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} 
t_{0,1}{4,3}  
Wythoff symbol  2 3  4 
Coxeter diagram  
Symmetry group  O_{h}, B_{3}, [4,3], (*432), order 48 
Rotation group  O, [4,3]^{+}, (432), order 24 
Dihedral angle  38: 125°15′51″ 88: 90° 
References  U_{09}, C_{21}, W_{8} 
Properties  Semiregular convex 
Colored faces 
3.8.8 (Vertex figure) 
Triakis octahedron (dual polyhedron) 
Net 
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 B_{2} and A_{2} Coxeter planes.
Centered by  Vertex  Edge 38 
Edge 88 
Face Octagon 
Face Triangle 

Truncated cube 

Triakis octahedron 

Projective symmetry 
[2]  [2]  [2]  [4]  [6] 
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.
octagoncentered 
trianglecentered 

Orthographic projection  Stereographic projections 

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 selfintersecting octagrammic faces.
If the selfintersected 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.
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]}
It shares the vertex arrangement with three nonconvex uniform polyhedra:
Truncated cube 
Nonconvex great rhombicuboctahedron 
Great cubicuboctahedron 
Great rhombihexahedron 
The truncated cube is related to other polyhedra and tlings in symmetry.
The truncated cube is one of a family of uniform polyhedra related to the cube and regular octahedron.
Uniform octahedral polyhedra  

Symmetry: [4,3], (*432)  [4,3]^{+} (432) 
[1^{+},4,3] = [3,3] (*332) 
[3^{+},4] (3*2) 

{4,3}  t{4,3}  r{4,3} r{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
= 

Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
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.
*n32 symmetry mutation of truncated spherical tilings: t{n,3}  

Symmetry *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 

Truncated figures 

Symbol  t{2,3}  t{3,3}  t{4,3}  t{5,3}  t{6,3}  t{7,3}  t{8,3}  t{∞,3}  
Triakis figures 

Config.  V3.4.4  V3.6.6  V3.8.8  V3.10.10  V3.12.12  V3.14.14  V3.16.16  V3.∞.∞ 
*n42 symmetry mutation of truncated tilings: n.8.8  

Symmetry *n42 [n,4] 
Spherical  Euclidean  Compact hyperbolic  Paracompact  
*242 [2,4] 
*342 [3,4] 
*442 [4,4] 
*542 [5,4] 
*642 [6,4] 
*742 [7,4] 
*842 [8,4]... 
*∞42 [∞,4] 

Truncated figures 

Config.  2.8.8  3.8.8  4.8.8  5.8.8  6.8.8  7.8.8  8.8.8  ∞.8.8  
nkis figures 

Config.  V2.8.8  V3.8.8  V4.8.8  V5.8.8  V6.8.8  V7.8.8  V8.8.8  V∞.8.8 
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 tilings.
The truncated cube, is second in a sequence of truncated hypercubes:
...  
Octagon  Truncated cube  Truncated tesseract  Truncated 5cube  Truncated 6cube  Truncated 7cube  Truncated 8cube  
Truncated cubical graph  

4fold symmetry schlegel diagram


Vertices  24 
Edges  36 
Automorphisms  48 
Chromatic number  3 
Properties  Cubic, Hamiltonian, regular, zerosymmetric 
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 
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