# Truncated cube

Template:Semireg polyhedron stat table
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 .

## 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 B_{2} and A_{2} Coxeter planes.

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)

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

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

Template:Octahedral truncations

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. Template:Truncated figure1 table

It is topologically related to a series of polyhedra and tilings with face configuration V*n*.8.8.
Template:Truncated figure4 table

### 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 cubical graph

Template:Infobox 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 |

## See also

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

## References

- ↑ B. M. Stewart,
*Adventures Among the Toroids*(1970) ISBN 978-0-686-11936-4 - ↑ http://www.doskey.com/polyhedra/Stewart05.html
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}

- Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
- Cromwell, P.
*Polyhedra*, CUP hbk (1997), pbk. (1999). Ch.2 p.79-86*Archimedean solids*

## External links

- Template:Mathworld2
- Template:KlitzingPolytopes
- 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"