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}
Conway notation	tC
Schläfli symbols	t{4,3}
	t0,1{4,3}
Wythoff symbol	2 3 | 4
Coxeter diagram
Symmetry group	Oh, BC3, [4,3], (*432), order 48
Rotation group	O, [4,3]+, (432), order 24
Dihedral Angle	3-8:125°15'51" 8-8:90°
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:

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

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

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
Triakis octahedron
Projective symmetry	[2]	[2]	[2]	[4]	[6]

Spherical tiling

The truncated cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

	octagon-centered	triangle-centered
Orthographic projection	Stereographic projections

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

The parameter ξ can be varied between ±1. A value of 1 produces a cube, 0 produces a cuboctahedron, and negative values produces self-intersecting octagrammic faces.

If the self-intersected portions of the octagrams are removed, leaving squares, and truncating the triangles into hexagons, truncated octahedrons are produced, and the sequence ends with the central squares being reduced to a point, and creating an octahedron.

Dissection

Dissected truncated cube, with elements expanded apart

The truncated cube can be dissected into a central cube, with six square cupola around each of the cube's faces, and 8 regular tetrahedral in the corners. This dissection can also be seen within the runcic cubic honeycomb, with cube, tetrahedron, and rhombicuboctahedron cells.

This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupola and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons.

Vertex arrangement

It shares the vertex arrangement with three nonconvex uniform polyhedra:

Truncated cube

Nonconvex great rhombicuboctahedron

Great cubicuboctahedron

Great rhombihexahedron

Related polyhedra

The truncated cube is related to other polyhedra and tlings in symmetry.

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

Octahedral truncations

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}
		=	=	=				= or	= or	=
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

Symmetry mutations

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, and a series of polyhedra and tilings n.8.8.

Symmetry mutations 3.2n.2n

*n32 symmetry mutation of truncated spherical tilings: t{n,3}
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Truncated figures
Symbol	t{2,3}	t{3,3}	t{4,3}	t{5,3}	t{6,3}	t{7,3}	t{8,3}	t{∞,3}
Triakis figures
Config.	V3.4.4	V3.6.6	V3.8.8	V3.10.10	V3.12.12	V3.14.14	V3.16.16	V3.∞.∞

Symmetry mutations n.8.8

*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
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
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

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.

Related polytopes

The truncated cube, is second in a sequence of truncated hypercubes:

Truncated hypercubes

							...
Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube

Truncated cubical graph

Truncated cubical graph
4-fold symmetry schlegel diagram
Vertices	24
Edges	36
Automorphisms	48
Chromatic number	2
Properties	Cubic, Hamiltonian, regular, zero-symmetric

In the mathematical field of graph theory, a truncated cubical graph is the graph of vertices and edges of the truncated cube, one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.

Orthographic

See also

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

References

• (Section 3-9)
• Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p.79-86 Archimedean solids

External links

• Eric W. Weisstein, Truncated cube (Archimedean solid) at MathWorld.
  • Weisstein, Eric W., "Truncated cubical graph", 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"