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In geometry, an octahedron () is a polyhedron with eight faces. 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. 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 (l_1) 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 A2 Coxeter 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. An octahedron with edge length can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then ( ±1, 0, 0 ); ( 0, ±1, 0 ); ( 0, 0, ±1 ). 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 The surface area A and the volume V of a regular octahedron of edge length a are: 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).

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