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.
Other Names
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.
It can also be called a rectified tetrahedron or tetratetrahedron, as it can be made by truncating a tetrahedron to the midpoints of the edges. In this form it can be represented as .
Symbols
The octahedron's Dynkin based symbols include:
- o4o3x (regular)
- o3x3o (rectified tetrahedron)
- s2s6o (triangular antiprism)
- s2s3s
- 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)
- oxox&#xr
Structure
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.
Sections
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.
Hypervolumes
Subfacets
- 6 points (0D)
- 12 line segments (1D)
- 8 triangles (2D)
- 1 octahedron (3D)
Radii
- Vertex radius:
- Edge radius:
- Face radius:
Angles
- Dihedral angle:
Equations
The surface of an octahedron with side
can be given by
Vertex coordinates
The coordinates of an octahedron of side
are all permuations of (±1,0,0).
Related shapes
- Dual: Cube
- Vertex figure: Square, side length 1
- Diminishings/Segments: Square pyramid (vertex first)
- Regiment members: 2 (other member: Tetrahemihexahedron)
See also
- Polytope Wiki. "Octahedron".
- Bowers, Jonathan. "Polyhedraon Category 1: Regulars".
Regular polyhedra |
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Convex regular polyhedra: tet · cube · oct · doe · ike
Self-intersecting regular polyhedra: gad · sissid · gike · gissid |
... | |||||||||
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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 |
Regular |
Rectified |
Birectified |
Truncated |
Bitruncated |
Cantellated |
Cantitruncated |
---|---|---|---|---|---|---|
Cube | Cuboctahedron | Octahedron | Truncated cube | Truncated octahedron | Rhombicuboctahedron | Great rhombicuboctahedron |
Regular |
Rectified |
Birectified |
Truncated |
Bitruncated |
Cantellated |
Cantitruncated |
---|---|---|---|---|---|---|
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 |