Octahedron

Octahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Oct
Coxeter diagram	o4o3x ()
Schläfli symbol	{3,4}
Bracket notation	<III>
Elements
Faces	8 triangles
Edges	12
Vertices	6
Vertex figure	Square, edge length 1
Petrie polygons	4 skew hexagons
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2} \approx 0.70711$
Edge radius	$\frac12 = 0.5$
Inradius	$\frac{\sqrt6}{6} \approx 0.40825$
Volume	$\frac{\sqrt2}{3} \approx 0.47140$
Dihedral angle	$\arccos\left(-\frac13\right) \approx 109.47122^\circ$
Height	$\frac{\sqrt6}{3} \approx 0.81650$
Central density	1
Number of external pieces	8
Level of complexity	1
Related polytopes
Army	Oct
Regiment	Oct
Dual	Cube
Petrie dual	Petrial octahedron
Conjugate	None
Abstract & topological properties
Flag count	48
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	Yes
Net count	11
Nature	Tame

The octahedron, or oct, is one of the five Platonic solids. It consists of 8 equilateral triangles, joined 4 to a square vertex. It is the 3 dimensional orthoplex. It also has 3 square pseudofaces. In fact, it can be built by joining two square pyramids by their square face, which makes it the square tegum.

It can also be constructed by rectifying the tetrahedron.

It is also the uniform triangular antiprism, and is a segmentohedron in this form.

It occurs as cells in one regular polychoron, namely the icositetrachoron.

Vertex coordinates

An octahedron of side length 1 has vertex coordinates given by all permutations of:

• $\left(±\frac{\sqrt2}{2},\,0,\,0\right).$

Representations

A regular octahedron can be represented by the following Coxeter diagrams:

• o4o3x (regular)
• o3x3o (A3 symmetry, tetratetrahedron)
• s2s3s (generally a triangular antiprism)
• s2s6o (similar to above, as alternated hexagonal prism)
• xo3ox&#x (A2 axial, generally a triangular antipodium)
• oxo4ooo&#xt (BC2 axial, generally a square bipyramid)
• oxo oxo&#xt (generally a rectangular bipyramid)
• xox oqo&#xt (A1×A1 axial, edge-first)
• oxox&#xr (single symmetry axis only)
• qo ox4oo&#xt (BC2 prism symmetry square bipyramid)
• qo ox ox&#xt (brick symmetry rectangle bipyramid)
• qoo oqo ooq&#zx (brick symmetry, rhombic bipyramid)

In vertex figures

Octahedra in vertex figures

Name	Picture	Schläfli symbol	Edge length
Hexadecachoron		{3,3,4}	$1$
Cubic honeycomb		{4,3,4}	$\sqrt{2}$
Dodecahedral honeycomb		{5,3,4}
Order-4 hexagonal tiling honeycomb		{6,3,4}

Variations

Other variants of the octahedron exist, using 8 triangular faces with 6 4-fold vertices. Some of these include:

• Tetratetrahedron - 2 sets of 4 triangles - just a coloring with no true variants in measures
• Triangular antiprism - 2 equilateral bases, 6 isosceles sides, vertex transitive
• Triangular antipodium - as above with 2 different sized bases and 2 sets of 3 isosceles sides
• Square tegum - 8 isosceles triangles, square prism symmetry
• Rectangular tegum - 2 sets of 4 isosceles triangles
• Rhombic tegum - 8 scalene triangles, digonal prism symmetry
• Digonal scalenohedron - 8 scalene triangles, digonal antiprism symmetry

Related polyhedra

The octahedron is the colonel of a two-member regiment that also includes the tetrahemihexahedron.

The octahedron is the regular-faced square bipyramid. If a cube, seen as a square prism, is inserted between the two haves, the result is an elongated square bipyramid.

A number of uniform polyhedron compounds are composed of octahedra, all but one of them featured octahedra in triangular antiprism symmetry:

• Small icosicosahedron (5)
• Snub octahedron (4)
• Inner disnub octahedron (8, with rotational freedom)
• Hexagrammic disnub octahedron (8)
• Outer disnub octahedron (8, with rotational freedom)
• Inner disnub tetrahedron (4, with rotational freedom)
• Hexagrammic disnub tetrahedron (4)
• Outer disnub tetrahedron (4, with rotational freedom)
• Snub icosahedron (10)
• Great snub icosahedron (10)
• Outer disnub icosahedron (20, with rotational freedom)
• Inner disnub icosahedron (20, with rotational freedom)
• Great disnub icosahedron (20, with rotational freedom)
• Disnub icosahedron (20)

There is also an infinite family of prismatic octahedron compounds, the antiprisms of compounds of triangles:

• Hexagrammic antiprism (2)
• Fissal enneagrammic antiprism (3)
• Tetratriangular antiprism (4)
• ...

The octahedron has one stellation, the stella octangula.

o4o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Cube	cube	{4,3}	x4o3o
Truncated cube	tic	t{4,3}	x4x3o
Cuboctahedron	co	r{4,3}	o4x3o
Truncated octahedron	toe	t{3,4}	o4x3x
Octahedron	oct	{3,4}	o4o3x
Small rhombicuboctahedron	sirco	rr{4,3}	x4o3x
Great rhombicuboctahedron	girco	tr{4,3}	x4x3x
Snub cube	snic	sr{4,3}	s4s3s

o3o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Tetrahedron	tet	{3,3}	x3o3o
Truncated tetrahedron	tut	t{3,3}	x3x3o
Tetratetrahedron = Octahedron	oct	r{3,3}	o3x3o
Truncated tetrahedron	tut	t{3,3}	o3x3x
Tetrahedron	tet	{3,3}	o3o3x
Small rhombitetratetrahedron = Cuboctahedron	co	rr{3,3}	x3o3x
Great rhombitetratetrahedron = Truncated octahedron	toe	tr{3,3}	x3x3x
Snub tetrahedron = Icosahedron	ike	sr{3,3}	s3s3s

The dihedral angle is a supplementary angle to that of the regular tetrahedron. Thus, augmenting one of the faces produces coplanar faces, disqualifying the resulting polyhedron from being a Johnson solid. If two opposite faces are augmented with tetrahedra, the result is a triangular antitegum with 6 identical 60°/120° rhombi for faces.

External links

• Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#3).

• Bowers, Jonathan. "Batch 1: Oct and Co Facetings" (#1 under oct).

• Klitzing, Richard. "Oct".

• Quickfur. "The Octahedron".

• Wikipedia Contributors. "Octahedron".
• McCooey, David. "Octahedron"

• Hi.gher.Space Wiki Contributors. "Octahedron".

• Hartley, Michael. "{3,4}*48".