Truncated cuboctahedron
In geometry, the truncated cuboctahedron 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 zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.
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  46: arccos(−√6/3) = 144°44′08″ 48: arccos(−√2/3) = 135° 68: arccos(−√3/3) = 125°15′51″ 
References  U_{11}, C_{23}, W_{15} 
Properties  Semiregular convex zonohedron 
Colored faces 
4.6.8 (Vertex figure) 
Disdyakis dodecahedron (dual polyhedron) 
Net 
Names
The name truncated cuboctahedron, given originally by Johannes Kepler, is misleading. An actual truncation of a cuboctahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid. Alternate interchangeable names are:


There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicuboctahedron.
Cartesian coordinates
The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all permutations of:
 (±1, ±(1 + √2), ±(1 + 2√2))
Area and volume
The area A and the volume V of the truncated cuboctahedron of edge length a are:
Dissection
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 square cupolas, the triangular cupolas or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing a subset of these dissected components. For example, removing half of the triangular cupolas creates a genus 3 torus, which (if they are chosen appropriately) has tetrahedral symmetry.[4][5]
Stewart toroids  

Genus 3  Genus 5  Genus 7  Genus 11 
Uniform colorings
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.
Orthogonal projections
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] 
Spherical tiling
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 
Full octahedral group
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:
 Those between octagons and squares correspond to the 3 reflections between opposite octagons.
 Hexagon edges correspond to the 6 reflections between opposite squares.
 (There are no reflections between opposite hexagons.)
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 truncated octahedron with full tetrahedral symmetry and a nonuniform rhombicuboctahedron with pyritohedral symmetry (the cantic snub octahedron).
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  

all 48 vertices  24 vertices  12 vertices 
Related polyhedra
Bowtie tetrahedron and cube contain two trapezoidal faces in place of the 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}} 
= 
= 
= 

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
*n32 symmetry mutations 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.∞ 
It is first in a series of cantitruncated hypercubes:
Truncated cuboctahedron  Cantitruncated tesseract  Cantitruncated 5cube  Cantitruncated 6cube  Cantitruncated 7cube  Cantitruncated 8cube 
Truncated cuboctahedral graph
Truncated cuboctahedral graph  

4fold symmetry  
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]
See also
Wikimedia Commons has media related to Truncated cuboctahedron. 
 Cube
 Cuboctahedron
 Octahedron
 Truncated icosidodecahedron
 Truncated octahedron – truncated tetratetrahedron
References
 Wenninger, Magnus (1974), Polyhedron Models, Cambridge University Press, ISBN 9780521098595, MR 0467493 (Model 15, p. 29)
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39, p. 82)
 Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999). (p. 82)
 B. M. Stewart, Adventures Among the Toroids (1970) ISBN 9780686119364
 Doskey, Alex. "Adventures Among the Toroids  Chapter 5  Simplest (R)(A)(Q)(T) Toroids of genus p=1". www.doskey.com.
 Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan
 Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
 Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0521554322.
External links
 Eric W. Weisstein, Great rhombicuboctahedron (Archimedean solid) at MathWorld.
 Klitzing, Richard. "3D convex uniform polyhedra x3x4x  girco".
 Editable printable net of a truncated cuboctahedron with interactive 3D view
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
 great Rhombicuboctahedron: paper strips for plaiting