Cuboctahedron  

(Click here for rotating model)  
Type  Archimedean solid Uniform polyhedron 
Elements  F = 14, E = 24, V = 12 (χ = 2) 
Faces by sides  8{3}+6{4} 
Conway notation  aC aaT 
Schläfli symbols  r{4,3} or rr{3,3} or 
t_{1}{4,3} or t_{0,2}{3,3}  
Wythoff symbol  2  3 4 3 3  2 
Coxeter diagram  or or 
Symmetry group  O_{h}, B_{3}, [4,3], (*432), order 48 T_{d}, [3,3], (*332), order 24 
Rotation group  O, [4,3]^{+}, (432), order 24 
Dihedral angle  125.26° arcsec(−√3) 
References  U_{07}, C_{19}, W_{11} 
Properties  Semiregular convex quasiregular 
Colored faces 
3.4.3.4 (Vertex figure) 
Rhombic dodecahedron (dual polyhedron) 
Net 
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertextransitive but also edgetransitive.^{[1]} It is radially equilateral.
Its dual polyhedron is the rhombic dodecahedron.
The cuboctahedron was probably known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares.^{[2]}
The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B_{2} and A_{2} Coxeter planes. The skew projections show a square and hexagon passing through the center of the cuboctahedron.
Square Face 
Triangular Face 
Vertex  Edge  Skew  

[4]  [6]  [2]  [2]  
Rhombic dodecahedron (Dual polyhedron)  
The 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.
orthographic projection  squarecentered  trianglecentered  Vertex centered 

Stereographic projection 
The Cartesian coordinates for the vertices of a cuboctahedron (of edge length √2) centered at the origin^{[4]} are:
An alternate set of coordinates can be made in 4space, as 12 permutations of:
This construction exists as one of 16 orthant facets of the cantellated 16cell.
The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A_{3}. With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B_{3}.
The area A and the volume V of the cuboctahedron of edge length a are:
The cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. This dissection is expressed in the tetrahedraloctahedral honeycomb where pairs of square pyramids are combined into octahedra.
The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron.^{[a]} If these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J_{27}, the triangular orthobicupola, is created.
In a cuboctahedron, the long radius (center to vertex) is the same as the edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Its center is like the apical vertex of a pyramid: one edge length away from all the other vertices. (In the case of the cuboctahedron, the center is in fact the apex of 6 square and 8 triangular pyramids). This radial equilateral symmetry is a property of only a few uniform polytopes, including the twodimensional hexagon, the threedimensional cuboctahedron, and the fourdimensional 24cell and 8cell (tesseract). Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra.
Each of these radially equilateral polytopes also occurs as cells of a characteristic spacefilling tessellation: the tiling of regular hexagons, the rectified cubic honeycomb (of alternating cuboctahedra and octahedra), the 24cell honeycomb and the tesseractic honeycomb, respectively. Each tessellation has a dual tessellation; the cell centers in a tessellation are cell vertices in its dual tessellation. The densest known regular spherepacking in two, three and four dimensions uses the cell centers of one of these tessellations as sphere centers.
A cuboctahedron has octahedral symmetry. Its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either.
A cuboctahedron can be obtained by taking an equatorial cross section of a fourdimensional 24cell or 16cell. A hexagon or a square can be obtained by taking an equatorial cross section of a cuboctahedron.
The cuboctahedron is a rectified cube and also a rectified octahedron.
It is also a cantellated tetrahedron. With this construction it is given the Wythoff symbol: 3 3  2.
A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains vertexuniform: the solid has the full tetrahedral symmetry group and its vertices are equivalent under that group.
The edges of a cuboctahedron form four regular hexagons. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a triangular cupola, one of the Johnson solids; the cuboctahedron itself thus can also be called a triangular gyrobicupola, the simplest of a series (other than the gyrobifastigium or "digonal gyrobicupola"). If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the triangular orthobicupola, also called an anticuboctahedron.
Both triangular bicupolae are important in sphere packing. The distance from the solid's center to its vertices is equal to its edge length. Each central sphere can have up to twelve neighbors, and in a facecentered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal closepacked lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's center.
Cuboctahedra appear as cells in three of the convex uniform honeycombs and in nine of the convex uniform 4polytopes.
The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron.
Because it is radially equilateral, the cuboctahedron's center can be treated as a 13th canonical apical vertex, one edge length distant from the 12 ordinary vertices, as the apex of a canonical pyramid is one edge length equidistant from its other vertices.
The cuboctahedron shares its edges and vertex arrangement with two nonconvex uniform polyhedra: the cubohemioctahedron (having the square faces in common) and the octahemioctahedron (having the triangular faces in common), both have four hexagons. It also serves as a cantellated tetrahedron, as being a rectified tetratetrahedron.
Cuboctahedron 
its equator 
Cubohemioctahedron 
Octahemioctahedron 
The cuboctahedron 2covers the tetrahemihexahedron,^{[5]} which accordingly has the same abstract vertex figure (two triangles and two squares: 3.4.3.4) and half the vertices, edges, and faces. (The actual vertex figure of the tetrahemihexahedron is 3.4.3/2.4, with the a/2 factor due to the cross.)
Cuboctahedron 
Tetrahemihexahedron 
When interpreted as a framework of rigid flat faces, connected along the edges by hinges, the cuboctahedron is a rigid structure, as are all convex polyhedra, by Cauchy's theorem. However, when the faces are removed, leaving only rigid edges connected by flexible joints at the vertices, the result is not a rigid system (unlike polyhedra whose faces are all triangles, to which Cauchy's theorem applies despite the missing faces).
Adding a central vertex, connected by rigid edges to all the other vertices, subdivides the cuboctahedron into square pyramids and tetrahedra, meeting at the central vertex. Unlike the cuboctahedron itself, the resulting system of edges and joints is rigid, and forms part of the infinite octet truss structure.
The 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} 
The cuboctahedron also has tetrahedral symmetry with two colors of triangles.
Family of uniform tetrahedral polyhedra  

