Octahedra

In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each .

A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations.

An octahedron is the three-dimensional case of the more general concept of a cross polytope.

A regular octahedron is a 3-ball in the Manhattan (ℓ₁) metric.

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

and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is

while the midradius, which touches the middle of each edge, is

The octahedron has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B₂ and A₂Coxeter planes.

The octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

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