Truncated octahedron

In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron.

Truncated octahedron

TypeArchimedean solid
Uniform polyhedron
ElementsF = 14, E = 36, V = 24 (χ = 2)
Faces by sides6{4}+8{6}
Conway notationtO
bT
Schläfli symbolst{3,4}
tr{3,3} or ${\displaystyle t{\begin{Bmatrix}3\\3\end{Bmatrix}}}$
t0,1{3,4} or t0,1,2{3,3}
Wythoff symbol2 4 | 3
3 3 2 |
Coxeter diagram
Symmetry groupOh, B3, [4,3], (*432), order 48
Th, [3,3] and (*332), order 24
Rotation groupO, [4,3]+, (432), order 24
Dihedral angle4-6: arccos(−1/√3) = 125°15′51″
6-6: arccos(−1/3) = 109°28′16″
ReferencesU08, C20, W7
PropertiesSemiregular convex parallelohedron
permutohedron
zonohedron

Colored faces

4.6.6
(Vertex figure)

Tetrakis hexahedron
(dual polyhedron)

Net

The truncated octahedron was called the "mecon" by Buckminster Fuller.[1]

Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths 9/82 and 3/22.