Truncated octahedron

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Truncated octahedron
Truncated octahedron
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 14, E = 36, V = 24 (χ = 2)
Faces by sides 6{4}+8{6}
Conway notation tO
bT
Schläfli symbols t{3,4}, tr{3,3}
t0,1{3,4} or t0,1,2{3,3}
Wythoff symbol 2 4 | 3
3 3 2 |
Coxeter diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group Oh, BC3, [4,3], (*432), order 48
Th, [3,3] and (*332), order 24
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 4-6:cos(-1/sqrt(3))=125°15'51"
6-6:cos(-1/3)=109°28'16"
References U08, C20, W7
Properties Semiregular convex zonohedron
permutohedron
Truncated octahedron.png
Colored faces
Truncated octahedron
4.6.6
(Vertex figure)
Tetrakishexahedron.jpg
Tetrakis hexahedron
(dual polyhedron)
Truncated Octahedron Net.svg
Net

In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tile (or "pack") 3-dimensional space, as a permutohedron.

If the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths \tfrac{9}{8}\scriptstyle {\sqrt{2}} and \tfrac{3}{2}\scriptstyle{\sqrt{2}}.

Construction[edit]

Truncated Octahedron with Construction.svg   Square Pyramid.svg

A truncated octahedron is constructed from a regular octahedron with side length 3a by the removal of six right square pyramids, one from each point. These pyramids have both base side length (a) and lateral side length (e) of a, to form equilateral triangles. The base area is then a2. Note that this shape is exactly similar to half an octahedron or Johnson solid J1.

From the properties of square pyramids, we can now find the slant height, s, and the height, h, of the pyramid:

h = \sqrt{e^2-\frac{1}{2}a^2}=\frac{\sqrt{2}}{2}a
s = \sqrt{h^2 + \frac{1}{4}a^2} = \sqrt{\frac{1}{2}a^2 + \frac{1}{4}a^2}=\frac{\sqrt{3}}{2}a

The volume, V, of the pyramid is given by:

V = \frac{1}{3}a^2h = \frac{\sqrt{2}}{6}a^3

Because six pyramids are removed by truncation, there is a total lost volume of \scriptstyle {\sqrt{2}a^3}.

Orthogonal projections[edit]

The truncated octahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: Hexagon, and square. The last two correspond to the B2 and A2 Coxeter planes.

Orthogonal projections
Centered by Vertex Edge
4-6
Edge
6-6
Face
Square
Face
Hexagon
Truncated
octahedron
Cube t12 v.png Cube t12 e46.png Cube t12 e66.png 3-cube t12 B2.svg 3-cube t12.svg
Hexakis
hexahedron
Dual cube t12 v.png Dual cube t12 e46.png Dual cube t12 e66.png Dual cube t12 B2.png Dual cube t12.png
Projective
symmetry
[2] [2] [2] [4] [6]

Spherical tiling[edit]

The truncated 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.

Uniform tiling 432-t12.png Truncated octahedron stereographic projection square.png
square-centered
Truncated octahedron stereographic projection hexagon.png
hexagon-centered
Orthographic projection Stereographic projections

Coordinates[edit]

Truncated octahedron in unit cube.png Triangulated truncated octahedron.png
Orthogonal projection in bounding box
(±2,±2,±2)
Truncated octahedron with hexagons replaced by 6 coplanar triangles. There are 6 new vertices at: (±1,±1,±1).

All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √ 2 centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes.

The edge vectors have Cartesian coordinates (0,± 1,±1) and permutations of these. The face normals (normalized cross products of edges that share a common vertex) of the 6 square faces are (0,0,±1), (0,±1,0) and (±1,0,0). The face normals of the 8 hexagonal faces are (± 1/√ 3, ± 1/√ 3, ± 1/√3). The dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either -1/3 or -1/√3. The dihedral angle is approximately 1.910633 rad (109.471 ° OEISA156546) at edges shared by two hexagons or 2.186276 rad (125.263 ° OEISA195698) at edges shared by a hexagon and a square.

