Truncated cuboctahedron

Truncated cuboctahedron

Type Archimedean solid
Uniform polyhedron
Elements F = 26, E = 72, V = 48 (χ = 2)
Faces by sides 12{4}+8{6}+6{8}
Schläfli symbols tr{4,3}
t0,1,2{4,3}
Wythoff symbol 2 3 4 |
Coxeter diagram
Symmetry group Oh, BC3, [4,3], (*432), order 48
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 4-6:cos(-sqrt(6)/3)=144°44'08"
4-8:cos(-sqrt(2)/3)=135°
6-8:cos(-sqrt(3)/3)=125°15'51"
References U11, C23, W15
Properties Semiregular convex zonohedron

Colored faces

4.6.8
(Vertex figure)

Disdyakis dodecahedron
(dual polyhedron)

Net

In geometry, the truncated cuboctahedron is an Archimedean solid. 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.

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:

$A = 12\left(2+\sqrt{2}+\sqrt{3}\right) a^2 \approx 61.7551724a^2$
$V = \left(22+14\sqrt{2}\right) a^3 \approx 41.7989899a^3.$

Vertices

To derive the number of vertices, we note that each vertex is the meeting point of a square, hexagon, and octagon.

• Each of the 12 squares with their 4 vertices contribute 48 vertices because 12 × 4 = 48.
• Each of the 8 hexagons with their 6 vertices contribute 48 vertices because 8 × 6 = 48.
• Each of the 6 octagons with their 8 vertices contribute 48 vertices because 6 × 8 = 48.

Therefore, there may seem to exist 48 + 48 + 48 = 144 vertices. However, we have over-counted the vertices thrice since a square, hexagon, and octagon meet at each vertex. Consequently, we divide 144 by 3 to correct for our over-counting: 144 / 3 = 48.

Dual

If the original truncated cuboctahedron has edge length 1, its dual disdyakis dodecahedron has edge lengths $\tfrac{2}{7}\scriptstyle{\sqrt{30-3\sqrt{2}}}$, $\tfrac{3}{7}\scriptstyle{\sqrt{6(2+\sqrt{2})}}$ and $\tfrac{2}{7}\scriptstyle{\sqrt{6(10+\sqrt{2})}}$.

Uniform colorings

There is only one uniform coloring of the faces of this polyhedron, one color for each face type.

A 2-uniform coloring also exists with alternately colored hexagons.

Other names

Alternate interchangeable names are:

The name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles. However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.

The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. Compare to small rhombicuboctahedron.

One unfortunate point of confusion: There is a nonconvex uniform polyhedron by the same name. See nonconvex great rhombicuboctahedron.

Orthogonal projections

The truncated cuboctahedron has two special orthogonal projections in the A2 and B2 Coxeter planes with [6] and [8] projective symmetry, and numerous [2] symmtries can be constructed from various projected planes relative to the polyhedron elements.

Orthogonal projections
Centered by Vertex Edge
4-6
Edge
4-8
Edge
6-8
Face normal
4-6
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] [8]

Related polyhedra

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{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{4,3}
s{31,1}

=

=

=
=
or
=
or
=

Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin 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.

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Hyperbolic
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]

*∞32
[∞,3]

Coxeter
Schläfli

tr{2,3}

tr{3,3}

tr{4,3}

tr{5,3}

tr{6,3}

tr{7,3}

tr{8,3}

tr{∞,3}
Omnitruncated
figure
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter
Omnitruncated
duals
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞
Dimensional family of omnitruncated polyhedra and tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Hyperbolic
*242
[2,4]
D4h
*342
[3,4]
Oh
*442
[4,4]
P4m
*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.∞
Coxeter
Schläfli

tr{2,4}

tr{3,4}

tr{4,4}

tr{5,4}

tr{6,4}

tr{7,4}

tr{8,4}

tr{∞,4}
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.∞
Coxeter