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}
Wythoff symbol 2 3 | 4
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Symmetry group Oh, 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
Polyhedron truncated 6 max.png
Colored faces
Truncated cube vertfig.png
(Vertex figure)
Polyhedron truncated 6 dual.png
Triakis octahedron
(dual polyhedron)
Polyhedron truncated 6 net.svg

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.

3D model of a truncated cube

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

Area and volumeEdit

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


Orthogonal projectionsEdit

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
[2] [2] [2] [4] [6]

Spherical tilingEdit

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

Cartesian coordinatesEdit

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 2ξ are all the permutations of

ξ, ±1, ±1),

where ξ = 2 − 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 octahedra are produced, and the sequence ends with the central squares being reduced to a point, and creating an octahedron.


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.[1][2]


Vertex arrangementEdit

It shares the vertex arrangement with three nonconvex uniform polyhedra:

Truncated cube
Nonconvex great rhombicuboctahedron
Great cubicuboctahedron
Great rhombihexahedron

Related polyhedraEdit

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.

Symmetry mutationsEdit

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.

Alternated truncationEdit

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 polytopesEdit

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 graphEdit

Truncated cubical graph
4-fold symmetry Schlegel diagram
Chromatic number3
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

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]


See alsoEdit


  1. ^ B. M. Stewart, Adventures Among the Toroids (1970) ISBN 978-0-686-11936-4
  2. ^
  3. ^ 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 linksEdit