Truncated Cube
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Truncated Cube
Truncated cube
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 14, E = 36, V = 24 (? = 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
3D model of a truncated cube

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 + .

Area and volume

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

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 A2Coxeter planes.

Orthogonal projections
Centered by Vertex Edge
Polyhedron truncated 6 from blue max.png
Polyhedron truncated 6 from red max.png Polyhedron truncated 6 from yellow max.png
Wireframe Cube t01 v.png Cube t01 e38.png Cube t01 e88.png 3-cube t01 B2.svg 3-cube t01.svg
Dual Dual truncated cube t01 v.png Dual truncated cube t01 e8.png Dual truncated cube t01 e88.png Dual truncated cube t01 B2.png Dual truncated cube t01.png
[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.

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 2? are all the permutations of

?, ±1, ±1),

where ? =  - 1.

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

Truncated cube sequence.png

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]

Excavated truncated cube.png

Vertex arrangement

It shares the vertex arrangement with three nonconvex uniform polyhedra:

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.

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.

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 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]

3-cube t01.svg

See also


  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 links

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.



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