Truncated cube

Truncated cubical graph
Truncated cubical graph
	<img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Truncated_cubic_graph.png/240px-Truncated_cubic_graph.png" decoding="async" width="240" height="233" class="mw-file-element" data-file-width="576" data-file-height="560"> 4-fold symmetry Schlegel diagram
Vertices	24
Edges	36
Automorphisms	48
Chromatic number	3
Properties	Cubic, Hamiltonian, regular, zero-symmetric
	Table of graphs and parameters

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}
Schläfli symbols	t_0,1{4,3}
Wythoff symbol	2 3 \| 4
Coxeter diagram
Symmetry group	O_h, B₃, [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 $δ S +1$ , where δ_S is the silver ratio, √2 +1.

Area and volume

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

{\begin{aligned}A&=2\left(6+6{\sqrt {2}}+{\sqrt {3}}\right)a^{2}&&\approx 32.434\,6644a^{2}\\V&={\frac {21+14{\sqrt {2}}}{3}}a^{3}&&\approx 13.599\,6633a^{3}.\end{aligned}}

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
Solid
Wireframe
Dual
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.

Orthographic projection	Stereographic projections
	octagon-centered	triangle-centered

Cartesian coordinates

A truncated cube with its octagonal faces pyritohedrally dissected with a central vertex into triangles and pentagons, creating a topological icosidodecahedron

Cartesian coordinates for the vertices of a truncated hexahedron centered at the origin with edge length 2⁠1/δ_S⁠ are all the permutations of

(±⁠1δ_S⁠, ±1, ±1),

where δ_S=√2+1.

If we let a parameter ξ= ⁠1/δ_S⁠, in the case of a Regular Truncated Cube, then 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 octahedra 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 cupolae around each of the cube's faces, and 8 regular tetrahedra 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 cupolae and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons.^[1]^[2]

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 tilings in symmetry.

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

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{3^1,1}	t{3,4} t{3^1,1}	{3,4} {3^1,1}	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{3^1,1}

		=	=	=				= 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.

*n32 symmetry mutation of truncated spherical tilings: t{n,3}
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.
Symmetry *n32 [n,3]	*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.∞.∞

*n42 symmetry mutation of truncated tilings: n.8.8 v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracompact
Symmetry *n42 [n,4]	*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

Tetrahedron, its edge truncation, and the truncated cube

Truncating alternating vertices of the cube gives the chamfered tetrahedron, i.e. the edge truncation of the tetrahedron.

The truncated triangular trapezohedron is another polyhedron which can be formed from cube edge truncation.

Related polytopes

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

Truncated hypercubes
Image								...
Name	Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube
Coxeter diagram
Vertex figure	( )v( )	( )v{ }	( )v{3}	( )v{3,3}	( )v{3,3,3}	( )v{3,3,3,3}	( )v{3,3,3,3,3}

Truncated cubical graph

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.^[3]

Orthographic

References

^ B. M. Stewart, Adventures Among the Toroids (1970) ISBN 978-0-686-11936-4
^ "Adventures Among the Toroids - Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1".
^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

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

External links

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

[1] B. M. Stewart, Adventures Among the Toroids (1970) ISBN 978-0-686-11936-4

[2] "Adventures Among the Toroids - Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1".

[3] Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

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