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Feel free to surf to my site - online psychic; mouse click for source, In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.

If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and $2+\sqrt{2}$.

Area and volume

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

$A=2(6+6\sqrt{2}+\sqrt{3})a^2\approx 32.4346644a^2$

$V=\frac{1}{3}(21+14\sqrt{2})a^3\approx 13.5996633a^3.$

Orthogonal projections

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

Orthogonal projections

Centered by	Vertex	Edge 3-8	Edge 8-8	Face Octagon	Face Triangle
Truncated cube	File:Cube t01 v.png	File:Cube t01 e38.png	File:Cube t01 e88.png	File:3-cube t01 B2.svg	File:3-cube t01.svg
Triakis octahedron	File:Dual truncated cube t01 v.png	File:Dual truncated cube t01 e8.png	File:Dual truncated cube t01 e88.png	File:Dual truncated cube t01 B2.png	File:Dual truncated cube t01.png
Projective symmetry	[2]	[2]	[2]	[4]	[6]

Cartesian coordinates

The following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ:

(±ξ, ±1, ±1),

(±1, ±ξ, ±1),

(±1, ±1, ±ξ)

where ξ = $\sqrt{2}-1$

Vertex arrangement

It shares the vertex arrangement with three nonconvex uniform polyhedra:

File:Truncated hexahedron.png Truncated cube

File:Uniform great rhombicuboctahedron.png Nonconvex great rhombicuboctahedron

File:Great cubicuboctahedron.png Great cubicuboctahedron

File:Great rhombihexahedron.png Great rhombihexahedron

Related polyhedra

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

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry. Template:Truncated figure1 table

It is topologically related to a series of polyhedra and tilings with face configuration Vn.6.6. Template:Truncated figure4 table

Alternated truncation

A cube can be alternately truncated producing tetrahedral symmetry, with six hexagonal faces, and four triangles at the truncated vertices. It is one of a sequence of alternate truncations of polyhedra and tiling.

File:Alternate truncated cube.png

Related polytopes

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See also

• Spinning truncated cube
• Cube-connected cycles, a family of graphs that includes the skeleton of the truncated cube

References

• Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
• Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p.79-86 Archimedean solids

External links

• Template:Mathworld2
• Template:KlitzingPolytopes
• Editable printable net of a truncated cube with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
  • VRML model
  • Conway Notation for Polyhedra Try: "tC"