Symmetry: [3,3], (*332)  [3,3]^{+}, (332)  
{3,3}  t{3,3}  r{3,3}  t{3,3}  {3,3}  rr{3,3}  tr{3,3}  sr{3,3} 
Duals to uniform polyhedra  
V3.3.3  V3.6.6  V3.3.3.3  V3.6.6  V3.3.3  V3.4.3.4  V4.6.6  V3.3.3.3.3 
The cuboctahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)^{2}, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are wythoff construction within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.^{[6]}^{[7]}
*n32 orbifold symmetries of quasiregular tilings: (3.n)^{2}  

Construction 
Spherical  Euclidean  Hyperbolic  
*332  *432  *532  *632  *732  *832...  *∞32  
Quasiregular figures 

Vertex  (3.3)^{2}  (3.4)^{2}  (3.5)^{2}  (3.6)^{2}  (3.7)^{2}  (3.8)^{2}  (3.∞)^{2} 
*n42 symmetry mutations of quasiregular tilings: (4.n)^{2}  

Symmetry *4n2 [n,4] 
Spherical  Euclidean  Compact hyperbolic  Paracompact  Noncompact  
*342 [3,4] 
*442 [4,4] 
*542 [5,4] 
*642 [6,4] 
*742 [7,4] 
*842 [8,4]... 
*∞42 [∞,4] 
[ni,4]  
Figures  
Config.  (4.3)^{2}  (4.4)^{2}  (4.5)^{2}  (4.6)^{2}  (4.7)^{2}  (4.8)^{2}  (4.∞)^{2}  (4.ni)^{2} 
This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertextransitive figures have (*n32) reflectional symmetry.
*n32 symmetry mutation of expanded tilings: 3.4.n.4  

Symmetry *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paracomp.  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3]  
Figure  
Config.  3.4.2.4  3.4.3.4  3.4.4.4  3.4.5.4  3.4.6.4  3.4.7.4  3.4.8.4  3.4.∞.4 
The cuboctahedron can be decomposed into a regular octahedron and eight irregular but equal octahedra in the shape of the convex hull of a cube with two opposite vertices removed. This decomposition of the cuboctahedron corresponds with the cellfirst parallel projection of the 24cell into three dimensions. Under this projection, the cuboctahedron forms the projection envelope, which can be decomposed into six square faces, a regular octahedron, and eight irregular octahedra. These elements correspond with the images of six of the octahedral cells in the 24cell, the nearest and farthest cells from the 4D viewpoint, and the remaining eight pairs of cells, respectively.
Cuboctahedral graph  

Vertices  12 
Edges  24 
Automorphisms  48 
Properties  
Table of graphs and parameters 
In the mathematical field of graph theory, a cuboctahedral graph is the graph of vertices and edges of the cuboctahedron, one of the Archimedean solids. It can also be constructed as the line graph of the cube. It has 12 vertices and 24 edges, is locally linear, and is a quartic Archimedean graph.^{[8]}
6fold symmetry 