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(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
Schläfli symbols r{4,3} or \begin{Bmatrix} 4 \\ 3 \end{Bmatrix}
rr{3,3} or r\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}
t1{4,3} or t0,2{3,3}
Wythoff symbol 2 | 3 4
3 3 | 2
Coxeter diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry group Oh, BC3, [4,3], (*432), order 48
Td, [3,3], (*332), order 24
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 125.26°
 \sec^{-1} \left(-\sqrt{3}\right)
References U07, C19, W11
Properties Semiregular convex quasiregular
Colored faces
Cuboctahedron vertfig.png
(Vertex figure)
Rhombic dodecahedron
(dual polyhedron)
Cuboctahedron flat.svg

In geometry, 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, being vertex-transitive and edge-transitive.

Its dual polyhedron is the rhombic dodecahedron.

Other names[edit]

Area and volume[edit]

The area A and the volume V of the cuboctahedron of edge length a are:

A = \left(6+2\sqrt{3}\right)a^2 \approx 9.4641016a^2
V = \frac{5}{3} \sqrt{2}a^3 \approx 2.3570226a^3.

Orthogonal projections[edit]

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 B2 and A2 Coxeter planes. The skew projections show a square and hexagon passing through the center of the cuboctahedron.

Cuboctahedron (orthogonal projections)
Vertex Edge Skew
3-cube t1 B2.svg 3-cube t1.svg Cube t1 v.png Cube t1 e.png Cuboctahedron B2 planes.png Cuboctahedron 3 planes.png
[4] [6] [2] [2]
Rhombic dodecahedron (Dual polyhedron)
Dual cube t1 B2.png Dual cube t1.png Dual cube t1 v.png Dual cube t1 e.png Dual cube t1 skew1.png Dual cube t1 skew2.png

Spherical tiling[edit]

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.

Uniform tiling 432-t1.png Cuboctahedron stereographic projection square.png
Cuboctahedron stereographic projection triangle.png
orthographic projection Stereographic projections

Cartesian coordinates[edit]

The Cartesian coordinates for the vertices of a cuboctahedron (of edge length √2) centered at the origin are:


An alternate set of coordinates can be made in 4-space, as 12 permutations of:


This construction exists as one of 16 orthant facets of the cantellated 16-cell.

Root vectors[edit]

The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B3.


The cuboctahedron can be dissected into two triangular cupola by a common hexagon passing through the center of the cuboctahedron. If these two triangular cupola are twisted so triangles and squares line up, Johnson solid J27, triangular orthobicupola is created.

Cuboctahedron 3 planes.pngTriangular cupola.pngTriangular orthobicupola.png

The cuboctahedron can also be dissected into 6 square pyramids, and 8 tetrahedra meeting at a central point. This dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra.


Geometric relations[edit]

Progression between a tetrahedron, expanded into a cuboctahedron, and reverse expanded into the dual tetrahedron
Progressions between an octahedron, pseudoicosahedron, and cuboctahedron

A cuboctahedron can be obtained by taking an appropriate cross section of a four-dimensional 16-cell.

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.

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. Cantellated tetrahedron.png

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 vertex-uniform: 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 face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a hexagonal close-packed 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 4-polytopes.

The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron.

Vertex arrangement[edit]

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). It also serves as a cantellated tetrahedron, as being a rectified tetratetrahedron.


The cuboctahedron 2-covers the tetrahemihexahedron,[2] which accordingly has the same abstract vertex figure (two triangles and two squares: 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.)


Related polyhedra[edit]

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) [3+,4], (3*2)
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg Uniform polyhedron-43-t12.svg Uniform polyhedron-43-t2.svg Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-43-h01.svg
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
{4,3} t{4,3} r{4,3} t{3,4} {3,4} rr{4,3} tr{4,3} sr{4,3} s{3,4}
Duals to uniform polyhedra
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Dodecahedron.svg
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V35

The cuboctahedron also has tetrahedral symmetry with two colors of triangles.

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
Uniform polyhedron-33-t0.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t1.png Uniform polyhedron-33-t12.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t02.png Uniform polyhedron-33-t012.png Uniform polyhedron-33-s012.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
{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
Tetrahedron.svg Triakistetrahedron.jpg Hexahedron.svg Triakistetrahedron.jpg Tetrahedron.svg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Dodecahedron.svg
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.

Related quasiregular polyhedra and tilings[edit]

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.[3][4]

*n32 orbifold symmetries of quasiregular tilings: (3.n)2
Quasiregular fundamental domain.png
Spherical Euclidean Hyperbolic
*332 *432 *532 *632 *732 *832... *∞32
Uniform tiling 332-t1-1-.png Uniform tiling 432-t1.png Uniform tiling 532-t1.png Uniform tiling 63-t1.png H2 tiling 237-2.png H2 tiling 238-2.png H2 tiling 23i-2.png
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
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
Figures Uniform tiling 432-t1.png Uniform tiling 44-t1.png H2 tiling 245-2.png H2 tiling 246-2.png H2 tiling 247-2.png H2 tiling 248-2.png H2 tiling 24i-2.png
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)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 vertex-transitive figures have (*n32) reflectional symmetry.

*n42 symmetry mutation of expanded tilings: 3.4.n.4
Spherical Euclid. Compact hyperb. Paracomp.
Figure Spherical triangular prism.png Uniform tiling 332-t02.png Uniform tiling 432-t02.png Uniform tiling 532-t02.png Uniform polyhedron-63-t02.png H2 tiling 237-5.png H2 tiling 238-5.png H2 tiling 23i-5.png
Config. 3.4.∞.4

Related polytopes[edit]

Orthogonal projections of 24-cell

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 cell-first parallel projection of the 24-cell 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 24-cell, the nearest and farthest cells from the 4D viewpoint, and the remaining eight pairs of cells, respectively.

Cultural occurrences[edit]

Two cuboctahedra on a chimney in Israel.
  • In the Star Trek episode "By Any Other Name", aliens seize the Enterprise by transforming crew members into inanimate cuboctahedra.
  • The "Geo Twister" fidget toy [1] is a flexible cuboctahedron.
  • The Coriolis space stations in the computer game Elite are cuboctahedron-shaped.

Cuboctahedral graph[edit]

Cuboctahedral graph
Cuboctahedral graph2.png
4-fold symmetry
Vertices 12
Edges 24
Automorphisms 48
Properties Quartic graph, Hamiltonian, regular

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 has 12 vertices and 24 edges, and is a quartic Archimedean graph.[5]

orthogonal projection
3-cube t1.svg
6-fold symmetry

See also[edit]


  1. ^ Vector Equilibrium: R. Buckminster Fuller
  2. ^ (Richter)
  3. ^ Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
  4. ^ Two Dimensional symmetry Mutations by Daniel Huson
  5. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 
  • Ghyka, Matila (1977). The geometry of art and life. ([Nachdr.] ed.). New York: Dover Publications. pp. 51–56, 81–84. ISBN 9780486235424. 
  • Richter, David A., Two Models of the Real Projective Plane 
  • Weisstein, Eric W. (2002). "Cuboctahedron". CRC Concise Encyclopedia of Mathematics. (2nd ed.). Hoboken: CRC Press. pp. 620–621. ISBN 9781420035223. 
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Cromwell, P. Polyhedra, CUP hbk (1997), pbk. (1999). Ch.2 p.79-86 Archimedean solids

External links[edit]