eight faces. In a previous example, the regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges.<ref>{{Cite web |url=http://www.uwgb.edu/dutchs/symmetry/polynum0.htm |title=Enumeration of Polyhedra |access-date=2 May 2006 |archive-url=https://web.archive.org/web/20111010185122/http://www.uwgb.edu/dutchs/symmetry/polynum0.htm |archive-date=10 October 2011 |url-status=dead }}</ref> There are 257 topologically distinct ''convex'' octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively.<ref>{{cite web| url = http://www.numericana.com/data/polycount.htm| title = Counting polyhedra}}</ref><ref>{{cite web |url=http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm |title=Polyhedra with 8 Faces and 6-8 Vertices |access-date=14 August 2016 |url-status=dead |archive-url=https://web.archive.org/web/20141117072140/http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm |archive-date=17 November 2014  }}</ref> (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) Some of the polyhedrons do have eight faces aside from being square bipyramids in the following:
* [[Hexagonal prism]]: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges.
* Heptagonal [[Pyramid (geometry)|pyramid]]: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). All triangular faces can't be equilateral.
* [[Truncated tetrahedron]]: The four faces from the tetrahedron are truncated to become regular hexagons, and there are four more equilateral triangle faces where each tetrahedron vertex was truncated.
* [[Tetragonal trapezohedron]]: The eight faces are congruent [[kite (geometry)|kites]].
* [[Gyrobifastigium]]: Two uniform [[triangular prisms]] glued over one of their square sides so that no triangle shares an edge with another triangle (Johnson solid 26).
* [[Truncated triangular trapezohedron]], also called Dürer's solid: Obtained by truncating two opposite corners of a cube or rhombohedron, this has six pentagon faces and two triangle faces.<ref>{{citation|last1=Futamura|first1=F.|author1-link=Fumiko Futamura|first2=M.|last2=Frantz|first3=A.|last3=Crannell|author3-link= Annalisa Crannell |title=The cross ratio as a shape parameter for Dürer's solid|journal=Journal of Mathematics and the Arts|volume=8|issue=3–4|year=2014|pages=111–119|doi=10.1080/17513472.2014.974483|arxiv=1405.6481|s2cid=120958490}}</ref>
* Octagonal [[hosohedron]]: degenerate in Euclidean space, but can be realized spherically.

[[File:Br2-anim.gif|thumb|upright=0.9|[[Bricard octahedron]] with an [[antiparallelogram]] as its equator. The axis of symmetry passes through the plane of the antiparallelogram.]]
The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it:
* Triangular [[antiprism]]s: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral.
* Tetragonal [[bipyramid]]s, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares.
* [[Schönhardt polyhedron]], a non-convex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.
* [[Bricard octahedron]], a non-convex self-crossing [[flexible polyhedron]]