Truncated cube

Truncated cube
(Click here for rotating model)
Type	Archimedean solid Uniform polyhedron
Elements	F = 14, E = 36, V = 24 (χ = 2)
Faces by sides	8{3}+6{8}
Schläfli symbol	t{4,3}
Wythoff symbol	2 3 | 4
Coxeter–Dynkin
Symmetry	Oh , [4,3], (*432)
Dihedral Angle
References	U09, C21, W8
Properties	Semiregular convex
Colored faces	3.8.8 (Vertex figure)
Triakis octahedron (dual polyhedron)	Net

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 $\scriptstyle {2+\sqrt{2}}$.

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
Image
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 = $\scriptstyle {\sqrt2 - 1}$

Related polyhedra

The truncated cube can be seen as a cube with its corners truncated, as shown in this truncation sequence:

Cube

Truncated cube

cuboctahedron

Truncated octahedron

Octahedron

It shares the vertex arrangement with three nonconvex uniform polyhedra:

Truncated cube

Nonconvex great rhombicuboctahedron

Great cubicuboctahedron

Great rhombihexahedron

A cube can be alternately truncated producing tetrahedral symmetry, with six hexagonal faces, and four triangles at the truncated vertices.

See also

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

References

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)

External links

• Eric W. Weisstein, Truncated cube (Archimedean solid) at MathWorld.
• Richard Klitzing, 3D convex uniform polyhedra, o3x4x - tic
• 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"