Dissection[edit]

The truncated octahedron can be dissected into a central octahedron, surrounded by 8 triangular cupola on each face, and 6 square pyramids above the vertices.[1]

Removing the central octahedron and 2 or 4 triangular cupola creates two Stewart toroids, with dihedral and tetrahedral symmetry:

Genus 2 Genus 3
D3d, [2+,6], (2*3), order 12 Td, [3,3], (*332), order 24
Excavated truncated octahedron1.png Excavated truncated octahedron2.png

Permutohedron[edit]

The truncated octahedron can also be represented by even more symmetric coordinates in four dimensions: all permutations of (1, 2, 3, 4) form the vertices of a truncated octahedron in the three-dimensional subspace x + y + z + w = 10. Therefore, the truncated octahedron is the permutohedron of order 4.

Permutohedron.svg

Area and volume[edit]

The area A and the volume V of a truncated octahedron of edge length a are:

A = \left(6+12\sqrt{3}\right) a^2 \approx 26.7846097a^2
V = 8\sqrt{2} a^3 \approx 11.3137085a^3.

Uniform colorings[edit]

There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry, and two 2-uniform coloring with dihedral symmetry as a truncated triangular antiprism.

Oh
Order 48
Td
Order 24
D4h
Order 16
D3d
Order 12
Truncated Octahedron 122 Colouring.svg
122 coloring
Truncated Octahedron 123 Colouring.svg
123 coloring
Truncated square bipyramid.png
122 & 322 colorings
Truncated octahedron prismatic symmetry.png
122 & 123 colorings
Truncated octahedron Omnitruncated tetrahedron Truncated square bipyramid Truncated triangular antiprism

Related polyhedra[edit]

The truncated octahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+
(432)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg POV-Ray-Dodecahedron.svg

It also exists as the omnitruncate of the tetrahedron family:

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-33-t0.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t1.png Uniform polyhedron-33-t12.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t02.png Uniform polyhedron-33-t012.png Uniform polyhedron-33-s012.png
{3,3} t{3,3} r{3,3} t{3,3} {3,3} rr{3,3} tr{3,3} sr{3,3}
Duals to uniform polyhedra
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Tetrahedron.svg Triakistetrahedron.jpg Hexahedron.svg Triakistetrahedron.jpg Tetrahedron.svg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg POV-Ray-Dodecahedron.svg
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.3.3.3.3
Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{2,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{4,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{5,3}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{6,3}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{7,3}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{8,3}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{∞,3}
Omnitruncated
figure
Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png H2 tiling 237-7.png H2 tiling 238-7.png H2 tiling 23i-7.png
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Omnitruncated
duals
Hexagonale bipiramide.png Tetrakishexahedron.jpg Disdyakisdodecahedron.jpg Disdyakistriacontahedron.jpg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Order-3 heptakis heptagonal tiling.png Order-3 octakis octagonal tiling.png H2checkers 23i.png
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞
Dimensional family of omnitruncated polyhedra and tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*222
[2,2]
D2h
*332
[3,3]
Td
*442
[4,4]
P4m
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
 
[∞,iπ/λ]
Figure Spherical square prism.png
4.4.4
Uniform tiling 332-t012.png
4.6.6
Uniform tiling 44-t012.png
4.8.8
H2 tiling 255-7.png
4.10.10
H2 tiling 266-7.png
4.12.12
H2 tiling 277-7.png
4.14.14
H2 tiling 288-7.png
4.16.16
H2 tiling 2ii-7.png
4.∞.∞
H2 tiling 2iu-7.png
4.∞.∞
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{2,2}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{4,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png
tr{5,5}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png
tr{6,6}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node 1.png
tr{7,7}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png
tr{8,8}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
tr{∞,∞}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel ultra.pngCDel node 1.png
Dual Octahedron.png
V4.4.4
Tetrakishexahedron.jpg
V4.6.6
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
V4.8.8
Order-4 bisected pentagonal tiling.png
V4.10.10
Hyperbolic domains 642.png
V4.12.12
Hyperbolic domains 742.png
V4.14.14
Hyperbolic domains 842.png
V4.16.16
H2checkers 24i.png
V4.∞.∞
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel ultra.pngCDel node f1.png

