In geometry, an octahedron (pl.: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex and non-convex shapes.
Regular
Main article: Regular octahedron
The regular octahedron has eight equilateral triangle sides, six vertices at which four sides meet, and twelve edges. Its dual polyhedron is a cube. It can be formed as the convex hull of the six axis-parallel unit vectors in three-dimensional Euclidean space. It is one of the five Platonic solids, and the three-dimensional case of an infinite family of regular polytopes, the cross polytopes. Although it does not tile space by itself, it can tile space together with the regular tetrahedron to form the tetrahedral-octahedral honeycomb.
It occurs in nature in certain crystals, in architecture as part of certain types of space frame, and in popular culture as the shape of certain eight-sided dice.
Combinatorially equivalent to the regular octahedron
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 antiprisms: 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 bipyramids, 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
Other convex polyhedra
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.1 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.23 (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.)
Notable eight-sided convex polyhedra include:
Wikimedia Commons has media related to Polyhedra with 8 faces.References
"Enumeration of Polyhedra". Archived from the original on 10 October 2011. Retrieved 2 May 2006. https://web.archive.org/web/20111010185122/http://www.uwgb.edu/dutchs/symmetry/polynum0.htm ↩
"Counting polyhedra". http://www.numericana.com/data/polycount.htm ↩
"Polyhedra with 8 Faces and 6-8 Vertices". Archived from the original on 17 November 2014. Retrieved 14 August 2016. https://web.archive.org/web/20141117072140/http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm ↩