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.

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.

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.

*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Hexagonal dihedron.png Uniform tiling 432-t01.png Uniform tiling 44-t12.png H2 tiling 245-6.png H2 tiling 246-6.png H2 tiling 247-6.png H2 tiling 248-6.png H2 tiling 24i-6.png
Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Spherical octagonal hosohedron.png Spherical triakis octahedron.png 1-uniform 2 dual.svg Order-4 pentakis pentagonal tiling.png Order4 hexakis hexagonal til.png Order4 heptakis heptagonal til.png Uniform tiling 83-t2.png Ord4 apeirokis apeirogonal til.png
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8

References[edit]

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

External links[edit]