Truncated cuboctahedron  

(Click here for rotating model)  
Type  Archimedean solid Uniform polyhedron 
Elements  F = 26, E = 72, V = 48 (χ = 2) 
Faces by sides  12{4}+8{6}+6{8} 
Conway notation  bC or taC 
Schläfli symbols  tr{4,3} or 
t_{0,1,2}{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  
References  U_{11}, C_{23}, W_{15} 
Properties  Semiregular convex zonohedron 
Colored faces 
4.6.8 (Vertex figure) 
Disdyakis dodecahedron (dual polyhedron) 
Net 
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a 9zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.
The name truncated cuboctahedron, given originally by Johannes Kepler, is misleading: an actual truncation of a cuboctahedron has rectangles instead of squares; however, this nonuniform polyhedron is topologically equivalent to the Archimedean solid unrigorously named truncated cuboctahedron. Alternate interchangeable names are:


There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron.
The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of:
The area A and the volume V of the truncated cuboctahedron of edge length a are:
The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2fold symmetry axes. The rest of its space can be dissected into 6 square cupolas below the octagons, and 8 triangular cupolas below the hexagons.
A dissected truncated cuboctahedron can create a genus 5, 7, or 11 Stewart toroid by removing the central rhombicuboctahedron, and either the 6 square cupolas, the 8 triangular cupolas, or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing the central rhombicuboctahedron, and a subset of the other dissection components. For example, removing 4 of the triangular cupolas creates a genus 3 toroid; if these cupolas are appropriately chosen, then this toroid has tetrahedral symmetry.^{[4]}^{[5]}
Stewart toroids  

Genus 3  Genus 5  Genus 7  Genus 11 
There is only one uniform coloring of the faces of this polyhedron, one color for each face type.
A 2uniform coloring, with tetrahedral symmetry, exists with alternately colored hexagons.
The truncated cuboctahedron has two special orthogonal projections in the A_{2} and B_{2} Coxeter planes with [6] and [8] projective symmetry, and numerous [2] symmetries can be constructed from various projected planes relative to the polyhedron elements.
Centered by  Vertex  Edge 46 
Edge 48 
Edge 68 
Face normal 46 

Image  
Projective symmetry 
[2]^{+}  [2]  [2]  [2]  [2] 
Centered by  Face normal Square 
Face normal Octagon 
Face Square 
Face Hexagon 
Face Octagon 
Image  
Projective symmetry 
[2]  [2]  [2]  [6]  [4] 
The truncated cuboctahedron 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.
Orthogonal projection  squarecentered  hexagoncentered  octagoncentered 

Stereographic projections 
Like many other solids the truncated octahedron has full octahedral symmetry  but its relationship with the full octahedral group is closer than that: Its 48 vertices correspond to the elements of the group, and each face of its dual is a fundamental domain of the group.
The image on the right shows the 48 permutations in the group applied to an example object (namely the light JF compound on the left). The 24 light elements are rotations, and the dark ones are their reflections.
The edges of the solid correspond to the 9 reflections in the group:
The subgroups correspond to solids that share the respective vertices of the truncated octahedron.
E.g. the 3 subgroups with 24 elements correspond to a nonuniform snub cube with chiral octahedral symmetry, a nonuniform rhombicuboctahedron with pyritohedral symmetry (the cantic snub octahedron) and a nonuniform truncated octahedron with full tetrahedral symmetry. The unique subgroup with 12 elements is the alternating group A_{4}. It corresponds to a nonuniform icosahedron with chiral tetrahedral symmetry.
Subgroups and corresponding solids  

Truncated cuboctahedron tr{4,3} 
Snub cube sr{4,3} 
Rhombicuboctahedron s_{2}{3,4} 
Truncated octahedron h_{1,2}{4,3} 
Icosahedron 
[4,3] Full octahedral 
[4,3]^{+} Chiral octahedral 
[4,3^{+}] Pyritohedral 
[1^{+},4,3] = [3,3] Full tetrahedral 
[1^{+},4,3^{+}] = [3,3]^{+} Chiral tetrahedral 
all 48 vertices  24 vertices  12 vertices 
Bowtie tetrahedron and cube contain two trapezoidal faces in place of each square.^{[6]} 
The truncated cuboctahedron 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 can be considered a member of a sequence of uniform patterns with vertex configuration (4.6.2p) and CoxeterDynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p < 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n  

Sym. *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3] 
*∞32 [∞,3] 
[12i,3] 
[9i,3] 
[6i,3] 
[3i,3]  
Figures  
Config.  4.6.4  4.6.6  4.6.8  4.6.10  4.6.12  4.6.14  4.6.16  4.6.∞  4.6.24i  4.6.18i  4.6.12i  4.6.6i 
Duals  
Config.  V4.6.4  V4.6.6  V4.6.8  V4.6.10  V4.6.12  V4.6.14  V4.6.16  V4.6.∞  V4.6.24i  V4.6.18i  V4.6.12i  V4.6.6i 
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n  

Symmetry *n42 [n,4] 
Spherical  Euclidean  Compact hyperbolic  Paracomp.  
*242 [2,4] 
*342 [3,4] 
*442 [4,4] 
*542 [5,4] 
*642 [6,4] 
*742 [7,4] 
*842 [8,4]... 
*∞42 [∞,4]  
Omnitruncated figure 
4.8.4 
4.8.6 4.8.8 
4.8.10 
4.8.12 
4.8.14 
4.8.16 
4.8.∞  
Omnitruncated duals 
V4.8.4 
V4.8.6 
V4.8.8 
V4.8.10 
V4.8.12 
V4.8.14 
V4.8.16 
V4.8.∞ 
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n  

Sym. *n43 [(n,4,3)] 
Spherical  Compact hyperbolic  Paraco.  
*243 [4,3] 
*343 [(3,4,3)] 
*443 [(4,4,3)] 
*543 [(5,4,3)] 
*643 [(6,4,3)] 
*743 [(7,4,3)] 
*843 [(8,4,3)] 
*∞43 [(∞,4,3)]  
Figures  
Config.  4.8.6  6.8.6  8.8.6  10.8.6  12.8.6  14.8.6  16.8.6  ∞.8.6  
Duals  
Config.  V4.8.6  V6.8.6  V8.8.6  V10.8.6  V12.8.6  V14.8.6  V16.8.6  V6.8.∞ 
It is first in a series of cantitruncated hypercubes:
Truncated cuboctahedron  Cantitruncated tesseract  Cantitruncated 5cube  Cantitruncated 6cube  Cantitruncated 7cube  Cantitruncated 8cube 
Truncated cuboctahedral graph  

Vertices  48 
Edges  72 
Automorphisms  48 
Chromatic number  2 
Properties  Cubic, Hamiltonian, regular, zerosymmetric 
Table of graphs and parameters 
In the mathematical field of graph theory, a truncated cuboctahedral graph (or great rhombcuboctahedral graph) is the graph of vertices and edges of the truncated cuboctahedron, one of the Archimedean solids. It has 48 vertices and 72 edges, and is a zerosymmetric and cubic Archimedean graph.^{[7]}