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 These reduce to the equations for the regular octahedron when Geometric relations The formulas for the surface area and volume expand to becomeĪdditionally the inertia tensor of the stretched octahedron is If an octahedron has been stretched so that it obeys the equation

Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 rather than 4 triangles). The surface area A and the volume V of a regular octahedron of edge length a are: In an x– y– z Cartesian coordinate system, the octahedron with center coordinates ( a, b, c) and radius r is the set of all points ( x, y, z) such that Area and volume Straight lines on the sphere are projected as circular arcs on the plane.Īn octahedron with edge length √ 2 can be placed with its center at the origin and its vertices on the coordinate axes the Cartesian coordinates of the vertices are then ( ☑, 0, 0 ) ( 0, ☑, 0 ) ( 0, 0, ☑ ). This projection is conformal, preserving angles but not areas or lengths.

The octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. The second and third correspond to the B 2 and A 2 Coxeter planes. The octahedron has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face. While the midradius, which touches the middle of each edge, is Orthogonal projections If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) isĪnd the radius of an inscribed sphere ( tangent to each of the octahedron's faces) 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. A regular octahedron is 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) is a polyhedron with eight faces. For the album by The Mars Volta, see Octahedron (album).

Octahedron vertices