Truncated cube
| Truncated cubical graph | |
|---|---|
4-fold symmetry schlegel diagram
|
|
| Vertices | 24 |
| Edges | 36 |
| Automorphisms | 48 |
| Chromatic number | 3 |
| Properties | Cubic, Hamiltonian, regular, zero-symmetric |
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 + √2.
The area A and the volume V of a truncated cube of edge length a are:
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.
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.
The following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ:
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 octahedrons are produced, and the sequence ends with the central squares being reduced to a point, and creating an octahedron.
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