The regular octahedron, often simply called "the" octahedron, is the Platonic solid with six polyhedron
vertices, 12 polyhedron edges, and eight equivalent
equilateral triangular faces,
denoted 
.
It is illustrated above together with a wireframe version and a net
that can be used for its construction.
The regular octahedron is also the uniform polyhedron with Maeder index 5 (Maeder 1997), Wenninger index 2 (Wenninger 1989), Coxeter index
17 (Coxeter et al. 1954), and Har'El index 10 (Har'El 1993). It is given by
the Schläfli symbol 
and Wythoff symbol 
. The octahedron
of unit side length is the antiprism of 
sides with height 
(i.e., the regular triangular antiprism). The octahedron
is also a square dipyramid with equal edge
lengths.
A number of symmetric projections of the regular octahedron are illustrated above.
The regular octahedron is implemented in the Wolfram Language as Octahedron[] or UniformPolyhedron["Octahedron"]. Precomputed properties are available as PolyhedronData["Octahedron", prop].
There are 11 distinct nets for the octahedron, the same as for the cube (Buekenhout and Parker 1998). Questions of polyhedron coloring of the octahedron can be addressed using the Pólya enumeration theorem.
The octahedron is the convex hull of the tetrahemihexahedron.
Reciprocation with respect to the midsphere of an octahedron with unit edge
lengths gives a dual polyhedron that is a cube
with edge lengths 
.
The illustration above shows an origami octahedron constructed from a single sheet of paper (Kasahara and Takahama 1987, pp. 60-61).
Like the cube, the regular octahedron has the 
octahedral group of
symmetries.
The connectivity of the vertices is given by the octahedral graph.
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The regular octahedron has a single stellation: the stella octangula. The intersection of the two tetrahedra of the stella octangula (left figure) is a regular octahedron (right figure; Ball and Coxeter 1987).
S. Wagon (pers. comm., Oct. 30, 2013) has constructed a closed loop of eight regular octahedra.
The following table gives polyhedra which can be constructed by augmentation of a regular octahedron of unit
edge length by pyramids
of given heights 
, where 
is the inradius of
the regular octahedron.
Three orientations of a regular octahedron are illustrated above. The left one has vertices 
, 
, 
, 
(for edge
lengths 
),
the middle one has vertices 
, 
, 
(for edge lengths
),
and the right one has vertices 
and 
(for edge lengths
).
In the middle orientation, the face planes are 
,
so a solid octahedron is given by the inequality
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(1)
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If the edges of a regular octahedron are divided in the golden ratio such that the points
of division for any face form an equilateral
triangle, then the twelve points of division form an
icosahedron (Wells 1991). In fact, there are two
ways in which the edges can be internally divided
in the golden ratio and two ways in which they can
be externally divided, resulting in four possible icosahedra.
For either internal construction, keeping the same connectivity but reversing the
long and short ends and changing the division ratio to 
gives Jessen's
orthogonal icosahedron (Jessen 1967).
A plane through the center and perpendicular to a 
axis of an octahedron cuts
the solid in a regular hexagonal cross
section (Holden 1991, pp. 22-23). Since there are four such axes, there
are four possible regular hexagonal cross
sections.
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The centers of the faces of a regular octahedron form a cube, and the centers of the faces of a cube form an octahedron (Steinhaus 1999, pp. 194-195). A faceted form of the regular octahedron is the tetrahemihexahedron.
Let a regular octahedron be length 
on a side. The height of the top
polyhedron vertex from the square plane is also the circumradius
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(2)
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where 
is half the diagonal length of the equatorial square,
so
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(3)
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To compute the inradius, let 
be half the base of an equilateral
triangle face, 
its height, and 
its inradius. Then
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(4)
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(5)
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so
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(6)
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With 
,
use similar triangles to obtain
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(7)
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(8)
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(9)
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so the inradius is
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(10)
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and twice the inradius gives the height of the octahedron viewed as a 3-sided antiprism. The midradius of the octahedron is
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(11)
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The area of one face of a regular octahedron is the area of an equilateral triangle
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(12)
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The volume is two times the volume of a square-base pyramid,
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(13)
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The dihedral angle is
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(14)
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and the Dehn invariant of a regular octahedron of unit edge length is
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(15)
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(16)
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(17)
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(OEIS A377296), where the first expression uses the basis of Conway et al. (1999, p. 331).
The regular octahedron can be built using a Haüy construction. The Haüy octahedral numbers
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(18)
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give another method for calculating the volume of the octahedron,
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(19)
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in agreement with the result derived above.
." Disc. Math. 186, 69-94, 1998. 
." §3.5.3 in