# 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 ${2+{\sqrt {2}}}$ .

## 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.

## 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

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. ## Vertex arrangement

It shares the vertex arrangement with three nonconvex uniform polyhedra:

## Related polyhedra

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

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 Vn.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.