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedra), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures n.6.6, extending into the hyperbolic plane:

Dimensional family of truncated polyhedra and tilings: n.6.6
Symmetry
*n42
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
Order 12 24 48 120
Truncated
figures
Hexagonal dihedron.png
2.6.6
Uniform tiling 332-t12.png
3.6.6
Uniform tiling 432-t12.png
4.6.6
Uniform tiling 532-t12.png
5.6.6
Uniform tiling 63-t12.png
6.6.6
Uniform tiling 73-t12.png
7.6.6
Uniform tiling 83-t12.png
8.6.6
H2 tiling 23i-6.png
∞.6.6
Coxeter
Schläfli
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node.png
t{3,2}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t{3,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t{3,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t{3,5}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
t{3,6}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 7.pngCDel node.png
t{3,7}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 8.pngCDel node.png
t{3,8}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel infin.pngCDel node.png
t{3,∞}
Uniform dual figures
n-kis
figures
Hexagonal Hosohedron.svg
V2.6.6
Triakistetrahedron.jpg
V3.6.6
Tetrakishexahedron.jpg
V4.6.6
Pentakisdodecahedron.jpg
V5.6.6
Uniform tiling 63-t2.png
V6.6.6
Order3 heptakis heptagonal til.png
V7.6.6
Uniform dual tiling 433-t012.png
V8.6.6
H2checkers 33i.png
V∞.6.6
Coxeter CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel infin.pngCDel node.png

The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:

Dimensional family of truncated polyhedra and tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
D4h
*342
[3,4]
Oh
*442
[4,4]
P4m
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Spherical square prism.png
4.4.4
Uniform tiling 432-t12.png
4.6.6
Uniform tiling 44-t01.png
4.8.8
Uniform tiling 54-t01.png
4.10.10
Uniform tiling 64-t01.png
4.12.12
Uniform tiling 74-t01.png
4.14.14
Uniform tiling 84-t01.png
4.16.16
H2 tiling 24i-3.png
4.∞.∞
Coxeter
Schläfli
CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{2,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t{3,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
t{4,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node 1.png
t{5,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node 1.png
t{6,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node 1.png
t{7,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node 1.png
t{8,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel infin.pngCDel node 1.png
t{4,∞}
Uniform dual figures
n-kis
figures
Spherical square bipyramid.png
V4.4.4
Tetrakishexahedron.jpg
V4.6.6
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
V4.8.8
Order-5 tetrakis square tiling.png
V4.10.10
Order-6 tetrakis square tiling.png
V4.12.12
Hyperbolic domains 772.png
V4.14.14
Order-8 tetrakis square tiling.png
V4.16.16
H2checkers 2ii.png
V4.∞.∞
Coxeter CDel node.pngCDel 4.pngCDel node f1.pngCDel 2.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 6.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 7.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 8.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel infin.pngCDel node f1.png

Related polytopes[edit]

The truncated octahedron (bitruncated cube), is first in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
3-cube t12.svgTruncated octahedron.png 4-cube t12.svgSchlegel half-solid bitruncated 8-cell.png 5-cube t12.svg5-cube t12 A3.svg 6-cube t12.svg6-cube t12 A5.svg 7-cube t12.svg7-cube t12 A5.svg 8-cube t12.svg8-cube t12 A7.svg ...
Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Tessellations[edit]

The truncated octahedron exists in three different convex uniform honeycombs (space-filling tessellations):

Bitruncated cubic Cantitruncated cubic Truncated alternated cubic
Bitruncated Cubic Honeycomb.svg Cantitruncated Cubic Honeycomb.svg Truncated Alternated Cubic Honeycomb.svg

The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centered cubic lattice. The truncated octahedron is one of five three-dimensional primary parallelohedra.

References[edit]

External links[edit]