An octahedron is the three-dimensional cross polytope. It is the dual of a cube - in other words, it is a cube where all vertices have been replaced with faces and all faces have been replaced with vertices. It is also the triangular antiprism, a fact that can clearly be seen from its graph. In addition it is a square bipyramid, meaning it is composed of two square pyramids stuck together at their bases.
Jonathan Bowers calls the octahedron an oct, a shortened form of its name. The Higher Dimensions wiki sometimes calls this shape an aerohedron, from aero- referring to a cross polytope and -hedron referring to three dimensions.
The octahedron's Dynkin based symbols include:
- o4o3x (regular)
- o3x3o (rectified tetrahedron)
- s2s6o (triangular antiprism)
- qo ox4oo&#zx (square dipyramid)
- qo ox ox&#zx (recangular dipyramid)
- qoo oqo ooq&#zx (rhombic dipyramid)
- oxo4ooo&#xt (join of two square pyramids)
- oxo oxo&#xt (two rectangle pyramids)
- xo3ox&#x (triangle antipodium)
- xox oqo&#xt (trapezoid bipyramid)
The octahedron is composed of 8 triangles, with four of them meeting at each vertex. It is formed of two square pyramids joined together. It is also a triangular antiprism, composed of two triangles, in dual orientations, joined by a band of 6 triangles.
When seen vertex first, it looks like a point expanding to a square of the same side as its edge length, then shrinking back. When seen face first, it appears as a triangle that expands through various hexagons, then back to a triangle in dual orientation.
- Vertex radius:
- Edge radius:
- Face radius:
- Dihedral angle:
The surface of an octahedron with side
can be given by
The coordinates of an octahedron of side
are all permuations of (±1,0,0).
- Dual: Cube
- Vertex figure: Square, side length 1
- Diminishings/Segments: Square pyramid (vertex first)
- Regiment members: 2 (other member: Tetrahemihexahedron)
|Convex regular polyhedra: tet · cube · oct · doe · ike|
|Square hosohedron||Octahedron||Square tiling||Order-4 pentagonal tiling||Order-4 hexagonal tiling||Order-4 heptagonal tiling||Order-4 octagonal tiling||...||Order-4 apeirogonal tiling||Order-4 pseudogonal tiling|
|Trigonal dihedron||Tetrahedron||Octahedron||Icosahedron||Triangular tiling||Order-7 triangular tiling||Order-8 triangular tiling||...||Infinite-order triangular tiling||Imaginary-order triangular tiling|
|Cube||Cuboctahedron||Octahedron||Truncated cube||Truncated octahedron||Rhombicuboctahedron||Great rhombicuboctahedron|
|Tetrahedron||Octahedron||Tetrahedron||Truncated tetrahedron||Truncated tetrahedron||Cuboctahedron||Truncated octahedron|
|Trigonal dihedron||Octahedron||Cuboctahedron||Icosidodecahedron||Trihexagonal tiling||Triheptagonal tiling||Trioctagonal tiling||...||Triapeirogonal tiling||Tripseudogonal tiling|