Triakis octahedron

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Triakis octahedron
Triakisoctahedron.jpg
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
Conway notation kO
Face type V3.8.8
DU09 facets.png

isosceles triangle
Faces 24
Edges 36
Vertices 14
Vertices by type 8{3}+6{8}
Symmetry group Oh, B3, [4,3], (*432)
Rotation group O, [4,3]+, (432)
Dihedral angle 147° 21' 0"
 \arccos ( -\frac{3 + 8\sqrt{2}}{17} )
Properties convex, face-transitive
Truncated hexahedron.png
Truncated cube
(dual polyhedron)
Triakis octahedron Net
Net

In geometry, a triakis octahedron (or kisoctahedron[1]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.

It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more fully, trigonal trisoctahedron. Both names reflect the fact that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron.

This convex polyhedron is topologically similar to the concave stellated octahedron. They have the same face connectivity, but the vertices are in different relative distances from the center.

If its shorter edges have length 1, its surface area and volume are:

A=3\sqrt{7+4\sqrt{2}}
V=\frac{1}{2}(3+2\sqrt{2}).

Orthogonal projections[edit]

The triakis octahedron has three symmetry positions, two located on vertices, and one mid-edge:

Orthogonal projections
Projective
symmetry
[2] [4] [6]
Triakis
octahedron
Dual truncated cube t01 e88.png Dual truncated cube t01 B2.png Dual truncated cube t01.png
Truncated
cube
Cube t01 e88.png 3-cube t01 B2.svg 3-cube t01.svg

Cultural references[edit]

Related polyhedra[edit]

Spherical triakis octahedron

The triakis octahedron is one of a family of duals to the 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 Dodecahedron.svg

The triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

Dimensional family of truncated spherical polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*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]
 
[12i,3] [9i,3] [6i,3] [3i,3]
Figures Spherical triangular prism.png Uniform tiling 332-t01-1-.png Uniform tiling 432-t01.png Uniform tiling 532-t01.png Uniform tiling 63-t01.png H2 tiling 237-3.png H2 tiling 238-3.png H2 tiling 23i-3.png H2 tiling 23j12-3.png H2 tiling 23j9-3.png H2 tiling 23j6-3.png H2 tiling 23j3-3.png
Schläfli t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3} t{12i,3} t{9i,3} t{6i,3} t{3i,3}
Coxeter CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel ultra.pngCDel node 1.pngCDel 3.pngCDel node.png
Dual
figures
Triakis
figures
Spherical trigonal bipyramid.png
V3.4.4
Spherical triakis tetrahedron.png
V3.6.6
Spherical triakis octahedron.png
V3.8.8
Spherical triakis icosahedron.png
V3.10.10
Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg
V3.12.12
Ord7 triakis triang til.png
V3.14.14
Ord8 triakis triang til.png
V3.16.16
Ord-infin triakis triang til.png
V3.∞.∞
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel ultra.pngCDel node f1.pngCDel 3.pngCDel node.png

The triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n42) reflectional symmetry.

Dimensional family of truncated polyhedra and tilings: n.8.8
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
2.8.8 Uniform tiling 432-t01.png
3.8.8
Uniform tiling 44-t12.png
4.8.8
H2 tiling 245-6.png
5.8.8
H2 tiling 246-6.png
6.8.8
H2 tiling 247-6.png
7.8.8
H2 tiling 248-6.png
8.8.8
H2 tiling 24i-6.png
∞.8.8
Coxeter
Schläfli
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node.png
t{4,2}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t{4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
t{4,4}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node.png
t{4,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node.png
t{4,6}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node.png
t{4,7}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
t{4,8}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel infin.pngCDel node.png
t{4,∞}
Uniform dual figures
n-kis
figures
Spherical octagonal hosohedron.png
V2.8.8
Triakisoctahedron.jpg
V3.8.8
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
V4.8.8
Order-4 pentakis pentagonal tiling.png
V5.8.8
Order4 hexakis hexagonal til.png
V6.8.8
Order4 heptakis heptagonal til.png
V7.8.8
Uniform tiling 83-t2.png
V8.8.8
Ord4 apeirokis apeirogonal til.png
V∞.8.8
Coxeter CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel infin.pngCDel node.png

References[edit]

  1. ^ Conway, Symmetries of things, p.284

External links[edit]