There are 3 uniform colorings of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111. The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. Truncation of two opposite vertices results in a square bifrustum. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces. The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Another is a tessellation of octahedra and cuboctahedra. This is the only such tiling save the regular tessellation of cubes, and is one of the 28 convex uniform honeycombs. Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound. 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. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron.

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. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e.

The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. The octahedron represents the central intersection of two tetrahedra The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is It is also a triangular antiprism in any of four orientations.Īn octahedron is the three-dimensional case of the more general concept of a cross polytope.Ī regular octahedron is a 3-ball in the Manhattan ( ℓ 1) metric. It is a square bipyramid in any of three orthogonal orientations.

The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.Ī regular octahedron is the dual polyhedron of a cube. In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces.

Tall octahedron template